Turnover Liquidity and the Transmission of Monetary Policy

Ricardo Lagos∗

Shengxing Zhang†

New York University

London School of Economics

May 2016

Abstract
We provide empirical evidence of a novel liquidity-based transmission mechanism throughwhich monetary policy influences asset markets, develop a model of this mechanism, andassess the ability of the quantitative theory to match the evidence.

Keywords: Asset prices; Liquidity; Monetary policy; Monetary transmission
JEL classification: D83, E52, G12∗
Lagos is thankful for the support from the C.V. Starr Center for Applied Economics at NYU, and forthe hospitality of University College London, the University of Minnesota, and the Federal Reserve Bank of
Minneapolis. The views expressed herein are those of the authors and not necessarily those of the Federal
Reserve Bank of Minneapolis or the Federal Reserve System.
†
Zhang is thankful for the support from the Centre for Macroeconomics at LSE.

1Introduction

In most modern economies, central banks implement monetary policy indirectly, by intervening in certain financial markets (e.g., in the United States, the federal funds market and themarket for treasury securities). The underlying idea is that the effects of those interventions
on asset prices are transmitted to the rest of the economy to help achieve the ultimate policy
objectives. Thus, the transmission mechanism of monetary policy to asset prices is importantfor understanding how monetary policy actually operates.
In this paper, we conduct an empirical, theoretical, and quantitative study of the effects ofmonetary policy on financial markets in general and the equity market in particular. Specifically,
we make three contributions. First, we provide empirical evidence of a novel channel throughwhich monetary policy influences financial markets: tight money increases the opportunity costof holding the nominal assets used routinely to settle financial transactions (e.g., bank reserves,money balances), making these payment instruments scarcer. In turn, this scarcity reduces theresalability and turnover of financial assets, and this increased illiquidity leads to a reduction in
price. We label this mechanism the turnover-liquidity (transmission) mechanism (of monetary
policy). Second, to gain a deeper understanding of this mechanism, we develop a theory of
trade in financial over-the-counter (OTC) markets (that nests the competitive benchmark asa special case) in which money is used as a medium of exchange in financial transactions.
The model shows how the details of the market microstructure and the quantity of money
shape the performance of financial markets (e.g., as gauged by standard measures of market
liquidity), contribute to the determination of asset prices (e.g., through the resale option valueof assets), and—consistent with the evidence we document—offer a liquidity-based explanation
for the negative correlation between real stock returns and unexpected increases in the nominalinterest rate that is used to implement monetary policy. Third, we bring the theory to thedata. We calibrate a generalized version of the basic model and use it to conduct a numberof quantitative theoretical exercises designed to assess the ability of the theory to match theempirical effects of monetary policy on asset prices, both on policy announcement days and at
longer horizons.
The rest of the paper is organized as follows. Section 2 presents the basic model. It
considers a setting in which a financial asset that yields a dividend flow of consumption goods(e.g., an equity or a real bond) is demanded by investors who have time-varying heterogeneous2

valuations for the dividend. To achieve the gains from trade that arise from their heterogeneous
valuations, investors participate in a bilateral market with random search that is intermediatedby specialized dealers who have access to a competitive interdealer market. In the dealerintermediated bilateral market, which has many of the stylized features of a typical OTC market
structure but also nests the perfectly competitive market structure as a special case, investors
and dealers seek to trade the financial asset using money as a means of payment. Periodically,
dealers and investors are also able to rebalance their portfolios in a conventional Walrasianmarket. Section 3 describes the efficient allocation. Equilibrium is characterized in Section 4.
Section 5 presents the main implications of the theory. Asset prices and conventional measures offinancial liquidity (e.g., spreads, trade volume, and dealer supply of immediacy) are determinedby the quantity of money and the details of the microstructure where the asset trades (e.g.,
the degree of market power of dealers and the ease with which investors find counterparties).
Generically, asset prices in the monetary economy exhibit a speculative premium (or speculative“bubble”) whose size varies systematically with the market microstructure and the monetary
policy stance. For example, a high anticipated opportunity cost of holding money reduces
equilibrium real balances and distorts the asset allocation by causing too many assets to remainin the hands of investors with relatively low valuations, which depresses real asset prices. Section
6 is purely empirical. In it we confirm the finding, documented in previous empirical work, thatsurprise increases in the nominal policy rate cause sizable reductions in real stock returns on
FOMC announcement days. A 1 basis point unexpected increase in the policy rate causes
a decrease of between 4 and 8 basis points in the stock market return on the day of thepolicy announcement. In addition, this section contains two new empirical findings. First,
we document that episodes of unexpected policy tightening are also associated with large andpersistent declines in stock turnover. Second, we find evidence that the magnitude of the
reduction in return caused by the policy tightening is significantly larger for stocks that arenormally traded more actively, e.g., stocks with higher turnover rates. For example, in responseto an unexpected increase in the policy rate, the announcement-day decline in the return of astock in the 95th percentile of turnover rates is about twelve times larger than that of a stockin the 5th percentile. The empirical evidence in this section suggests a mechanism whereby
monetary policy affects asset prices through a reduction in turnover liquidity. In Section 7
we formulate, calibrate, and simulate a generalized version of the basic model and use it toassess the ability of the theory to fit the empirical evidence on the effects of monetary shocks3

on aggregate stock returns as well as the new cross-sectional evidence on the turnover-liquidity
transmission mechanism. This paper is related to four areas of research: search-theoretic models
of money, search-theoretic models of financial trade in OTC markets, resale option theories ofasset price bubbles, and an extensive empirical literature that studies the effects of monetary
policy on asset prices. Section 8 places our contribution in the context of these literatures.
Section 9 concludes. Appendix A contains all proofs. Appendix B contains supplementarymaterial.

2

Model

Time is represented by a sequence of periods indexed by t = 0, 1, .... Each time is divided
into two subperiods where different activities take place. There is a continuum of infinitely
lived agents called investors, each identified with a point in the set I = [0, 1]. There is alsoa continuum of infinitely lived agents called dealers, each identified with a point in the set
D = [0, 1]. All agents discount payoffs across periods with the same factor, β ∈ (0, 1). In
every period, there is a continuum of active production units with measure As ∈ R++ . Every
active unit yields an exogenous dividend yt ∈ R+ of a perishable consumption good at the endof the first subperiod of period t. (Each active unit yields the same dividend as every otheractive unit, so yt As is the aggregate dividend.) At the beginning of every period, every active
unit is subject to an independent idiosyncratic shock that renders it permanently unproductivewith probability 1 − δ ∈ [0, 1). If a production unit remains active, its dividend in period t is
yt = γt yt−1 where γt is a nonnegative random variable with cumulative distribution function Γ,i.e., Pr (γt ≤ γ) = Γ (γ), and mean γ̄ ∈ (0, (βδ)−1 ). The time t dividend becomes known to allagents at the beginning of period t, and at that time each failed production unit is replaced bya new unit that yields dividend yt in the initial period and follows the same stochastic process
as other active units thereafter (the dividend of the initial set of production units, y0 ∈ R++ ,is given at t = 0). In the second subperiod of every period, every agent has access to a linear
production technology that transforms effort into a perishable homogeneous consumption good.
For each active production unit, there is a durable and perfectly divisible equity sharethat represents the bearer’s ownership of the production unit and confers him the right tocollect dividends. At the beginning of every period t ≥ 1, each investor receives an endowmentof (1 − δ) As equity shares corresponding to the new production units. (When a production
unit fails, its equity share disappears.) There is a second financial instrument, money, that4

is intrinsically useless (it is not an argument of any utility or production function, and unlike
equity, ownership of money does not constitute a right to collect any resources). The stockmof money at time t is denoted Am
t . The initial stock of money, A0 ∈ R++ , is given andmAm
t+1 = µAt , with µ ∈ R++ . A monetary authority injects or withdraws money via lump-sumtransfers or taxes to investors in the second subperiod of every period. At the beginning of
period t = 0, each investor is endowed with a portfolio of equity shares and money. All financial
instruments are perfectly recognizable, cannot be forged, and can be traded in every subperiod.
In the second subperiod of every period, all agents can trade the consumption good producedin that subperiod, equity shares, and money in a spot Walrasian market. In the first subperiodof every period, trading is organized as follows. Investors can trade equity shares and money
in a random bilateral OTC market with dealers, while dealers can also trade equity shares andmoney with other dealers in a spot Walrasian interdealer market. We use α ∈ [0, 1] to denotethe probability that an individual investor is able to make contact with a dealer in the OTC
market. (The probability that a dealer contacts an investor is also α.) Once a dealer andan investor have contacted each other, the pair negotiates the quantity of equity shares andmoney that the dealer will trade in the interdealer market on behalf of the investor and anintermediation fee for the dealer’s intermediation services. We assume the terms of the trade
between an investor and a dealer in the OTC market are determined by Nash bargaining whereθ ∈ [0, 1] is the investor’s bargaining power. The timing is that the round of OTC trade takes
place in the first subperiod and ends before production units yield dividends. Hence equity is
traded cum dividend in the OTC market (and in the interdealer market) of the first subperiod
and ex dividend in the Walrasian market of the second subperiod.1 Asset purchases in the
OTC market cannot be financed by borrowing (e.g., due to anonymity and lack of commitment
and enforcement). This assumption and the structure of preferences described below create theneed for a medium of exchange in the OTC market.
An individual dealer’s preferences are represented by
Ed0

∞
X

β t (cdt − hdt ),t=0where cdt is his consumption of the homogeneous good that is produced, traded, and consumed
in the second subperiod of period t, and hdt is the utility cost from exerting hdt units of effort
1

As in previous search models of OTC markets, e.g., see Duffie et al. (2005) and Lagos and Rocheteau (2009),
an investor must own the equity share in order to consume the dividend.

5to produce this good. The expectation operator Ed0 is with respect to the probability measureinduced by the dividend process and the random trading process in the OTC market. Dealers
get no utility from the dividend good.2 An individual investor’s preferences are represented by
E0∞X

β t (εit yit + cit − hit ) ,t=0

where yit is the quantity of the dividend good that investor i consumes at the end of the first
subperiod of period t, cit is his consumption of the homogeneous good that is produced, traded,
and consumed in the second subperiod of period t, and hit is the utility cost from exerting hit
units of effort to produce this good. The variable εit denotes the realization of a valuation shockthat is distributed independently over time and across agents, with a differentiable cumulativeR
distribution function G on the support [εL , εH ] ⊆ [0, ∞], and ε̄ = εdG (ε). Investor i learns
his realization εit at the beginning of period t, before the OTC trading round. The expectation
operator E0 is with respect to the probability measure induced by the dividend process, theinvestor’s valuation shock, and the random trading process in the OTC market.

3

Efficiency

Consider a social planner who wishes to maximize the sum of all agents’ expected discounted
utilities subject to the same meeting frictions that agents face in the decentralized formulation.
Specifically, in the first subperiod of every period, the planner can only reallocate assets among
all dealers and the measure α of investors who contact dealers at random. We restrict attention
to symmetric allocations (identical agents receive equal treatment). Let cDt and hDt denotea dealer’s consumption and production of the homogeneous consumption good in the second
subperiod of period t. Let cIt and hIt denote an investor’s consumption and production of
the homogeneous consumption good in the second subperiod of period t. Let ãDt denote thebeginning-of-period t (before depreciation) equity holding of a dealer, and let a0Dt denote theequity holding of a dealer at the end of the first subperiod of period t (after OTC trade). LetãIt denote the beginning-of-period t (before depreciation and endowment) asset holding of aninvestor. Finally, let a0It denote a measure on F ([εL , εH ]), the Borel σ-field defined on [εL , εH ].
The measure a0It is interpreted as the distribution of post-OTC-trade asset holdings among
2

This assumption implies that dealers have no direct consumption motive for holding the equity share. It iseasy to relax, but we adopt it because it is the standard benchmark in the search-based OTC literature, e.g., seeDuffie et al. (2005), Lagos and Rocheteau (2009), Lagos, Rocheteau, and Weill (2011), and Weill (2007).

6

investors with different valuation types who contacted a dealer in the first subperiod of periodt. With this notation, the planner’s problem consists of choosing a nonnegative allocation,no∞
ãjt , a0jt , cjt , hjt j∈{D,I},t=0

to maximize
E0

∞
X

" Z
#
Z εH
εH
βt α
εyt a0It (dε) + (1 − α)εyt aIt dG (ε) + cDt + cIt − hDt − hIt

t=0

εL

εL

(the expectation operator E0 is with respect to the probability measure induced by the dividend
process) subject toãDt + ãIt ≤ As
Z εH
a0Dt + αa0It (dε) ≤ aDt + αaIt

(1)
(2)

εL

cDt + cIt ≤ hDt + hIt(3)

aDt = δãDt(4)

aIt = δãIt + (1 − δ) As .

(5)

Proposition 1 The efficient allocation satisfies the following two conditions for every t: (a)ãDt = As − ãIt = As and (b) a0It (E) = I{εH ∈E} [δ + α (1 − δ)] As /α, where I{εH ∈E} is anindicator function that takes the value 1 if εH ∈ E, and 0 otherwise, for any E ∈ F ([εL , εH ]).
According to Proposition 1, the efficient allocation is characterized by the following two
properties: (a) only dealers carry equity between periods, and (b) among those investors who
have a trading opportunity with a dealer, only those with the highest valuation type hold equity
shares at the end of the first subperiod.

4

Equilibrium

Consider the determination of the terms of trade in a bilateral meeting in the OTC roundof period t between a dealer with portfolio adt and an investor with portfolio ait and vals
uation type ε. Let at = (amt , at ) denote the investor’s post-trade portfolio and let kt de-note the intermediation fee the dealer charges for his intermediation services. We assume
the fee is expressed in terms of the general good and paid by the investor in the second
7subperiod.3

We assume (at , kt ) is determined by the Nash bargaining solution where theinvestor has bargaining power θ ∈ [0, 1]. Let ŴtD (adt , kt ) denote the maximum expecteddiscounted payoff of a dealer with portfolio adt and earned fee kt when he reallocates his
portfolio in the interdealer market of period t. Let WtI (ait , −kt ) denote the maximum expected discounted payoff at the beginning of the second subperiod of period t (after the production units have borne dividends) of an investor who is holding portfolio ait and has topay a fee kt . For each t, define a pair of functions akt : R2+ × [εL , εH ] → R+ for k = m, ssand a function kt : R2+ × [εL , εH ] → R, and let at (ait , ε) = (amt (ait , ε) , at (ait , ε)) for each

(ait , ε) ∈ R2+ × [εL , εH ]. We use [at (ait , ε) , kt (ait , ε)] to represent the bargaining outcome for abilateral meeting at time t between an investor with portfolio ait and valuation ε, and a dealer
with portfolio adt . That is, [at (ait , ε) , kt (ait , ε)] solves
max(at ,kt )∈R2+ ×Rθ
εyt ast + WtI (at , −kt ) − εyt asit − WtI (ait , 0) [ŴtD (adt , kt ) − ŴtD (adt , 0)]1−θ (6)smss.t. amt + pt at ≤ ait + pt ait

ŴtD (adt , 0) ≤ ŴtD (adt , kt )εyt asit + WtI (ait , 0) ≤ εyt ast + WtI (at , −kt ) ,where pt is the dollar price of an equity share in the interdealer market of period t.
Let WtD (at , kt ) denote the maximum expected discounted payoff of a dealer who has earned
fee kt in the OTC round of period t and, at the beginning of the second subperiod of period t,
is holding portfolio at . Then the dealer’s value of trading in the interdealer market is
ŴtD (at , kt ) = max WtD (ât , kt )ât ∈R2+

(7)

s
m
s
s.t. âmt + pt ât ≤ at + pt at ,sk2
where ât ≡ (âmt , ât ). For each t, define a pair of functions, ât : R+ → R+ for k = m, s, and letsât (at ) = (âmt (at ) , ât (at )) denote the solution to (7).

Let VtD (at ) denote the maximum expected discounted payoff of a dealer who enters thesm smOTC round of period t with portfolio at ≡ (amt , at ). Let φt ≡ (φt , φt ), where φt is the real3

In the working paper version of this model (Lagos and Zhang, 2015), we instead assume that the investor
must pay the intermediation fee on the spot, i.e., with money or equity. The alternative formulation we use
here makes the analysis and the exposition much simpler while the main economic mechanisms are essentiallyunchanged.

8

price of money and φst the real ex dividend price of equity in the second subperiod of period t(both expressed in terms of the second subperiod consumption good). Then,
WtD (at , kt ) =max(ct ,ht ,ãt+1 )∈R4+Dct − ht + βEt Vt+1
(at+1 )

(8)

s.t. ct + φt ãt+1 ≤ ht + kt + φt at ,


s
m
s
where ãt+1 ≡ ãm
t+1 , ãt+1 , at+1 = ãt+1 , δãt+1 , Et is the conditional expectation over the nextperiod realization of the dividend, and φt at denotes the dot product of φt and at . Similarly,
let VtI (at , ε) denote the maximum expected discounted payoff of an investor with valuation εsand portfolio at ≡ (amt , at ) at the beginning of the OTC round of period t. Then,


Z

I
WtI (at , −kt ) =max
ct − ht + βEt Vt+1
at+1 , ε0 dG(ε0 )(ct ,ht ,ãt+1 )∈R4+(9)s.t. ct + φt ãt+1 ≤ ht − kt + φt at + Tt ,sswhere at+1 = (ãm
t+1 , δãt+1 + (1 − δ) A ) and Tt ∈ R is the real value of the time t lump-sum

monetary transfer.
The value function of an investor who enters the OTC round of period t with portfolio atand valuation ε is
VtI (at , ε) = α εyt ast (at , ε) + WtI [at (at , ε) , −kt (at , ε)] + (1 − α) εyt ast + WtI (at , 0) .
The value function of a dealer who enters the OTC round of period t with portfolio at isZ
VtD (at ) = α ŴtD [at , kt (ait , ε)] dHIt (ait , ε) + (1 − α) ŴtD (at , 0) ,where HIt is the joint cumulative distribution function over the portfolios and valuations of theinvestors the dealer may contact in the OTC market of period t.
Let j ∈ {D, I} denote the agent type, i.e., “D” for dealers and “I” for investors. Then forsj ∈ {D, I}, let Am
jt and Ajt denote the quantities of money and equity shares, respectively, heldby all agents of type j at the beginning of the OTC round of period t (after production units have
R mR
depreciated and been replaced). That is, Am
at dFjt (at ) and Asjt = ast dFjt (at ), wherejt =s
Fjt is the cumulative distribution function over portfolios at = (amt , at ) held by agents of type jsat the beginning of the OTC round of period t. Let Ãm
jt+1 and Ãjt+1 denote the total quantities
Rof money and shares held by all agents of type j at the end of period t, i.e., ÃkDt+1 = D ãkjt+1 dj
R
m
s
s
m
m
and ÃkIt+1 = I ãkit+1 di for k ∈ {s, m}, with Am
Dt+1 = ÃDt+1 , ADt+1 = δ ÃDt+1 , AIt+1 = ÃIt+1 ,9s
and AsIt+1 = δ ÃsIt+1 + (1 − δ) As . Let Ām
Dt and ĀDt denote the quantities of money and sharessheld after the OTC round of trade of period t by all the dealers, and let Ām
It and ĀIt denote thequantities of money and shares held after the OTC round of trade of period t by all the investors
R
who are able to trade in the first subperiod. For asset k ∈ {s, m}, ĀkDt = âkt (at ) dFDt (at )R
and ĀkIt = α akt (at , ε)dHIt (at , ε). We are now ready to define an equilibrium.
s ∞Definition 1 An equilibrium is a sequence of prices, {1/pt , φmt , φt }t=0 , bilateral terms of trade∞in the OTC market, {āt , kt }∞t=0 , dealer portfolios, {hâdt , ãdt+1 , adt+1 id∈D }t=0 , and investor port∞
folios, {hãit+1 , ait+1 ii∈I }∞
t=0 , such that for all t: (i) the bilateral terms of trade {āt , kt }t=0 solve(6), (ii) taking prices and the bargaining protocol as given, the portfolios hâdt , ãdt+1 , adt+1 i solve
the individual dealer’s optimization problems (7) and (8), and the portfolios hãit+1 , ait+1 i solves ∞the individual investor’s optimization problem (9), and (iii) prices, {1/pt , φmt , φt }t=0 , are suchthat all Walrasian markets clear, i.e., ÃsDt+1 + ÃsIt+1 = As (the end-of-period t Walrasian marmm
ket for equity clears), Ãm
Dt+1 + ÃIt+1 = At+1 (the end-of-period t Walrasian market for money

clears), and ĀkDt + ĀkIt = AkDt + αAkIt for k = s, m (the period t OTC interdealer markets forequity and money clear). An equilibrium is “monetary” if φmt > 0 for all t and “nonmonetary”
otherwise.
The following result characterizes the equilibrium post-trade portfolios of dealers and investors in the OTC market, taking beginning-of-period portfolios as given.
Lemma 1 Define ε∗t ≡sp t φm
t −φt
yt

and

if ε∗t < ε =1∈ [0, 1] if ε∗t = εχ (ε∗t , ε)=0
if ε < ε∗t .

Consider a bilateral meeting in the OTC round of period t between a dealer and an investorswith portfolio at and valuation ε. The investor’s post-trade portfolio, [amt (at , ε) , at (at , ε)], is

given by∗ms
amt (at , ε) = [1 − χ (εt , ε)] (at + pt at )s
ast (at , ε) = χ (ε∗t , ε) (1/pt ) (amt + pt at ) ,and the intermediation fee charged by the dealer is


1
∗
s
kt (at , ε) = (1 − θ) (ε − ε∗t ) χ (ε∗t , ε) am
−
[1
−
χ
(ε
,
ε)]
a
t
t yt .
pt t
10

A dealer who enters the OTC market with portfolio adt exits the OTC market with portfolio
s
m
s
[âm
t (adt ) , ât (adt )] = [at (adt , 0) , at (adt , 0)].

Lemma 1 offers a full characterization of the post-trade portfolios of investors and dealers in
the OTC market. First, the bargaining outcome depends on whether the investor’s valuation,ε, is above or below a cutoff, ε∗t . If ε∗t < ε, the investor uses all his cash to buy equity. If ε < ε∗t ,he sells all his equity holding for cash. The intermediation fee earned by the dealer is equal toa share 1 − θ of the investor’s gain from trade. The dealer’s post-trade portfolio is the same asthat of an investor with ε = 0.
We focus the analysis on stationary equilibria, that is, equilibria in which aggregate equity
holdings are constant over time, i.e., AsDt = AsD and AsIt = AsI for all t, and real asset pricesss
are time-invariant linear functions of the aggregate dividend, i.e., φst = φs yt , pt φmt ≡ φ̄t = φ̄ yt ,m
m m
φm
t AIt = Zyt , and φt ADt = ZD yt , where Z, ZD ∈ R+ Hence, in a stationary equilibrium,mε∗t = φ̄s −φs ≡ ε∗ , φst+1 /φst = φ̄st+1 /φ̄st = γt+1 , φmt /φt+1 = µ/γt+1 , and pt+1 /pt = µ. Throughout

the analysis, we let β̄ ≡ βγ̄ and maintain the assumption µ > β̄ (but we consider the limitingcase µ → β̄).
For the analysis that follows, it is convenient to define
#
#
"
"


αθ 1 − β̄δ (ε̄ − εL )(1 − αθ) 1 − β̄δ (ε̂ − ε̄)and µ̄ ≡ β̄ 1 +,µ̂ ≡ β̄ 1 +ε̂
β̄δ ε̄ + 1 − β̄δ εL(10)where ε̂ ∈ [ε̄, εH ] is the unique solution toZ ε̂(ε̂ − ε) dG(ε) = 0.

ε̄ − ε̂ + αθ

(11)

εL

Lemma 4 (in Appendix A) establishes that µ̂ < µ̄. The following proposition characterizes theequilibrium set.
Proposition 2 (i) A nonmonetary equilibrium exists for any parametrization. (ii) There is
no stationary monetary equilibrium if µ ≥ µ̄. (iii) In the nonmonetary equilibrium, AsI =As − AsD = As (only investors hold equity shares), there is no trade in the OTC market, andthe equity price in the second subperiod is
φst = φs yt , with φs =

11

β̄δ
ε̄.
1 − β̄δ(12)

(iv) If µ ∈ (β̄, µ̄), then there is one stationary monetary equilibrium; asset holdings of dealersmm
and investors at the beginning of the OTC round of period t are Am
Dt = At − AIt = 0 and
if β̄ < µ < µ̂
 = δAssss∈ [0, δAs ] if µ = µ̂
AD = A − AI=0if µ̂ < µ < µ̄

and asset prices are
φst = φs yt , with φs =





β̄δ
ε∗
1−β̄δ hiR ε∗ ∗
β̄δ
ε̄
+
αθ
(ε
−
ε)
dG(ε)
εL
1−β̄δ

if β̄ < µ ≤ µ̂
if µ̂ < µ < µ̄φ̄st = φ̄s yt , with φ̄s = ε∗ + φsyt
φmt =Z mAtsφ̄p t = Am
,
Z t
where
Z=

(13)
(14)
(15)
(16)

αG (ε∗ ) AsI + AsD ∗(ε + φs )α [1 − G (ε∗ )]and for any µ ∈ β̄, µ̄ , ε∗ ∈ (εL , εH ) is the unique solution toRε1 − β̄δ ε∗H (ε − ε∗ ) dG(ε)µ − β̄
i
h
−
= 0.
R ε∗
β̄αθ
ε∗ + β̄δ ε̄ − ε∗ + αθ εL (ε∗ − ε) dG(ε) I{µ̂<µ}

(17)

(18)

(v) (a) As µ → µ̄, ε∗ → εL and φst → 1−β̄δβ̄δ ε̄yt . (b) As µ → β̄, ε∗ → εH and φst → 1−β̄δβ̄δ εH yt .
In the nonmonetary equilibrium, dealers are inactive and equity shares are held only by
investors. With no valued money, investors and dealers cannot exploit the gains from tradethat arise from the heterogeneity in investor valuations in the first subperiod, and the real asset
price is φs = 1−β̄δβ̄δ ε̄y, i.e., equal to the expected discounted value of the dividend stream since
the equity share is not traded. (Shares can be traded in the Walrasian market of the second
subperiod, but gains from trade at that stage are nil.) The stationary monetary equilibrium
exists only if the inflation rate is not too high, i.e., if µ < µ̄. In the monetary equilibrium, themarginal valuation type, ε∗ , which according to Lemma 1 partitions the set of investors into
those who buy and those who sell the asset when they meet a dealer in the OTC market, ischaracterized by (18) in part (iv ) of Proposition 2. Unlike what happens in the nonmonetary
equilibrium, the OTC market is active in the monetary equilibrium, and it is easy to show that12

∗

the marginal type, ε∗ , is strictly decreasing in the rate of inflation, i.e., ∂ε∂µ < 0 (see Corollary
3 in Appendix A). Intuitively, the real value of money falls as µ increases, so the marginal
investor type, ε∗ , decreases, reflecting the fact that under the higher inflation rate, the investor
that was marginal under the lower inflation rate is no longer indifferent between carrying cash
and equity out of the OTC market—he prefers equity.
According to Proposition 2, 0 ≤ εL < ε∗t in the monetary equilibrium, so Lemma 1 implies
that dealers hold no equity shares at the end of the OTC round: all equity is held by investors,
in particular, by those investors who carried equity into the period but were unable to contact
a dealer, and by those investors who purchased equity shares in bilateral trades with dealers.
After the round of OTC trade, all the money supply is held by the investors who carried cash
into the period but were unable to contact a dealer, by the investors who sold equity sharesthrough dealers, and by those dealers who carried equity into the OTC market.
A feature of the monetary equilibrium is that dealers never hold money overnight: at the
beginning of every period t, the money supply is all in the hands of investors, i.e., Am
Dt = 0 andmAm
It = At . The reason is that access to the interdealer market allows dealers to intermediate

assets without cash. Whether it is investors or dealers who hold the equity shares overnight
depends on the inflation rate: if it is low, i.e., if µ ∈ (β̄, µ̂), then only dealers hold equity
overnight, that is, ÃsDt+1 = As and ÃsIt+1 = 0 for all t. Conversely, if the inflation rate ishigh, i.e., if µ ∈ (µ̂, µ̄), then at the end of every period t, all equity shares are in the hands ofinvestors, i.e., ÃsDt+1 = 0 and ÃsIt+1 = As , so strictly speaking, in this case dealers only provide
brokerage services in the OTC market. The intuition for this result is as follows.4 For dealers,
the return from holding equity overnight is given by the resale price in the OTC market. If
inflation is low, ε∗t is high (the asset is priced by relatively high valuation investors), and thismeans the resale price in the OTC market is high. Since dealers are sure to trade in the OTC
market every period while investors only trade with effective probability αθ, the former are
in a better position to reap the capital gains and end up holding all equity shares overnight.
Conversely, if inflation is high then ε∗t is low, so the capital gain to a dealer from carrying theasset to sell in the OTC market is small. The benefit to investors from holding equity includes
not only the resale value in the OTC market (which is small at high inflation) but also theirown expected valuation of the dividend good, so for high inflation, the return that investors
obtain from holding equity overnight is higher than it is for dealers. For example, as µ → µ̄ we
4

See Lagos and Zhang (2015) for a more detailed discussion.

13

s

)γ̄
have ε∗t → εL , so the dealer’s expected return from holding equity overnight is (εL +φ, whileφss)γ̄
the investor’s is (ε̄+φ.
φsGiven the marginal valuation type, ε∗ , part (iv ) of Proposition 2 gives all asset prices inclosed form. The real ex dividend price of equity (in terms of the second subperiod consumptiongood), φst , is given by (13). The cum dividend dollar price of equity in the OTC market, pt , isgiven by (16). The real price of money (in terms of the second subperiod consumption good),φmt , is given by (15). The real cum dividend price of equity (in terms of the second subperiodsconsumption good) in the OTC market, pt φm
t = φ̄ yt , is given by (14).

Finally, part (v )(a) states that as the rate of money creation increases toward µ̄, ε∗ approaches the lower bound of the distribution of valuations, εL , so no investor wishes to sellequity in the OTC market, and as a result the allocations and prices of the monetary equilibrium approach those of the nonmonetary equilibrium. Part (v )(b) states that as µ decreases
toward β̄, ε∗ increases toward the upper bound of the distribution of valuations, εH , so only
investors with the highest valuation purchase equity in the OTC market (all other investors
wish to sell it). Moreover, since β̄ < µ̂, as µ → β̄ only dealers hold equity overnight. Thus, wehave the following result.
Corollary 1 The allocation implemented by the stationary monetary equilibrium converges tothe efficient allocation as µ → β̄.
B denote the nominal price in the second subperiod of period t of an N -period risk-free
Let qt,kpure discount nominal bond that matures in period t+k, for k = 0, 1, 2, ..., N (so k is the numberof periods until the bond matures). Imagine the bond is illiquid in the sense that it cannot be
B = (β̄/µ)k , andtraded in the OTC market. Then in a stationary monetary equilibrium, qt,k

ι=

µ
−1
β̄

(19)

is the time t nominal yield to maturity of the bond with k periods until maturity. Thus, theoptimal monetary policy described in Corollary 1 and part (v )(b) of Proposition 2 in which themoney supply grows at rate β̄ can be interpreted as a policy that implements the Friedman
rule, i.e., ι = 0 for all contingencies at all dates.
Example 1 Suppose that the probability distribution over investor valuations is concentrated
on two points: εL with probability πL and εH with probability πH , with ε̄ = πH εH + πL εL . Then14

(18) implies
∗

ε =



 1+

εH
µ−β̄
αθ β̄ (1−β̄δ )πH

if β̄ < µ ≤ µ̂

β̄αθ(1−β̄δ )πH εH −(µ−β̄ )β̄δ(ε̄−αθπL εL )β̄αθ(1−β̄δ )πH +(µ−β̄ )[1−β̄δ(1−αθπL )]if µ̂ < µ < µ̄

with
#

1 − β̄δ (1 − αθ) αθπL (ε̄ − εL )µ̂ = β̄ 1 +ε̄ − αθπL εL

"

"

and

#

1 − β̄δ αθ (ε̄ − εL )µ̄ = β̄ 1 +.
β̄δ ε̄ + 1 − β̄δ εLGiven ε∗ , the closed-form expressions for the equilibrium allocation are given in Proposition 2.
Example 2 Suppose that the probability distribution over investor valuations is distributed
uniformly on [0, 1]. Then (18) implies
q

2
2

αθ
1−
β̄δ
+ι−
(
)
[αθ(1−β̄δ)+ι] −[αθ(1−β̄δ)]αθ 1−β̄δ )
q (
ε∗ =
2

1−β̄δ )(αθ+ι)− [(1−β̄δ )(αθ+ι)] −αθβ̄δ [1−β̄δ(1+ι)](ῑ−ι) (αθ[1−β̄δ(1+ι)]with

if β̄ < µ ≤ µ̂
if µ̂ < µ < µ̄

"

#
"

#
1 − β̄δ (1 − αθ) (ε̂ − 1/2)αθ 1 − β̄δµ̂ = β̄ 1 +and µ̄ = β̄ 1 +
ε̂
β̄δ
√

and where ῑ ≡ µ̄ − β̄ /β̄ and ε̂ = (1 − 1 − αθ)/ (αθ). Given ε∗ , the closed-form expressions
for the equilibrium allocation are given in Proposition 2.

5

Implications

In this section, we discuss the main implications of the theory. Specifically, we show how asset
prices and conventional measures of financial liquidity (spreads, trade volume, and dealer supplyof immediacy) are determined by the quantity of money (the inflation regime) and the detailsof the microstructure where the asset trades (e.g., the degree of market power of dealers andthe ease with which investors find counterparties). We also show that generically, asset prices
in the monetary economy exhibit a speculative premium (or speculative “bubble”) whose size
varies systematically with the inflation regime and the market microstructure.

5.1

Asset prices

In this subsection, we study the asset-pricing implications of the theory. We focus on how theasset price depends on monetary policy and on the degree of OTC frictions as captured by the15parameters that regulate trading frequency and the relative bargaining strengths of the various
traders.5
5.1.1

Inflation

The real price of equity in a monetary equilibrium is in part determined by the option available
to low-valuation investors to resell the equity to high-valuation investors. As the nominal rate ι(or equivalently, the inflation rate µ) increases, equilibrium real money balances decline and themarginal investor type, ε∗ , decreases, reflecting the fact that under the higher nominal rate, theinvestor type that was marginal under the lower nominal rate is no longer indifferent between
carrying cash and equity out of the OTC market (he prefers equity). Since the marginal investor
who prices the equity in the OTC market has a lower valuation, the value of the resale option
is smaller, i.e., the turnover liquidity of the asset is lower, which in turn makes the real equity
price (both φs and φ̄s ) smaller. Naturally, the real value of money, φmt , is also decreasing in
the nominal interest rate. All this is formalized in Proposition 3. The top row of Figure 1millustrates the typical time paths of the ex dividend equity price, φst , real balances, φmt At , andthe price level, φm
t , for different values of µ.
Proposition 3 In the stationary monetary equilibrium: (i) ∂φs /∂µ < 0 and ∂φs /∂ι < 0, (ii)∂ φ̄s /∂µ < 0, (iii) ∂Z/∂µ < 0 and ∂φmt /∂µ < 0.
5.1.2

OTC frictions: trading delays and market power

In the OTC market, αθ is an investor’s effective bargaining power in negotiations with dealers.
A larger αθ implies a larger gain from trade for low-valuation investors when they sell the asset
to dealers. In turn, this makes investors more willing to hold equity shares in the previous
period, since they anticipate larger gains from selling the equity in case they were to draw a
relatively low valuation type in the following OTC round. Hence, real equity prices, φs andφ̄s , are increasing in α and θ.6 If α increases, money becomes more valuable (both Z and
7
φmt increase), provided we focus on a regime in which only investors carry equity overnight.

Proposition 4 formalizes these ideas. The bottom row of Figure 1 illustrates the time paths of
5

In Appendix A (Proposition 8) we also establish the effect of a mean-preserving spread in the distributionof valuations on the equity price.
6
This finding is consistent with the behavior of the illiquidity premia in response to variations in the measuresof liquidity documented by Ang et al. (2013).
7
Real balances can actually fall with α for µ ∈ (β̄, µ̂).

16

m
m
the ex dividend equity price, φst , real balances φmt At , and the price level, φt , for two different

values of α.
Proposition 4 In the stationary monetary equilibrium: (i) ∂φs /∂ (αθ) > 0, (ii) ∂ φ̄s /∂ (αθ) >0, (iii) ∂Z/∂α > 0 and ∂φmt /∂α > 0, for µ ∈ (µ̂, µ̄).

5.2

Financial liquidity

In this subsection, we use the theory to study the determinants of standard measures of market
liquidity: liquidity provision by dealers, trade volume, and bid-ask spreads.
5.2.1

Liquidity provision by dealers

Broker-dealers in financial markets provide liquidity (immediacy) to investors by finding themcounterparties for trade, or by trading with them out of their own account, effectively becoming
their counterparty. The following result characterizes the effect of inflation on dealers’ provisionof liquidity by accumulating assets.
Proposition 5 In the stationary monetary equilibrium: (i) dealers’ provision of liquidity byaccumulating assets, i.e., AsD , is nonincreasing in the inflation rate. (ii) For any µ close to β̄,dealers’ provision of liquidity by accumulating assets is nonmonotonic in αθ, i.e., AsD = 0 forαθ close to 0 and close to 1, but AsD > 0 for intermediate values of αθ.
Part (i ) of Proposition 5 is related to the discussion that followed Proposition 2. The expected
return from holding equity is larger for investors than for dealers with high inflation (µ > µ̂)
because in that case the expected resale value of equity in the OTC market is relatively low
and dealers only buy equity to resell in the OTC market, while investors also buy it withthe expectation of getting utility from the dividend flow. For low inflation (µ < µ̂), dealers
value equity more than investors because the OTC resale value is high and they have a higher
probability of making capital gains from reselling than investors, and this trading advantage
more than compensates for the fact that investors enjoy the additional utility from the dividend
flow. Part (ii ) of Proposition 5 states that given a low enough rate of inflation, dealers’ incentive
to hold equity inventories overnight is nonmonotonic in the degree of OTC frictions as measuredby αθ. In particular, dealers will not hold inventories if αθ is either very small or very large. If
αθ is close to zero, few investors contact the interdealer market, and this makes the equity price17

in the OTC market very low, which in turn implies too small a capital gain to induce dealers tohold equity overnight. Conversely, if αθ is close to one, a dealer has no trading advantage overan investor in the OTC market and since the investor gets utility from the dividend while thedealer does not, the willingness to pay for the asset in the centralized market is higher for theinvestor than for the dealer, and therefore it is investors and not dealers who carry the asset
overnight into the OTC market.
5.2.2

Trade volume

Trade volume is commonly used as a measure of market liquidity because it is a manifestationof the ability of the market to reallocate assets across investors. According to Lemma 1, any
investor with ε < ε∗t who has a trading opportunity in the OTC market sells all his equity
holding. Hence, in a stationary equilibrium, the quantity of assets sold by investors to dealers
in the OTC market is Qs = αG (ε∗ ) AsI . From Lemma 1, the quantity of assets purchased byinvestors from dealers is Qb = α [1 − G (ε∗ )] Am
t /pt . Thus, the total quantity of equity shares
traded in the OTC market is V = Qb + Qs , or equivalently8
V = 2αG (ε∗ ) AsI + AsD .

(20)

Trade volume, V, depends on the nominal rate ι (or equivalently, inflation µ), and dealers’market power θ indirectly, through the general equilibrium effect on ε∗ . A decrease in ι oran increase in θ increases the expected return to holding money, which makes more investors
willing to sell equity for money in the OTC market, i.e., ε∗ increases and so does trade volume.
In other words, the increase in turnover liquidity caused by the increase in ι or θ manifests
itself through an increase in trade volume. The indirect positive effect on V (through ε∗ ) of anincrease in the investors’ trade probability α is similar to an increase in θ, but in addition, αdirectly increases trade volume, since with a higher α more investors are able to trade in the
OTC market. These results are summarized in the following proposition.
Proposition 6 In the stationary monetary equilibrium: (i) ∂V/∂µ < 0 and ∂V/∂ι < 0, and(ii) ∂V/∂θ > 0 and ∂V/∂α > 0.
8

To obtain (20) we used the clearing condition for the interdealer market, Qb = Qs + AsDt . Also, note that V
is trade volume in the OTC market, but since every equity share traded in the first subperiod gets retraded in
the second subperiod, total trade volume in the whole time period equals 2V.

18

5.2.3

Bid-ask spreads

Bid-ask spreads and intermediation fees are a popular measure of market liquidity as they
constitute the main out-of-pocket transaction cost that investors bear in OTC markets. Lemma
1 shows that when dealers execute trades on behalf of their investors, they charge a fee kt (at , ε)that is linear in the trade size. This means that when an investor with ε > ε∗t wants to∗buy equity, the dealer charges him an ask price, pat (ε) = pt φm
t + (1 − θ) (ε − εt ) yt per share.

When an investor with ε < ε∗t wants to sell, the dealer pays him a bid price, pbt (ε) = pt φmt −(1 − θ) (ε∗t − ε) yt per share. Define Sta (ε) =mpat (ε)−pt φtm
p t φtand Stb (ε) =bp t φmt −pt (ε), i.e., the askmp t φtspread and bid spread, respectively, expressed as fractions of the price of the asset in theinterdealer market. Then in a stationary equilibrium, the ask spread earned by a dealer when∗)
trading with an investor with ε > ε∗ is S a (ε) = (1−θ)(ε−εand the bid spread earned by aε∗ +φs∗−ε)dealer when trading with an investor with ε < ε∗ is S b (ε) = (1−θ)(ε. The average real spreadε∗ +φsR abearned by dealers is S̄ =S (ε) I{ε∗ <ε} + S (ε) I{ε<ε∗ } dG (ε). The change S̄ in response tochanges in µ or α is ambiguous in general.9

5.3

Speculative premium

According to Proposition 2, in a monetary equilibrium the equity price, φs , is larger thanthe expected present discounted value that any agent assigns to the dividend stream, i.e.,φ̂st ≡ β̄δ/(1 − β̄δ) ε̄yt . We follow Harrison and Kreps (1978) and call the equilibrium value ofthe asset in excess of the expected present discounted value of the dividend, i.e., φst − φ̂st , thespeculative premium that investors are willing to pay in anticipation of the capital gains they
will reap when reselling the asset to investors with higher valuations in the future.10 Thus, we
say investors exhibit speculative behavior if the prospect of reselling a stock makes them willing∗
The reason is that the spread S a (ε) charged to buyers
the spread S b (ε) charged to is decreasing in ε while∗asellers may be increasing in ε . For example, if µ ∈ β̄, µ̂ , it is easy to show ∂S (ε) /∂ε∗ = −∂S b (ε) /∂ε∗ < 0.
10
It is commonplace to define the fundamental value of the asset as the expected present discounted value ofthe dividend stream and to call any transaction value in excess of this benchmark a bubble. In fact, our notionof speculative premium corresponds to the notion of speculative bubble that is used in the modern literature on
bubbles. See, e.g., Barlevy (2007), Brunnermeier (2008), Scheinkman and Xiong (2003a, 2003b), Scheinkman(2013), and Xiong (2013), who discuss Harrison and Kreps (1978) in the context of what is generally known
as the resale option theory of bubbles. One could argue, of course, that the relevant notion of “fundamental
value” should be calculated through market aggregation of diverse investor valuations and taking into accountthe monetary policy stance as well as all the details of the market structure in which the asset is traded (suchas the frequency of trading opportunities and the degree of market power of financial intermediaries), which
ultimately also factor into the asset price in equilibrium. We adopt the label used by Harrison and Kreps (1978)
to avoid semantic controversies.
919

to pay more for it than they would if they were obliged to hold it forever. Investors exhibit
speculative behavior in the sense that they buy with the expectation to resell, and naturally
the asset price incorporates the value of this option to resell.
The speculative premium in a monetary equilibrium is Pt = Pyt , where( β̄δ(ε∗ − ε̄)if β̄ < µ ≤ µ̂
1−β̄δ
P=
R ε∗
β̄δ
αθ εL G (ε) dε if µ̂ < µ < µ̄.
1−β̄δThe speculative premium is nonnegative in any monetary equilibrium, i.e., Pt ≥ 0, with “=”only if µ = µ̄. Since ∂ε∗ /∂µ < 0 (Corollary 3), it is immediate that the speculative premiumis decreasing in the rate of inflation. Intuitively, anticipated inflation reduces the real money
balances used to finance asset trading, which limits the ability of high-valuation traders topurchase the asset from low-valuation traders. As a result, the speculative premium is decreasing in µ. Since ∂ε∗ /∂ (αθ) > 0 (see the proof of Proposition 4), the speculative premium isincreasing in α and θ. Intuitively, the speculative premium is the value of the option to resell
the equity to a higher valuation investor in the future, and the value of this resale option to
the investor increases with the probability α that the investor gets a trading opportunity in
an OTC trading round and with the probability θ that he can capture the gains from trade in
those trades. So in low-inflation regimes, the model predicts large trade volume and a large
speculative premium. The following proposition summarizes these results.
Proposition 7 In the stationary monetary equilibrium: (i) ∂P/∂µ < 0 and ∂P/∂ι < 0, and(ii) ∂P/∂ (αθ) > 0.
Together, Proposition 6 and Proposition 7 imply that changes in the trading probability will
generate a positive correlation between trade volume and the size of the speculative premium.
The same is true of changes in the bargaining power. The positive correlation between tradevolume and the size of the speculative premium is a feature of historical episodes that areusually regarded as bubbles—a point emphasized by Scheinkman and Xiong (2003a, 2003b)
and Scheinkman (2013).

6

Empirical results

According to the theory, the real asset price decreases in response to an entirely unanticipatedand permanent increase in the nominal interest rate (part (i ) of Proposition 3). The mechanism
20

through which the increase in the nominal rate is transmitted to the asset price is the reduction
in turnover liquidity evidenced in a reduction in trade volume (part (i ) of Proposition 6).
These two theoretical results suggest two hypotheses that can be tested with asset pricing andturnover data: (a) surprise increases in the nominal rate reduce real stock returns, and (b)the mechanism operates through a reduction in turnover liquidity (e.g., as measured by tradevolume).

6.1

Data

We use daily time series for all individual common stocks in the New York Stock Exchange(NYSE) from the Center for Research in Security Prices (CRSP). The daily stock return fromCRSP takes into account changes in prices and accrued dividend payment, i.e., the return of

 s s
P +D
stock s on day t is Rst = Pt s t − 1 × 100, where Pts is the ex dividend dollar price of stock s
t−1

on day t and Dts denotes the dollar dividend paid per share of stock s on day t. As a measure of
trade volume for each stock, we construct the daily turnover rate from CRSP, i.e., Tts = Vts /Ast ,where Vts is the trade volume of stock s on day t (measured as the total number of shares
traded) and Ast is the number of outstanding shares of stock s on day t. Whenever we use an
average, e.g., of equity returns or turnover rates across a set of stocks, we use the arithmetic
Xn
Xn
average, e.g., RIt = n1
Rst and TtI = n1
Tts are the average return and the average
s=1

s=1

turnover rate for the universe of n common stocks listed in the NYSE.
As a proxy for the policy (nominal interest) rate, we use the rate on the nearest Eurodollar
futures contract due to mature after the FOMC (Federal Open Market Committee) policy
announcement, as in Rigobon and Sack (2004).11 Specifically, we use the 3-month Eurodollar
11

Eurodollar futures are based on a $1 million face value 3-month maturity Eurodollar time deposit. Thesefutures contracts mature during the conventional IMM (International Monetary Market) dates in the months ofMarch, June, September, or December, extending outward 10 years into the future. In addition, at any point
in time, there are so-called 3-month Eurodollar serial contracts extending 4 months into the future that mature
in months that are not conventional IMM dates. For example, at the beginning of January 2016, there arecontracts maturing in mid-March, mid-June, mid-September, and mid-December of 2016, through 2025. There
are also serial contracts maturing in mid-January, mid-February, mid-April, and mid-May of 2016. Thus, depending on the timing of the FOMC announcement, the nearest contract to mature may expire between zero and
30 days after the announcement. Current quotes are available at http://www.cmegroup.com/trading/interestrates/stir/eurodollar quotes settlements futures.html. An advantage of using a futures rate as a proxy for the“policy rate” is that its movement on dates of FOMC policy announcements reflects policy surprises only anddoes not reflect anticipated policy changes. The importance of focusing on the surprise component of policy
announcements (rather than on the anticipated component) in order to identify the response of asset prices tomonetary policy was originally pointed out by Kuttner (2001) and has been emphasized by the literature sincethen, e.g., Bernanke and Kuttner (2005) and Rigobon and Sack (2004). Gürkaynak et al. (2007) offer empiricalevidence supporting the use of futures contracts as an effective proxy for policy expectations and discuss their21

futures rate produced by the CME Group (Chicago Mercantile Exchange Group) and supplied
by Datastream. In some of our empirical exercises, we use the tick-by-tick nominal interest
rate implied by 30-day federal funds futures and consider a high-frequency measure of theunexpected change in the nominal policy rate on FOMC announcement days. The sample weanalyze runs from January 3, 1994 to November 26, 2001.12 The sample includes between 1300
and 1800 stocks (depending on the time period) and 73 policy dates.13
In the following subsections, we use the data described above to estimate the sign andmagnitude of the effect of monetary policy on stock returns and turnover. In Subsection 6.2, weestimate these effects for policy announcement days for a broad index of stocks. In Subsection
6.3, we document that the strength of the effect of monetary policy on stock returns differs
systematically with the turnover liquidity of the stock. In Subsection 6.4, we go a step further
and estimate the dynamic effects of the policy announcement on returns and turnover.

6.2

Aggregate announcement-day effects

The empirical literature has followed several approaches to estimate the impact of monetary
policy surprises on the stock market on the day of a policy announcement. A popular one,
known as event-study analysis, consists of estimating the market reaction to monetary policy
surprises on a subsample of trading days consisting exclusively of the days of monetary policy
announcements (we denote this subsample S1 ). Let it denote the day t “policy rate” (in ourcase, the CME Group 3-month Eurodollar future with closest expiration date at or after day t,
expressed in percentage terms) and define ∆it ≡ it − it−1 . The event-study analysis consists ofuse to define policy shocks.
12
We choose to end the sample period in 2001 because our theory abstracts from credit, and credit conditions in
the U.S. financial market appear to have eased dramatically in the six years leading up to the 2007 financial crisis.
For example, the private-label securitization market grew in issuance from under $500 billion to over $2 trillion in
2006, the year before the crisis; see e.g., Gorton and Metrick (2012). However, in Appendix B (Section B.6), we
extend the empirical analysis to the sample period 1994-2007. We start our sample period in 1994 because prior to
1994, policy changes in the federal funds target were unannounced and frequently occurred between FOMC meetings. From 1994 onward, all changes are announced and most coincided with FOMC meetings, so as policy announcement dates we use the dates of FOMC meetings obtained from the website of the Board of Governors of theFederal Reserve System. The web address is http://www.federalreserve.gov/monetarypolicy/fomccalendars.htm.
See Bernanke and Kuttner (2005) for more discussion on the exact timing of policy announcements.
13
Our full sample contains 78 policy dates, but we follow Rigobon and Sack (2004) and discard five of thesepolicy dates because they are preceded by either one or two holidays in financial markets. Those observations
are needed because one of our procedures requires data involving first differences in variables on the policy day
and on the day preceding the policy day.

22

running the following regression:
YtI = a + b∆it + t

(21)

for t ∈ S1 , with YtI = RIt , where t is an exogenous shock to the asset price.14 We refer to theestimator b as the event-study estimator (or “E-based” estimator, for short).
A concern with (21) is that it does not take into account the fact that the policy rate on
the right side may itself be reacting to asset prices (a simultaneity bias) and that a number ofother variables (e.g., news about economic outlook) are likely to have an impact on both thepolicy rate and asset prices (an omitted variables bias). This concern motivates us to consider
the heteroskedasticity-based estimator (“H-based” estimator, for short) proposed by Rigobon
and Sack (2004). The derivation of the H-based estimator is discussed in Appendix B (SectionB.1). In Appendix B (Section B.2), we also consider a version of the event-study estimatorthat relies on an instrumental variable identification strategy that uses intraday high-frequency
tick-by-tick interest rate data. By focusing on changes in a proxy for the policy rate in a very
narrow 30-minute window around the time of the policy announcement, this high-frequency
instrumental variable estimator (“HFIV” estimator, for short) addresses the omitted variablebias and the concern that the Eurodollar futures rate may itself respond to market conditions
on policy announcement days.
Table 1 presents the baseline results. The first column corresponds to the event-based
estimation, the second column corresponds to the heteroskedasticity-based estimation, and thethird column corresponds to the high-frequency instrumental variable estimation. Returns are
expressed in percentage terms. The first row presents estimates of the reaction of equity returns
to monetary policy. The point estimate for b in (21) is −3.77. This means that a 1 basis point(bp) increase in the policy rate causes a decrease of 3.77 basis points (bps) in the stock market
return on the day of the policy announcement.15 The analogous H-based point estimate is−6.18, implying that a 25 bp increase in the policy rate causes a decrease in the stock market
return of 1.54 percentage points (pps) on the day of the policy announcement. These results
are in line with those reported in previous studies.16 The HFIV point estimate is also negative,
14
In the context of monetary policy, this approach was originally used by Cook and Hahn (1989) and has been
followed by a large number of papers, e.g., Bernanke and Kuttner (2005), Cochrane and Piazzesi (2002), Kuttner(2001), and Thorbecke (1997).
15
The R2 indicates that 16 percent of the variance of equity prices in days of FOMC policy announcements is
associated with news about monetary policy.
16
The comparable event-based estimates in Bernanke and Kuttner (2005), who focus on a different sample
period and measure stock returns using the value-weighted return from CRSP, range between −2.55 and −4.68.

23

and the magnitude is larger than the E-based and H-based estimates.17
Previous studies have not clearly identified the specific economic mechanism that transmits
monetary policy shocks to the stock market. Conventional asset-pricing theory suggests threebroad immediate reasons why an unexpected policy nominal rate increase may lead to a decline
in stock prices. It may be associated with a decrease in expected dividend growth, with a risein the future real interest rates used to discount dividends, or with an increase in the expectedexcess returns (i.e., equity premia) associated with holding stocks. Our theory formalizes a newmechanism: the reduction in turnover liquidity caused by the increase in the opportunity costof holding the nominal assets that are routinely used to settle financial transactions. To assess
this theoretical mechanism, we again estimate b in (21) (and the analogous H-based and HFIV
I , i.e., we use the change in the daily turnover rate averaged
estimates), but with YtI = TtI − Tt−1

over all traded stocks as the dependent variable (instead of the average stock return, RIt ).
The estimated effects of monetary policy announcements on the turnover rate are reported
in the second row of Table 1. According to the event-based estimate, a 1 bp increase in thepolicy rate causes a change in the level of the marketwide turnover rate on the day of the policy
announcement equal to −.000025.18 The daily marketwide turnover rate for our sample period
is .0037 (i.e., on average, stocks turn over .94 times during a typical year composed of 252
trading days), which means that an increase in the policy rate of 25 bps causes a reduction in
the marketwide turnover rate on the day of the policy announcement of about 17 percent of
its typical level. The heteroskedasticity-based estimate for a 1 bp increase in the policy rate is−.000045, implying that a 25 bp increase in the policy rate causes a reduction in the marketwide
turnover rate of about 30 percent of its typical level. The HFIV estimate is similar.

6.3

Disaggregative announcement-day effects

Another way to provide direct evidence of the turnover-liquidity transmission mechanism ofmonetary policy is to exploit the cross-sectional variation in turnover rates across stocks. Our
theory implies that the magnitude of the change in the stock return induced by a change in the
The comparable heteroskedasticity-based estimates in Rigobon and Sack (2004), who use a different series for theEurodollar forward rate, are −6.81 for the S&P 500 index, −6.5 for the WIL5000 index, −9.42 for the NASDAQ,
and −4.85 for the DJIA.
17
Note, however, that due to data availability, the HFIV estimation is based on 64 policy dates used for the
E-based and H-based estimations.
18
The R2 indicates that 15 percent of the variance of the daily turnover rate in days of FOMC policy announcements is associated with unexpected changes in monetary policy.

24policy rate will tend to be larger for more liquid stocks (i.e., stocks with a higher turnover rate).
To test this prediction, we sort stocks into portfolios according to their turnover liquidity, as
follows. For each policy announcement date, t, we calculate Tts as the average turnover rate of
an individual stock s over all the trading days during the four weeks prior to the day of the policy
announcement. We then sort all stocks into 20 portfolios by assigning stocks with Tts ranked
between the [5 (i − 1)]th percentile and (5i)th percentile to the ith portfolio, for i = 1, ..., 20.
Hence, the average turnover rate over the four-week period prior to the announcement date fora stock in ith portfolio is at least as large as that of a stock in (i − 1)th portfolio. In Table 2, thecolumn labeled “Turnover” reports the annual turnover rate (based on a year with 252 tradingdays) corresponding to each of the 20 portfolios. For example, portfolio 1 turns over .11 times
per year while portfolio 20 turns over 3.11 times per year.19
For each of the 20 portfolios, the columns in Table 2 labeled “E-based” report the eventstudy estimates of the responses (on the day of the policy announcement) of the return andturnover of the portfolio to a 1 percentage point (pp) increase in the policy rate. All theestimates in the column labeled “Return” are negative, as predicted by the theory. Also as
predicted by the quantitative theory, the magnitude of the (statistically significant) estimates
increases with the turnover liquidity of the portfolio. For example, a 1 bp increase in the policy
rate causes a decrease of 2.03 bps in the return of portfolio 1 and a decrease of 6.27 bps in the
return of portfolio 20. Fourteen of the E-based estimates in the column labeled “Turnover” arenegative and statistically significant (at the 5 percent level), as predicted by the theory. Also as
predicted by the quantitative theory, the magnitude of the (statistically significant) estimates
increases with the turnover liquidity of the portfolio. For example, based on the point estimates,
the magnitude of the response of the turnover rate of portfolio 20 is about twelve times larger
than that of portfolio 2.
In Table 2, the columns labeled “H-based” report the H-based estimates of the responses(on the day of the policy announcement) of the return and turnover of each of the 20 portfolios
to a 1 pp increase in the policy rate. The magnitudes of the H-based estimates tend to be larger
19

Our motivation for constructing these liquidity-based portfolios is twofold. First, at a daily frequency,
individual stock returns are extremely noisy; by grouping stocks into portfolios based on some characteristic(s)related to returns, it becomes possible to see average return differences. Second, stock-specific turnover measures
are time-varying, i.e., the turnover rate of a particular stock may change over time. Bernanke and Kuttner(2005) also examine the responses of more disaggregated indices to monetary policy shocks. Specifically, they
estimate the responses of 10 industry portfolios constructed from CRSP returns as in Fama and French (1988)
but find that the precision of their estimates is not sufficient to reject the hypothesis of an equal reaction for all
10 industries.

25

than the E-based estimates. The sign and ranking of the H-based estimates across portfolios
are roughly in line with the predictions of the theory. All the coefficients in the column labeled“Return” are negative, and the magnitude of the (statistically significant) estimates tends toincrease with the turnover liquidity of the portfolio. For example, a 1 bp increase in the policy
rate causes a decrease of 3.4 bps in the return of portfolio 1 and a decrease of 12 bps in the returnof portfolio 20. Fifteen of the coefficients in the column labeled “Turnover” are negative andstatistically significant (at the 5 percent level), as predicted by the theory. Also, the magnitudeof the (statistically significant) coefficients tends to increase with the turnover liquidity of theportfolio.20 For example, the response of the turnover rate of portfolio 20 is about twelve times
larger than the response of the turnover rate of portfolio 2.
As an alternative way to estimate the heterogeneous responses of returns to monetary policy
shocks for stocks with different turnover liquidity, we ran an event-study regression of individual
stock returns (for the universe of stocks listed in the NYSE) on changes in the policy rate, aninteraction term between the change in the policy rate and individual stock turnover rate, andseveral controls. As before, ∆it denotes the monetary policy shock on policy announcement
day t (measured by the change between day t and day t − 1 in the 3-month Eurodollar futurescontract with nearest expiration after the day t FOMC policy announcement), and Tts is theaverage turnover rate of the individual stock s over all the trading days during the four weeks
prior to the day of the policy announcement of day t. Let ∆i and T denote the sample averagesof ∆it and Tts , respectively, and define Tts ≡ (Tts − T ) and ∆it ≡ (∆it − ∆i). The regression
we fit isRst = β0 + β1 ∆it + β2 Tts + β3 Tts × ∆it+ Ds + Dt + β4 (∆it )2 + β5 (Tts )2 + εst ,(22)

where Ds is a stock fixed effect, Dt is a quarterly time dummy, and εst is the error term
corresponding to stock s on policy announcement day t. The time dummies control for omitted
variables that may affect the return of all stocks in the NYSE over time. The stock fixed
effects control for the effects that permanent stock characteristics not included explicitly in the
regression may have on individual stock returns. We include the interaction term Tts × ∆it to20

In Appendix B (Section B.3), we report similar results from an alternative procedure that sorts stocks into
portfolios according to the strength of individual stock returns to changes in an aggregate (marketwide) measureof turnover. This alternative sorting criterion allows us to control for other differences across stocks, such as theconventional risk factors used in empirical asset-pricing models.

26

estimate how the effect of changes in the policy rate on individual stock returns varies acrossstocks with different turnover liquidity. The coefficient of interest is β3 , i.e., we want to test
whether changes in the policy rate affect individual stock returns through the stock-specificturnover-liquidity channel. The estimate of β3 can help us evaluate the theoretical predictionthat increases (reductions) in the policy rate cause larger reductions (increases) in returns ofstocks with a larger turnover rate, i.e., the quantitative theory predicts β3 < 0.
Table 3 reports the results from estimating five different specifications based on (22). Specification (I) excludes Ds , Dt , the interaction term, Tts × ∆it , and the squared terms, (∆it )2 and(Tts )2 . Specification (II) adds the interaction term to specification (I). Specification (III) adds
Ds to specification (II). Specification (IV) adds Dt to specification (II). Specification (V) adds
Ds to specification (IV). Specifications (VI), (VII), (VIII), and (IX) each add the squared terms(∆it )2 and (Tts )2 to specifications (II), (III), (IV), and (V), respectively. In all specifications,
all estimates are significant at 1 percent level.
The estimate of β1 is about −2.4 in specifications (I)-(V), implying that a 1 bp increase in
the policy rate reduces the return of a stock with average turnover by 2.4 bps on the day of thepolicy announcement.21 Combined, the estimates of β1 and β4 in specifications (VI)-(IX) imply
a stronger response of stock returns to the policy rate. For example, according to specification(IX), a 1 bp increase in the policy rate reduces the return of a stock with average turnover by
5.6 bps on the day of the policy announcement. The sign and magnitude of the estimates lie
within the range of responses reported in Table 1.
The estimate of interest, β3 , is large and negative in all specifications, ranging from −415
(specification (VII)) to −100 (specification (IV)). The negative and statistically significant
estimates of β3 indicate that the magnitude of the negative effect of changes in the policy rateon announcement-day equity returns is larger for stocks with higher turnover liquidity. To
interpret the magnitude of the estimates, consider an equity A with a turnover rate equal to94.679 bps (i.e., an equity in the 95th percentile of turnover rates of the sample) and an equity
B with a turnover rate equal to 5.975 bps (i.e., in the 5th percentile of turnover rates). Then,for example according to specification (IX), the estimate of β3 is −410, implying that a 1 bp
increase in the policy rate reduces the announcement-day return by β1 +2β4 +β3 TtA − T ≈ −4bps for equity A and by β1 + 2β4 + β3 TtB − T ≈ −.34 bps for equity B. Together with thefindings reported in Table 1 and Table 2, these results provide additional evidence that turnover
21

Recall the average daily turnover in our sample is .0037.

27

liquidity is a quantitatively important channel that transmits monetary policy shocks to asset
prices.

6.4

Dynamic effects

In the previous section, we have documented the effect of monetary policy shocks on equity
returns and turnover on the day the policy announcement takes place. While the turnover
liquidity channel highlighted by our theory can generate the effects on announcement days
documented in the previous section, the theoretical channel is eminently dynamic. In thetheory, persistent changes in the nominal rate affect stock returns because they imply persistentchanges in future stock turnover. To study the dynamic effects of monetary policy on prices
and turnover rates, we conduct a vector autoregression (VAR) analysis on the sample consistingof all trading days during 1994-2001.

The baseline VAR we estimate consists of three variables, i.e., it , RIt , TtI , where it , RIt ,and TtI are the daily measures of the policy rate, the stock return, and turnover describedin Section 6.1 and Section 6.2.22 The lag length is set to 10.23 To identify the effects ofmonetary policy shocks, we apply an identification scheme based on an external high-frequency
instrument.24
Figure 2 reports the impulse responses of the policy rate, the average stock return, and theaverage turnover rate, to a 1 bp increase in the policy rate. The 95 percent confidence intervals
are computed using a recursive wild bootstrap based on 10,000 replications.25 The top andbottom rows show responses for forecast horizons of 30 days and 120 days, respectively. The
22

In Section 6.2, we used the change in the 3-month Eurodollar futures rate on the day of the FOMC announcement as a proxy for the unexpected component of the change in the true policy rate, i.e., the effective
federal funds rate. In this section, we instead regard the 3-month Eurodollar futures rate as the policy rate itself.
We do this because, at a daily frequency, the effective federal funds rate is very volatile for much of our sample,
e.g., due to institutional considerations, such as “settlement Wednesdays.” The path of the 3-month Eurodollar
futures rate is quite similar to the effective federal funds rate, but it does not display the large regulation-induced
weekly swings. In any case, we have also performed the estimation in this section using the daily effective federal
funds rate instead of the Eurodolar futures rate, and the results for returns and turnover are quite similar.
23
The Akaike information criterion (AIC) suggests 10 lags, while Schwarz’s Bayesian information criterion(SBIC) and the Hannan and Quinn information criterion (HQIC) suggest 5 lags. We adopted the formulation
with 10 lags, but both formulations deliver similar estimates of the theoretical impulse responses implied by ourquantitative theory (see Appendix B, Section B.4.3, for details).
24
See Appendix B (Section B.4.1) for details. The basic idea of structural vector autoregression (SVAR)
identification using instruments external to the VAR can be traced back to Romer and Romer (1989) and has
been adopted in a number of more recent papers, including Cochrane and Piazzesi (2002), Hamilton (2003),
Kilian (2008a, 2008b), Stock and Watson (2012), Mertens and Ravn (2013), and Gertler and Karadi (2015).
25
The procedure to calculate the confidence intervals is described in Appendix B (Section B.4.2). See Gonçalves
and Kilian (2004) for a formal econometric analysis of this method.

28

path of the policy rate is quite persistent; it remains significantly above the level prevailing
prior to the shock for about one year. The middle panels in Figure 2 show the response of dailystock returns. On impact, in response to the 1 bp unexpected increase in the nominal rate, thestock return falls by about 7 bps (the 95 percent confidence band ranges from −8.03 to −6.24bps). Notice that the size and magnitude of this decrease in the stock return on the day of thepolicy shock are in line with the estimates reported in Table 1. The negative effect on the stock
return is relatively short-lived: it becomes statistically insignificant about two days after thepolicy shock, and according to the point estimates, it takes about 1 day to recover half of theinitial drop. The right panels in Figure 2 show the response of the level of the daily turnoverrate. On impact, a 1 bp surprise increase in the nominal rate causes a change in the level ofthe turnover rate equal to −.000038 (the 95 percent confidence band ranges from −.000047
to −.000035), which is similar to the H-based and HFIV point estimates reported in Table 1.
According to the point estimates, it takes about 1 day for the turnover rate to recover half of
the initial drop. However, beyond that point, the negative effect of the increase in the policy
rate is quite persistent (e.g., it takes about 50 days for it to become statistically insignificant).
Bernanke and Kuttner (2005) is one of the few papers that tries to identify the economic
forces behind the negative effect of nominal rate increases on stock returns. They use a VAR
to decompose excess equity returns into components attributable to news about dividends, realinterest rates, and future excess returns. They find that the component associated with futureexcess returns accounts for the largest part of the response of stock prices to changes in thenominal rate. This means that an increase in the policy rate lowers stock prices mostly byincreasing the expected equity premium. Bernanke and Kuttner speculate that this could comeabout via some unspecified mechanism through which tight money increases the riskiness ofstocks or decreases the investor’s willingness to bear risk. In this section, we have provided
empirical evidence for the novel mechanism suggested by our theory, i.e., that (at least partof) this increase in future stock returns comes about because tight money reduces the turnover
liquidity of stocks (as measured by the stock turnover rate). Specifically, a higher nominal rate
makes the payment instruments (e.g. real bank reserves, real money balances) more scarce,
which reduces the resalability and turnover, and this increased illiquidity is reflected in a reduction of the equity price. In Appendix B (Section B.4.5), we carry out the same VAR analysisof this section on each of the 20 liquidity portfolios constructed in Section 6.3 and find that—aspredicted by the quantitative theory—the strength and persistence of the responses to nominal

29

rate shocks are larger for portfolios with larger turnover liquidity.

7

Quantitative analysis

The theoretical result we used to motivate these regressions of Section 6 (i.e., part (i ) of
Proposition 3 and part (i ) of Proposition 6) is akin to an experiment consisting of a permanentand entirely unanticipated increase in the nominal rate, so while suggestive, it is not a theoreticalexperiment that corresponds exactly to the empirical estimates we reported in Section 6. Inorder to properly assess the predictions and quantitative performance of the theory, in thissection we formulate, calibrate, and simulate a generalized version of the model of Section
2. Specifically, we simultaneously extend the model in two directions. First, we incorporate
aggregate uncertainty regarding the path of monetary policy (implemented through changes in
the nominal interest rate). This extension allows us to consider theoretical experiments that
resemble more closely what goes on in financial markets, in the sense that while investors may
be surprised by the timing and size of changes in the nominal rate, they take into accounta probability distribution over future paths of the monetary policy so these changes are notentirely unexpected. Second, we extend the model to the case of multiple assets that differ
in their liquidity properties. This extension allows us to provide additional evidence for theturnover liquidity mechanism by exploiting the cross-sectional heterogeneity and using it toassess the quantitative theoretical effects of monetary policy on the cross section of asset returns
and turnover.

7.1

Generalized model

There are N asset classes (or types), each indexed by s ∈ N = {1, 2, ..., N }. The outstanding
quantity of equity shares of type s is As . Since the focus is on the implication of liquidity
differences across asset classes, we assume each asset gives the same dividend yt , which follows
the same stochastic process described in the one-asset model of Section 2. An investor’s periodt valuation of the dividend of any asset is distributed independently over time and acrossinvestors, with cumulative distribution function G, just as in the one-asset setup.
We model liquidity differences as follows. In each round of OTC trade, each investor cantrade asset type s ∈ N with probability αs ∈ [0, 1]. The event that the investor is able to tradeasset type s is independent of the event that he is able to trade any other asset type n ∈ N.
We interpret αs as the probability that an individual investor contacts a dealer with whom he30can trade asset type s. This captures the idea that dealers are somehow specialized in tradinga particular asset class.26 In each OTC trading round, there are N competitive interdealer
markets, one for each asset class. These markets are segmented in the following sense: (i ) assets can only be traded in market s, and (ii ) in the second subperiod of every period t, investors
choose equity holdings and partition their money holdings into N portfolios, i.e., each agent
sms
chooses ast+1 , ams
t+1 s∈N , where at+1 denotes the holding of asset s, and at+1 is the amount ofmoney the investor will have available to trade in the OTC market for asset class s of periodt + 1. For simplicity, in this section we assume dealers do not hold asset inventories overnight(and without loss of generality, also that they do not hold money overnight).
m
In Section 2, we assumed a constant growth rate of the money supply, i.e., Am
t+1 = µAt ,∞mwhere µ ∈ R++ . Here we instead assume Am
t+1 = µt At , where {µt }t=1 follows a Markov chain

with σij = Pr (µt+1 = µj |µt = µi ) and µi ∈ R++ for all i, j ∈ M = {0, 1, . . . , M }. The realizationof µt is known at the beginning of period t. As before, money is injected via lump-sum transfers
to investors in the second subperiod of every period.
We specialize the analysis to recursive equilibria in which prices are time-invariant functions
3
s
s
s
s
m
of the aggregate state, xt = (Amt , yt , µt ) ∈ R+ . That is, φt = φ (xt ), φ̄t = φ̄ (xt ), φt =s∗
φm (xt ), pst = ps (xt ), and εs∗t = ε (xt ). We conjecture that the recursive equilibrium has

the property that real prices are linear functions of the aggregate dividend, i.e., suppose xt =
s∗
s
s
s
s
m
s
m
m
(Am
t , yt , µi ), then φ (xt ) = φi yt , φ̄ (xt ) ≡ p (xt ) φ (xt ) = φ̄i yt , φ (xt ) At = Zi yt , ε (xt ) ≡
 s

m
ms
s
ms
φ̄ (xt ) − φs (xt ) /yt = φ̄si −φsi ≡ εs∗i , and φ (xt ) At = Zi yt , where At denotes the amountof money that investors have chosen to have available to trade asset type s in the OTC roundof period t. In Appendix B (Section B.5), we show that an equilibrium is characterized byssa vector {εs∗i , φi , Zi , Zi }i∈M,s∈N of M (3N + 1) unknowns that solves the following system of
26

In the theory, differences in α, θ, or G all give rise to differences in turnover across assets. We focus on
differences in α because it is conceptually the simplest and analytically the most direct way to construct asset
classes that differ in turnover liquidity. However, one could carry out the theoretical analysis by constructing asset
classes based on differences in G and θ. Differences in G work similarly to differences in α (see the equivalence
result proved in Proposition 9, Appendix A). With regard to differences in θ, in a large class of models that
includes this one, Duffie et al. (2005) and Lagos and Rocheteau (2009), the equilibrium asset price does notdepend on α and θ independently, but on their product, αθ. Thus, for asset-pricing purposes, differences in α canbe interpreted as capturing differences in the trading probability or in the bargaining power. The quantitative
response of turnover to money shocks will typically be different depending on whether assets differ in α or θ,
however.

31M (3N + 1) equations:"
φsi = β̄δ

X
j∈M

σij

ε̄ + φsj + αs θ

Z εs∗
j

#
(εs∗
j − ε)dG(ε)

for all (i, s) ∈ M × N

(23)

εL

"
#
Z εH
Z
β̄ X
j
(ε − εs∗for all (i, s) ∈ M × N
Zi =σij Zj + αs θj )dG(ε) s∗
s∗
µi
ε
+
φsj
εj
j

j∈M
s
G (εs∗
s
i )A
Zis =
(εs∗
i + φi ) for all (i, s) ∈ M × N
1 − G (εs∗
)
i
X
s
Zi =
Zi for all i ∈ M.
s∈N

(24)
(25)
(26)

In the following subsections, we calibrate and simulate this model to assess the ability ofthe theory to account for the empirical findings reported in Section 6. Before doing so, it isuseful to define the theoretical analogues to the variables we studied in the empirical section.
∗s
The return of stock s at date t + 1 is Rst+1 = φ̄st+1 /φst − 1, where φ̄st ≡ pt φm
t = φt + εt yt

is the cum dividend price of equity at date t defined in Section 4. In a recursive equilibrium,m
suppose the state at date t is xt = (Amt , yt , µi ) and at date t + 1 is xt+1 = (µi At , yt+1 , µj ),then
Rst+1 =

φsj + εs∗j yt+1
− 1.
s
φi
yt

So far we have implicitly assumed that As , i.e., all outstanding equity shares of type s areactively traded every day. In actual markets, however, a fraction of the outstanding equity
shares are seldom traded (stocks held in 401(k) accounts, for example). Our theory remains
unchanged if we replace As with κAs for some κ ∈ [0, 1] that represents the proportion of theuniverse of outstanding stocks that are actively traded and think of the remaining (1 − κ) As
as being held by nontraders outside the model. In an equilibrium in which dealers do nothold assets (as is the case in this section), trade volume for asset type s at date t is Vts =s
2αs G (εs∗t ) κA . A conventional measure of trade volume is the turnover rate used in theempirical work of Section 6.1. According to the theory, the turnover rate on date t is
Tts = Vts /As = 2αs G (εs∗t ) κ.
Naturally, a nonzero fraction of inactive stocks (i.e., κ < 1) lowers the measured turnover rate.27
27
The first column labeled “Turnover” in Table 2 reports the annual turnover rates corresponding to each of
the 20 portfolios we studied in Section 6.3. Notice that the turnover rates appear to be quite low: even the top
5 percent most traded stocks are only traded about 3 times per year, on average, which suggests that the model
should allow for the possibility of κ < 1.

32

In a recursive equilibrium, suppose the state at date t is xt = (Amt , yt , µi ), then the turnoverrate can be written as Tis = 2αs G (εs∗i ) κ.
In the theory as in our empirical work, whenever we use an average, e.g., of equity returns
XRs
or turnover rates across a set of stocks, we use the arithmetic average, e.g., RIt = N1s∈N t
Xand TtI = N1
Tts are the average return and the average turnover rate for the universe ofs∈Nstocks in the theory.
Let ψ b (xt ) denote the state xt real price of an illiquid one-period pure discount nominalbond in the second subperiod of any period (the bond is illiquid in the sense that it cannot be
traded in the OTC market). The Euler equation for this asset is ψ b (xt ) = βE [ φm (xt+1 )| xt ].
The dollar price of the asset in the second subperiod of period t is q (xt ) ≡ ψ b (xt ) /φm (xt ), somt)
the Euler equation can be written as q (xt ) = β/π̄ (xt ), where π̄ (xt ) ≡ E[ φmφ(x(x. Then thet+1 )|xt ](net) nominal rate on this bond (between period t and period t + 1) is r (xt ) ≡ q (xt )−1 − 1 =π̄ (xt ) /β − 1. Suppose xt = (Amt , yt , µi ), then in a recursive equilibrium, π̄ (xt ) ≡ π̄i andr (xt ) ≡ ri = π̄i /β − 1, where π̄i = µγ̄i P

Zij∈M σij Zj. So the one-period risk-free nominal interest

rate between time t in state xt = (Amt , yt , µi ) and time t + 1 is
ri =

7.2

µi
Zi
P
− 1.
β̄ j∈M σij Zj(27)

Calibration

We think of one model period as being one day. The discount factor, β, is chosen so that theannual real risk-free rate equals 3 percent, i.e., β = (0.97)1/365 . Idiosyncratic valuation shocks
are distributed uniformly on [0, 1], i.e., G (ε) = I{εL ≤ε≤εH } ε + I{εH <ε} with εL = 0 and εH = 1.
The dividend growth rate is independently lognormally distributed over time, with mean .04 andstandard deviation .12 per annum (e.g., as documented in Lettau and Ludvigson (2005), Table
1). That is, yt+1 = ext+1 yt , with xt+1 ∼ N γ̄ − 1, Σ2 , where γ̄ − 1 = E (log yt+1 − log yt ) =√.04/365 and Σ = SD (log yt+1 − log yt ) = .12/ 365. The parameter δ can be taken as a proxyof the riskiness of stocks; a relatively low value ensures the monetary equilibrium exists even at
relatively high inflation rates. We choose δ = (.7)1/365 , i.e., a productive unit has a 70 percent
probability of remaining productive each year. The number of outstanding shares of stocks ofevery class is normalized to 1, i.e., As = 1 for all s ∈ N. We set N = 20 so the number of asset
classes in the theory matches the number of synthetic empirical liquidity portfolios we consideredin the cross-sectional analysis of Section 6.3. We normalize α20 = 1 and calibrate {αs }19
s=133

and the fraction of actively traded stocks, κ, so that the following twenty moment conditions
are satisfied: (a) for s = 1, ..., 19, the long-run time-average of the equilibrium turnover rate,XTi.e., T̄ s ≡ limT →∞ T1
Tts , equals the average turnover rate of the sth synthetic empirical
t=1

liquidity portfolio in our sample, and (b) the average equilibrium turnover rate across asset
classes and time (under the invariant distribution of monetary policy shocks) equals the average
X20
1
turnover rate across all stocks in our sample, i.e., 20
T̄ s = .0037.28
s=1

We set θ = 1 in our baseline and abstract from micro-level pricing frictions induced bybargaining. Finally, we estimate the parameters that determine the monetary policy process,i.e., {µi , σij }i,j∈M and M , as follows. We estimate the stochastic process for the policy rateusing the rate of the 3-month Eurodollar future contract. The sample runs from January 3,
1994, to November 26, 2001. We formulate that the logarithm of the policy rate follows an
AR(1) process, we estimate this process at a daily frequency, and approximate it with a 7-state
Markov chain, {ri , σij }7i,j=1 .29 From (27), there is a one-to-one mapping between the nominal
rate and the money growth rate in the theory, and since the policy rate is estimated to be very
persistent, we have ri ≈ µi /β̄ − 1. Thus, the money supply process we use to simulate themodel is {µi , σij }7i,j=1 , where σij are the estimated transition probabilities for the policy rate,and µi ≈ β̄(1 + ri ).
In the remainder of the section, we conduct three experiments to assess the ability of thetheory to match the evidence documented in Section 6. In all experiments, we simulate thecalibrated model as follows. First, compute the equilibrium functions characterized by (23)(26). Second, simulate 1,000 samples of the dividend, each of length equal to our data sample.
Then set the path of the nominal rate in the model equal to the actual empirical path of theproxy for the policy rate. Finally, compute the equilibrium path of the model 1,000 times (onefor each realization of the simulated dividend path) and for each simulated equilibrium path,compute the average daily equity return and turnover rate for each asset class.
28

This procedure delivers α1 = .0952, α2 = .1171, α3 = .1344, α4 = .1506, α5 = .1665, α6 = .1822, α7 = .1977,
α = .2135, α9 = .2296, α10 = .2467, α11 = .2654, α12 = .2849, α13 = .3076, α14 = .3341, α15 = .3658,
α16 = .4051, α17 = .4561, α18 = .5287, α19 = .6520, and κ = .03.
29
Specifically, the process we estimate is ln it = (1 − ξ) ln i0 + ξ ln it−1 + εt , where εt is Gaussianp white noise.
With it denominated in bps, the estimates are ξ = .9997652, E (ln it ) = ln i0 = 5.7362, and pE (ε2t ) = .0102.
Hence the estimated mean and standard deviation of the nominal rate, it , are E (it ) = 346 and V ar (it ) = 172.
The estimated AR(1) process is very persistent so, as suggested by Galindev and Lkhagvasuren (2010), we use theRouwenhorst method to compute the approximating Markov matrix and states. The code for the Rouwenhorst
method is also from Galindev and Lkhagvasuren (2010).
8

34

7.3

Experiment 1: Aggregate announcement-day effects

The first experiment we conduct is the model analogue of the event-study regression analysis
of Section 6.2. For each simulated equilibrium path for average daily stock return and turnoverrate (i.e., the arithmetic average across the 20 stock classes), we run the event-study regression(21) for returns as well as for turnover.
Figure 3 displays the distribution of point estimates for the response of the daily marketwide
average return to the policy rate implied by these 1,000 regressions on simulated data. The
average point estimate is −2.23 (with standard deviation 1.08), which is quite close to theempirical (E-based) estimate of −3.77 obtained in Section 6.2. In fact, Figure 3 shows that
−3.77 is within the 95 percent confidence interval of the point estimates generated by the theory.
Regarding the model response of the turnover rate, the average point estimate in response toa 1 bp increase in the policy rate is −.000001, about two orders of magnitude smaller than theempirical point (E-based) estimate of −.000025 obtained in Section 6.2. These comparisons arelimited to the announcement-day responses. As we show below, although the model response
for turnover on the announcement day is relatively small, it is very persistent and tends toconverge to the empirical response in subsequent days.

7.4

Experiment 2: Disaggregative announcement-day effects

The second experiment is the model analogue of the cross-sectional event-study analysis of
Section 6.2. For each of the 20 asset classes, we run an event-study regression for returns
and an event-study regression of turnover using the simulated equilibrium path for daily stock
return and turnover rate for that particular asset class.
The results for returns are illustrated in Figure 4, which reports the empirical E-based
estimates from Table 2 along with the estimates from the simulated model. For each theoretical
portfolio, the value displayed in Figure 4 is the average E-based estimate over the model 1,000
simulations. The 95% confidence intervals for the theoretical estimates are constructed using
the distribution of estimates from the 1,000 model simulations. The 95 percent confidence
intervals for the empirical estimates are from the oridnary least squares (OLS) regressions from
Section 6.3. The magnitudes of the model estimates are somewhat smaller than their empiricalcounterparts, but they all fall within the 95 percent confidence bands of the empirical estimates.
Also, the slope of the response appears to be somewhat steeper in the data.
The results for turnover are illustrated in Figure 5, which reports the empirical E-based35

estimates from Table 2 along with the estimates from the simulated model. All estimates
shown in the figure have been normalized by the average of the estimates across portfolios.
That is, the response of each portfolio is expressed as a multiple of the average response.30
This allows us to focus on the magnitude of the relative response of turnover across asset
classes. For example, in both the empirical and the model regressions, the magnitude of thedrop in turnover of portfolio 13 is similar to that of the average portfolio. The magnitude of
the response for portfolios with turnover lower (higher) than portfolio 13 is lower (higher) than
the average. The relative magnitudes of the model responses are to a large extent in line withthose estimated from the data (the model only misses the particularly large relative responsesof portfolios 17 through 20).

7.5

Experiment 3: VAR analysis

The third experiment is the model analogue of the VAR analysis of Section 6.4. For eachsimulated equilibrium path for average daily stock return and turnover rate (i.e., the arithmetic
average across the 20 stock classes), we estimate the impulse response to a 1 bp increase in thepolicy rate using exactly the same procedure used to estimate the impulse responses from thedata, as described in Section 6.4 and Appendix B (Section B.4).31
Figure 6 reports the model-generated impulse responses for the policy rate, the averagestock return, and the average turnover rate to a 1 bp increase in the policy rate, along withthe corresponding empirical impulse responses estimated from actual data (those described in
Section 6.4). For each variable, the model-generated impulse response reported in the figureis the median of the 1,000 impulse responses estimated from the 1,000 simulated equilibrium
paths. The 95 percent confidence intervals for the theoretical responses are constructed using
the distribution of estimates from the 1,000 model simulations. The 95 percent confidence
intervals for the empirical impulse responses are based on the recursive wild bootstrap proceduredescribed in Appendix B (Section B.4.2). The top and bottom panels show responses for forecast
30
For each theoretical portfolio, the value displayed in Figure 5 is the average E-based estimate over the model
1,000 simulations divided by the average response across simulations and portfolios. The 95 percent confidence
intervals for the theoretical estimates are constructed using the distribution of estimates from the 1,000 model
simulations. The 95 percent confidence intervals for the empirical estimates are from the OLS regressions from
Section 6.3.
31
In the model-based VAR exercises, we add a small noise term (drawn i.i.d. from a uniform distribution on[−5 × 10−7 , 5 × 10−7 ]) to the simulated turnover rate. We do this to ensure that return and turnover are not
perfectly collinear in the simulated equilibrium. Adding this noise term to the turnover rate does not alter theequilibrium conditions in any way.

36

horizons of 30 days and 120 days, respectively.
The path of the policy rate from the model is quite close to the empirical path. The panelsin the middle show the response of daily stock returns. On impact, in response to the 1 bp
unexpected increase in the nominal rate, the model stock return falls by −2.4 bps, which isabout 35 percent of the size of the empirical estimate. Just as in the data, the negative effectof the policy rate increase on the stock return is relatively short-lived: it takes about 1 day
to recover half of the initial drop. The right panels of Figure 6 show the response of thelevel of the daily turnover rate. On impact, in response to a 1 bp unexpected increase in thenominal rate, the turnover rate falls by −.000001 in the model. As in Section 7.3, the model
response for turnover is much smaller than the empirical estimate (−.000038 according to theempirical impulse response). However, although the model response for turnover is smaller
on impact, it is very persistent and tends to converge to the empirical response in subsequentdays. For example, the difference between the empirical path for the turnover rate and thetheoretical path becomes statistically insignificant after about 30 days. This persistent effect ofpolicy on the turnover rate allows the model to generate a short-run response in return that isquantitatively roughly in line with the data, although the announcement-day effect on turnoveris much smaller than in the data.32

8Related literature

The empirical component of our paper (Section 6) is related to a large empirical literaturethat studies the effect of monetary policy shocks on asset prices. Like many of these studies,
we identify monetary policy shocks by focusing on the reaction of asset prices in a narrow
time window around FOMC monetary policy announcements. Cook and Hahn (1989), forexample, use this kind of event-study identification strategy (with an event window of oneday) to estimate the effects of changes in the federal funds rate on bond rates. Kuttner (2001)
conducts a similar analysis but shows the importance of focusing on unexpected policy changes,
which he proxies for with federal funds futures data. Cochrane and Piazzesi (2002) estimate
the effect of monetary policy announcements on the yield curve using a one-day window aroundthe FOMC announcement and the daily change in the one-month Eurodollar rate to proxy for
32

Figure 7, discussed in Appendix B.4.3, shows that according to the model, the VAR procedure may beoverestimating the announcement-day effect of the policy rate on turnover and underestimating the persistenceof the effect.

37

unexpected changes in the federal funds rate target. Bernanke and Kuttner (2005) use daily
event windows around FOMC announcements to estimate the effect of unexpected changes in
the federal funds rate (measured using federal funds futures data) on the return of broad stock
indices. Gürkaynak, Sack and Swanson (2005) focus on intraday event windows around FOMC
announcements (30 minutes or 60 minutes wide) to estimate the effects on the S&P500 returnand several Treasury yields of unexpected changes in the federal funds target and “forward
guidance” (i.e., information on the future path of policy contained in the announcement). Morerecently, Hanson and Stein (2015) estimate the effect of monetary policy shocks on the nominal
and real Treasury yield curves using a two-day window around the announcement. Nakamura
and Steinsson (2015) also estimate the effects of monetary policy shocks on the nominal and real
Treasury yield curves, but they use a 30-minute window around the announcement. Gertler andKaradi (2015) also use a 30-minute window around the announcement to estimate the responseof bond yields and credit spreads to monetary policy shocks. Rigobon and Sack (2004) propose
a heteroskedasticity-based estimator to correct for possible simultaneity biases remaining inthese event-study regressions.
Relatively fewer papers have attempted to identify the precise mechanism through whichsurprise increases in the federal funds rate lead to a reduction in stock prices. Bernanke andKuttner (2005), for example, take one step in this direction by analyzing the response of moredisaggregated indices, in particular 10 industry-based portfolios. They find that the precision of
their estimates is not sufficient to reject the hypothesis of an equal reaction for all 10 industries.
Firms differ along many dimensions, however, and a number of studies have focused on how
these may be related to different responses of their stock prices to policy shocks. Ehrmann
and Fratzscher (2004), for example, find that firms with low cash flows, small firms, firms
with low credit ratings, firms with high price-earnings multiples, or firms with high Tobin’s q
exhibit a higher sensitivity to monetary policy shocks. Ippolito et al. (2013) find that the stock
prices of bank-dependent firms that borrow from financially weaker banks display a stronger
sensitivity to monetary policy shocks, while bank-dependent firms that hedge against interest
rate risk display a lower sensitivity to monetary policy shocks. Gorodnichenko and Weber (2016)
document that after monetary policy announcements, the conditional volatility of stock market
returns rises more for firms with stickier prices than for firms with more flexible prices. Relative
to this literature, our contribution is to document and offer a theory of the turnover-liquidity
transmission mechanism that channels monetary policy to asset prices.

38

From a theoretical standpoint, the model we develop in this paper bridges the searchtheoretic monetary literature that has largely focused on macro issues and the search-theoretic
financial OTC literature that focuses on microstructure considerations. Specifically, we embed
an OTC financial trading arrangement similar to Duffie et al. (2005) into a Lagos and Wright(2005) economy. Despite several common ingredients with those papers, our formulation isdifferent from previous work along two important dimensions.
In the standard formulations of the Lagos-Wright framework, money (and sometimes otherassets) are used as payment instruments to purchase consumption goods in bilateral markets
mediated by search. We instead posit that money is used as a medium of exchange in OTC
markets for financial assets. In the standard monetary model, money and other liquid assets
help to allocate goods from producers to consumers, while in our current formulation, money
helps to allocate financial assets among traders with heterogeneous valuations. This shift in
the nature of the gains from trade offers a different perspective that delivers novel insights into
the interaction between monetary policy and financial markets. For example, from a normative
standpoint, the new perspective emphasizes a new angle on the welfare cost of inflation that is
associated with the distortion of the optimal allocation of financial assets across investors withhigh and low valuations when real balances are scarce. From a positive perspective, it explains
the positive correlation between nominal bond yields and real equity yields, something that theconventional formulation in which monetary or real assets are used to buy consumption goods
cannot do.
As a model of financial trade, an appealing feature of Duffie et al. (2005) is its realistic OTC
market structure consisting of an interdealer market and bilateral negotiated trades between
investors and between investors and dealers. In Duffie et al. (2005), agents who wish to buy
assets pay sellers with linear-utility transfers. In addition, utility transfers from buyers to sellers
are unconstrained, so effectively there is no bound on what buyers can afford to purchase infinancial transactions. Our formulation keeps the appealing market structure of Duffie et al.
(2005) but improves on its stylized model of financial transactions by considering traders whoface standard budget constraints and use fiat money to purchase assets. These modifications
make the standard OTC formulation amenable to general equilibrium analysis and deliver anatural transmission mechanism through which monetary policy influences asset prices and thestandard measures of financial liquidity that are the main focus of the microstructure strand of
the OTC literature.

39

Our theoretical work is related to several previous studies, e.g., Geromichalos et al. (2007),
Jacquet and Tan (2012), Lagos and Rocheteau (2008), Lagos (2010a, 2010b, 2011), Lester etal. (2012), and Nosal and Rocheteau (2013), which introduce a real asset that can (at least to
some degree) be used along with money as a medium of exchange for consumption goods invariants of Lagos and Wright (2005). These papers identify the liquidity value of the asset with
its usefulness in exchange and find that when the asset is valuable as a medium of exchange,
this manifests itself as a “liquidity premium” that makes the real asset price higher than the
expected present discounted value of its financial dividend. High anticipated inflation reduces
real money balances; this tightens bilateral trading constraints, which in turn increases theliquidity value and the real price of the asset. In contrast, we find that real asset prices are
decreasing in the rate of anticipated inflation. There are some models that also build on Lagos
and Wright (2005) where agents can use a real asset as collateral to borrow money that they
subsequently use to purchase consumption goods. In those models, anticipated inflation reduces
the demand for real balances, which in turn can reduce the real price of the collateral asset
needed to borrow money (see, e.g., He et al., 2012, and Li and Li, 2012). The difference isthat in our setup, inflation reduces the real asset price by constraining the reallocation of thefinancial asset from investors with low valuations to investors with relatively high valuations.33
We share with two recent papers, Geromichalos and Herrenbrueck (2016) and Trejos andWright (2016), the general interest in bringing models of OTC trade in financial markets within
the realm of modern monetary general equilibrium theory. Trejos and Wright (2016) offer anin-depth analysis of a model that nests Duffie et al. (2005) and the prototypical “second
generation” monetary search model with divisible goods, indivisible money, and a unit upperbound on individual money holdings (e.g., Shi, 1995 or Trejos and Wright, 1995). Trejos andWright (2016) emphasize the different nature of the gains from trade in both classes of models.
In monetary models, agents value consumption goods differently and use assets to buy goods,
while in Duffie et al. (2005), agents trade because they value assets differently, and goods thatare valued the same by all investors are used to pay for asset purchases. In our formulation,
there are gains from trading assets, as in Duffie et al. (2005), but agents pay with money, as
in standard monetary models. Another difference with Trejos and Wright (2016) is that rather
33
In the model that we have developed here, money is the only asset used as means of payment. It would bestraightforward, however, to enrich the asset structure so that investors may choose to carry other real assetsthat can be used as means of payment in the OTC market, e.g., along the lines of Lagos and Rocheteau (2008)or Lagos (2010a, 2010b, 2011). As long as money is valued in equilibrium, we anticipate that the main results
emphasized here would continue to hold.

40

than assuming indivisible assets and a unit upper bound on individual asset holdings, as in Shi(1995), Trejos and Wright (1995), and Duffie et al. (2005), we work with divisible assets andunrestricted portfolios, as in Lagos and Wright (2005) and Lagos and Rocheteau (2009).
Geromichalos and Herrenbrueck (2016) extend Lagos and Wright (2005) by incorporatinga real asset that by assumption cannot be used to purchase goods in the decentralized market(as usual, at the end of every period agents choose next-period money and asset portfolios ina centralized market). The twist is that at the very beginning of every period, agents learn
whether they will want to buy or sell consumption goods in the subsequent decentralized market,
and at that point they have access to a bilateral search market where they can retrade money
and assets. This market allows agents to rebalance their positions depending on their need formoney, e.g., those who will be buyers seek to buy money and sell assets. So although assets
cannot be directly used to purchase consumption goods as in Geromichalos et al. (2007) or Lagos
and Rocheteau (2008), agents can use assets to buy goods indirectly, i.e., by exchanging them forcash in the additional bilateral trading round at the beginning of the period. Geromichalos andHerrenbrueck use the model to revisit the link between asset prices and inflation. Mattesini andNosal (2016) study a related model that combines elements of Geromichalos and Herrenbrueck(2016) and elements of Lagos and Zhang (2015) but considers a new market structure for theinterdealer market.
The fact that the equilibrium asset price is larger than the expected present discounted valuethat any agent assigns to the dividend stream is reminiscent of the literature on speculative
trading that can be traced back to Harrison and Kreps (1978). As in Harrison and Kreps
and more recent work, e.g., Scheinkman and Xiong (2003a, 2003b) and Scheinkman (2013),
speculation in our model arises because traders have heterogeneous asset valuations that changeover time: investors are willing to pay for the asset more than the present discounted value thatthey assign to the dividend stream, in anticipation of the capital gain they expect to obtainwhen reselling the asset to higher-valuation investors in the future. In terms of differences, in thework of Harrison and Kreps or Scheinkman and Xiong, traders have heterogeneous stubborn
beliefs about the stochastic dividend process, and their motive for trading is that they all
believe (at least some of them mistakenly) that by trading the asset they can profit at theexpense of others. In our formulation, traders simply have stochastic heterogeneous valuations
for the dividend, as in Duffie et al. (2005). Our model offers a new angle on the speculative
premium embedded in the asset price, by showing how it depends on the underlying financial

41

market structure and the prevailing monetary policy that jointly determine the likelihood andprofitability of future resale opportunities. Through this mechanism, our theory can generate
a positive correlation between trade volume and the size of speculative premia, a key stylized
fact that the theory of Scheinkman and Xiong (2003b) also explains.
Piazzesi and Schneider (2016) also emphasize the general idea that the cost of liquidity canaffect asset prices. In their model, the cost of liquidity to end users depends on the cost ofleverage to intermediaries, while our model and our empirical work instead center on the role ofthe nominal policy rate, which represents the cost of holding the nominal assets used routinely
to settle financial transactions (e.g., bank reserves, real money balances).

9Conclusion

We conclude by mentioning what we think are three promising avenues for future work. First,
in the model we have presented, all asset purchases are paid for with outside money. In otherwords, it focuses on the relevant margin for settings, transactions, or traders for which credit
limits have become binding. While arguably stark, we think this formulation is a useful benchmark to contrast with the traditional asset-pricing literature that abstracts from the role ofcostly or scarce payment instruments. Having said this, we think it would be interesting toextend the theory to allow for credit arrangements. The possibility of “buying on margin,” forexample, could very well magnify some of the monetary mechanisms we have emphasized here.
Second, given that trading frictions in the exchange process are at the center of the analysis(e.g., the likelihood of finding a counterparty, or the market power of dealers who intermediatetransactions), it would be interesting to endogenize them (see Lagos and Zhang (2015) for somework in this direction). Third, while our empirical work has focused on stocks, the transmission
mechanism we have identified is likely to be operative—and possibly even stronger and more
conspicuous—in markets for other assets, such as Treasury securities and assets that trade in
more frictional over-the-counter markets.

42

A

Proofs

Proof of Proposition 1. The choice variable a0Dt does not appear in the planner’s objectivefunction, so a0Dt = 0 at an optimum. Also, (3) must bind for every t at an optimum, so theplanner’s problem is equivalent tomax{ãDt ,ãIt ,a0It }∞
t=0

∞
X

E0

 Z εH0β αεaIt (dε) + (1 − α) ε̄aIt yt
t

εL

t=0

Z εH
s.t. (1), (4), (5), and αa0It (dε) ≤ aDt + αaIt .

εL

Let W ∗ denote the maximum value of this problem. Then clearly, W ∗ ≤ W̄ ∗ , whereW̄ ∗ =

max ∞ E0{ãDt ,ãIt }t=0∞X

β t [εH (ãDt + αãIt ) + (1 − α) ε̄ãIt ] δyt + w

t=0

s.t. (1), where w ≡ [αεH + (1 − α) ε̄] (1 − δ) As E0P∞t∗
t=0 β yt . Rearrange the expression for W̄

and substitute (1) (at equality) to obtainW̄ ∗ = max
E0∞{ãIt }t=0∞X

β t {εH As − (1 − α) (εH − ε̄) ãIt } δyt + w

t=0= {δεH + (1 − δ) [αεH + (1 − α) ε̄]} As E0∞X

β t yt .

t=0

The allocation consisting of ãDt = As , ãIt = 0, and the Dirac measure defined in the statementof the proposition achieve W̄ ∗ and therefore solve the planner’s problem.
Proof of Lemma 1. Notice that (8) implies
WtD (at , kt ) = φt at + kt + W̄tD ,where
W̄tD ≡ max

ãt+1 ∈R2+



D
s
−φt ãt+1 + βEt Vt+1ãm
t+1 , δãt+1 ,so (7) implies
ŴtD (at , kt ) = kt + W̄tD + max φt âtât ∈R2+smss.t. âmt + pt ât ≤ at + pt at .

43

(28)

Hence,s
if 0 < ε∗t = amt + pt atmms∈ [0, at + pt at ] if 0 = ε∗tât (at )=0
if ε∗t < 0,smâst (at ) = (1/pt ) [amt + pt at − ât (at )] ,

and
s
m
s
D
ŴtD (at , kt ) = max (φmt , φt /pt ) (at + pt at ) + kt + W̄t .

(29)

Also, notice that (9) implies
WtI (at , −kt ) = φt at − kt + W̄tI ,

(30)

where
W̄tI ≡ Tt +



Z
 m

I
s
s
max −φt ãt+1 + βEt Vt+1 ãt+1 , δãt+1 + (1 − δ) A , ε dG(ε) .

ãt+1 ∈R2+

(31)

With (29) and (30), (6) can be written as

max

am
t ,kt

m 1
(ε∗t − ε) (amyt − ktt − ait )ptθ

ms.t. 0 ≤ kt ≤ (ε∗t − ε) (amt − ait )

kt1−θ

1
yt
pt

m
with ast = asit + (1/pt ) (amit − at ). Hence,s
if ε < ε∗t = amit + pt aitmms∈ [0, ait + pt ait ] if ε = ε∗tat (ait , ε)
=0
if ε∗t < ε,mast (ait , ε) = asit + (1/pt ) [amit − at (ait , ε)] ,andkt (ait , ε) = (1 − θ) (ε − ε∗t )




1 m
s
I{ε∗t <ε} ait − I{ε<ε∗t } ait yt .
ptThis concludes the proof.


s
m
s
Lemma 2 Let ãm
dt+1 , ãdt+1 and ãit+1 , ãit+1 denote the portfolios chosen by a dealer andan investor, respectively, in the second subperiod of period t. These portfolios must satisfy the44

following first-order necessary and sufficient conditions:

m
s
m
φm
t ≥ βEt max φt+1 , φt+1 /pt+1 , with “ = ” if ãdt+1 > 0ss
φst ≥ βδEt max pt+1 φm
t+1 , φt+1 , with “ = ” if ãdt+1 > 0
"
#
Z εH

1
m
ε − ε∗t+1 yt+1 dG(ε), with “ = ” if ãm
φm
it+1 > 0t ≥ βEt φt+1 + αθ
pt+1
ε∗t+1
"
#
Z ε∗
t+1

φst ≥ βδEt ε̄yt+1 + φst+1 + αθε∗t+1 − ε yt+1 dG(ε) , with “ = ” if ãsit+1 > 0.

(32)
(33)
(34)
(35)

εL

Proof. With Lemma 1, we can write VtI (at , ε) as


1
I
∗
m
Vt (at , ε) = αθ (ε − εt ) I{ε∗t <ε} yt + φt am
t
pt
nh
i
o
+ ε + αθ (ε∗t − ε) I{ε<ε∗t } yt + φst ast + W̄tI

(36)

and VtD (at ) as
VtD (at ) = α

Z

s
m
s
D
kt (ait , ε) dHIt (ait , ε) + max (φmt , φt /pt ) (at + pt at ) + W̄t .

Since ε is i.i.d. over time, WtI (at ) is independent of ε, and the portfolio that each investor
chooses to carry into period t + 1 is independent of ε. Therefore, we can write dHIt (at , ε) =dFIt (at ) dG (ε), where FIt is the joint cumulative distribution function of investors’ money andequity holdings at the beginning of the OTC round of period t. Thus,smsD
VtD (at ) = max (φmt , φt /pt ) (at + pt at ) + Vt (0) ,

(37)

where
VtD (0) = α (1 − θ)Z

(ε − ε∗t )



1 m
s
I{ε∗t <ε} AIt − I{ε<ε∗t } AIt dG (ε) yt + W̄tD .
pt

From (37) we have

 m

D
s
m
s
D
Vt+1
ãm
ãt+1 + pt+1 δãst+1 + Vt+1(0) ,t+1 , δãt+1 = max φt+1 , φt+1 /pt+1
and from (36) we have
#
" Z
Z
εH



1
m
I
s
s
yt+1 + φm
Vt+1ãm
ε − ε∗t+1 dG(ε)
t+1 ãt+1
t+1 , δãt+1 + (1 − δ) A , ε = αθ
∗
p
t+1
εt+1
("
#
)
Z ε∗
t+1

∗
s
+δ
ε̄ +
αθ εt+1 − ε dG(ε) yt+1 + φt+1 ãst+1 + ζt+1 ,
εL

45

nh
i
o

I . Thus, the necesε + αθ ε∗t+1 − ε I{ε<ε∗ } yt+1 + φst+1 (1 − δ) As + W̄t+1
t+1
sary and sufficient first-order conditions corresponding to the maximization problems in (28)where ζt+1 ≡and (31) are as in the statement of the lemma.
Lemma 3 In period t, the interdealer market-clearing condition for equity is∗m{α [1 − G (ε∗t )] Am
It + χ (εt , 0) ADt }Proof. Recall ĀsDt =R

1= αG (ε∗t ) AsIt + [1 − χ (ε∗t , 0)] AsDt .
pt

(38)

âst (at ) dFDt (at ), so from Lemma 1, we haveĀsDt = χ (ε∗t , 0) (AsDt + Am
Dt /pt ) .

Similarly, ĀsIt = αRast (at , ε)dHIt (at , ε), so from Lemma 1, we haveĀsIt = α [1 − G (ε∗t )] (AsIt + Am
It /pt ) .

With these expressions, the market-clearing condition for equity in the interdealer market of
period t, i.e., ĀsDt + ĀsIt = AsDt + αAsIt , can be written as in the statement of the lemma.
s ∞Corollary 2 A sequence of prices, {1/pt , φmt , φt }t=0 , together with bilateral terms of trade in∞
the OTC market, {āt , kt }∞t=0 , dealer portfolios, {hâdt , ãdt+1 , adt+1 id∈D }t=0 , and investor port-

folios, {hãit+1 , ait+1 ii∈I }∞
t=0 , constitute an equilibrium if and only if they satisfy the following
conditions for all t:(i) Intermediation fee and optimal post-trade portfolios in OTC market


1 m
∗
∗
∗
s
kt (at , ε) = (1 − θ) (ε − εt ) χ (εt , ε) at − [1 − χ (εt , ε)] at yt
pt
m
∗
m
s
at (at , ε) = [1 − χ (εt , ε)] (at + pt at )s
ast (at , ε) = χ (ε∗t , ε) (1/pt ) (amt + p t at )ât (at ) = at (at , 0) .
(ii) Interdealer market clearing∗m{α [1 − G (ε∗ )] Am
It + χ (εt , 0) ADt }where Am
jt ≡

R

s
am
t dFjt (at ) and Ajt ≡R

1= αG (ε∗ ) AsIt + [1 − χ (ε∗t , 0)] AsDt ,pt

ast dFjt (at ) for j ∈ {D, I}.

46

(iii) Optimal end-of-period portfolios:msφmt ≥ βEt max φt+1 , φt+1 /pt+1s
φst ≥ βδEt max pt+1 φm
t+1 , φt+1
#
"
Z εH

1
m
ε − ε∗t+1 dG(ε)
yt+1φmt ≥ βEt φt+1 + αθ
pt+1
ε∗t+1
"
#
Z ε∗
t+1

ε∗t+1 − ε yt+1 dG(ε)φst ≥ βδEt ε̄yt+1 + φst+1 + αθ
εL

with
 m
 m
s
φt − βEt max φm
t+1 , φt+1 /pt+1 ãdt+1 = 0
 s
 s
s
φt − βδEt max pt+1 φm
t+1 , φt+1 ãdt+1 = 0
(
"
#)
Z εH

1
m
m
∗
φt − βEt φt+1 + αθε − εt+1 dG(ε)
yt+1ãmit+1 = 0
∗
p
t+1
εt+1
(
"
#)
Z ε∗
t+1

s
s
∗
φt − βδEt ε̄yt+1 + φt+1 + αθεt+1 − ε yt+1 dG(ε)
ãsit+1 = 0
εL

for all d ∈ D and all i ∈ I, andmam
jt+1 = ãjt+1

asjt+1 = δãsjt+1 + I{j∈I} (1 − δ) Asãkjt+1 ∈ R+ for k ∈ {s, m}for all j ∈ D ∪ I.
(iv) End-of-period market clearing
ÃsDt+1 + ÃsIt+1 = Asmm
Ãm
Dt+1 + ÃIt+1 = At+1 ,where ÃkDt+1 ≡Rkk
D ãxt+1 dx and ÃIt+1 ≡Rk
I ãxt+1 dx for k ∈ {s, m}.

Proof. Follows immediately from Definition 1 together with Lemma 1, Lemma 2, andLemma 3.
Lemma 4 Consider µ̂ and µ̄ as defined in (10). Then µ̂ < µ̄.

47



Proof of Lemma 4. Define Υ (ζ) : R → R by Υ (ζ) ≡ β̄ 1 + αθ(1 − β̄δ)ζ . Let ζ̂ ≡ (1−αθ)(ε̂−ε̄)
αθε̂
ε̄−εL
, so that µ̂ = Υ(ζ̂) and µ̄ = Υ(ζ̄). Since Υ is strictly increasing, µ̂ < µ̄ ifand ζ̄ ≡ β̄δε̄+(1−β̄δ)εLRεRεand only if ζ̂ < ζ̄. With (11) and the fact that ε̄ ≡ εLH εdG (ε) = εH − εLH G (ε) dε,

R εH

ζ̂ =
so clearly,

R εH
ζ̂ <

εL

[1 − G (ε)] dε,R ε̂ε̄ + αθ εL G (ε) dεε̂

[1 − G (ε)] dεε̄=ε̄ − εL< ζ̄.
ε̄Hence, µ̂ < µ̄.
Proof of Proposition 2. In an equilibrium with no money (or no valued money), there is notrade in the OTC market. From Lemma 2, the first-order conditions for a dealer d ∈ D and an
investor i ∈ I in the time t Walrasian market are
φst ≥ βδEt φst+1 , “ = ” if ãst+1d > 0
φst ≥ βδEt ε̄yt+1 + φst+1 , “ = ” if ãst+1i > 0.
In a stationary equilibrium, Et (φst+1 /φst ) = γ̄, and βδγ̄ < 1 is a maintained assumption, so
no dealer holds equity. The Walrasian market for equity can only clear if φs = 1−β̄δβ̄δ ε̄. Thisestablishes parts (i ) and (iii ) in the statement of the proposition.
Next, we turn to monetary equilibria. In a stationary equilibrium, the Euler equations(32)-(35) becomeµ ≥ β̄, “ = ” if ãm
dt+1 > 0(39)

φs ≥ β̄δ φ̄s , “ = ” if ãsdt+1 > 0


Z εH
β̄
αθ
∗
1≥
(ε − ε ) dG(ε) , “ = ” if ãm
1+ ∗it+1 > 0µε + φs ε ∗
"
#
Z ε∗
β̄δ
s
∗
φ ≥
ε̄ + αθ(ε − ε) dG(ε) , “ = ” if ãsit+1 > 0.
1 − β̄δ
εL

(40)
(41)
(42)

(We have used the fact that, as will become clear below, φ̄s ≡ ε∗ + φs ≥ εL + φs > φs inany equilibrium.) Under our maintained assumption β̄ < µ, (39) implies ãm
dt+1 = ZD = 0,
so (41) must hold with equality for some investor in a monetary equilibrium. Thus, in order
to find a monetary equilibrium, there are three possible equilibrium configurations to consider48

depending on the binding patterns of the complementary slackness conditions associated with(40) and (42). The interdealer market-clearing condition, ĀsDt + ĀsIt = AsDt + αAsIt , must holdfor all three configurations. Lemma 3 shows that this condition is equivalent to (38) and in astationary equilibrium (38) reduces toZ=ε∗ + φ s{αG (ε∗ ) AsI + [1 − χ (ε∗ , 0)] AsD } .
α [1 − G (ε∗ )]This condition in turn reduces to (17) if, as shown below, the equilibrium has 0 < ε∗ . The restof the proof proceeds in three steps.
Step 1: Try to construct a stationary monetary equilibrium with ãsdt+1 = 0 for all d ∈ D
and ãsit+1 > 0 for some i ∈ I. The equilibrium conditions for this case are (17) together withφs > β̄δ φ̄s


Z εH
β̄
αθ
∗
1=
1+ ∗
(ε − ε ) dG(ε)µε + φs ε∗
"
#
Z ε∗
β̄δ
φs =
ε̄ + αθ(ε∗ − ε) dG(ε)
1 − β̄δ
εL

(43)
(44)
(45)

and
ãm
dt+1 = 0 for all d ∈ D

(46)

ãm
it+1 ≥ 0, with “ > ” for some i ∈ I(47)ãsdt+1 = 0 for all d ∈ D(48)ãsit+1 ≥ 0, with “ > ” for some i ∈ I.

(49)

Conditions (44) and (45) are to be solved for the two unknowns ε∗ and φs . Substitute (45) into(44) to obtain



R εH
∗ ) dG(ε)
(ε
−
ε
β̄
∗
hε
i ,
1 = 1 + αθ
R ε∗
β̄δ
µ
∗
ε + 1−β̄δ ε̄ + αθ εL (ε∗ − ε) dG(ε)(50)

which is a single equation in ε∗ . Define
R εH
T (x) ≡

µ − β̄x (ε − x) dG(ε)
−
β̄δ
1
β̄αθ
x + 1−β̄δ T̂ (x)
1−β̄δ

with

(51)

Z x
T̂ (x) ≡ ε̄ − x + αθ(x − ε) dG(ε),
εL

49

(52)

and notice that ε∗ solves (50) if and only if it satisfies T (ε∗ ) = 0. T is a continuous real-valued
function on [εL , εH ], withT (εL ) =ε̄ − εL
εL + 1−β̄δβ̄δ ε̄

T (εH ) = −and−µ − β̄,β̄αθ

µ − β̄
< 0,
β̄αθ

n
h
io R
n
o
R
β̄δ
β̄δ
[1−G(x)] x+ 1−ε̄+αθ εx G(ε)dε +[ xεH [1−G(ε)]dε] 1+ 1−
αθG(x)
β̄δ
β̄δ
L
io2
n
h
< 0.
T 0 (x) = −
R
β̄δ
x+ 1−
ε̄+αθ εx G(ε)dεβ̄δLHence, if T (εL ) > 0, or equivalently, if µ < µ̄ (with µ̄ is as defined in (10)), then there exists a
unique ε∗ ∈ (εL , εH ) that satisfies T (ε∗ ) = 0 (and ε∗ ↓ εL as µ ↑ µ̄). Once we know ε∗ , φs isgiven by (45). Given ε∗ and φs , the values of Z, φ̄s , φmt , and pt are obtained using (17) (withAsI = As and AsD = 0), (14), (15), and (16). To conclude this step, notice that for this caseto be an equilibrium, (43) must hold, or equivalently, using φ̄s = ε∗ + φs and (45), it must bethat T̂ (ε∗ ) > 0, where T̂ is the continuous function on [εL , εH ] defined in (52). Notice thatT̂ 0 (x) = − [1 − αθG (x)] < 0, and T̂ (εH ) = − (1 − αθ) (εH − ε̄) < 0 < ε̄ − εL = T̂ (εL ), so there
exists a unique ε̂ ∈ (εL , εH ) such that T̂ (ε̂) = 0. (Since T̂ (ε̄) > 0, and T̂ 0 < 0, it follows thatε̄ < ε̂.) Then T̂ 0 (x) < 0 implies T̂ (ε∗ ) ≥ 0 if and only if ε∗ ≤ ε̂, with “=” for ε∗ = ε̂. With(51), we know that ε∗ < ε̂ if and only if T (ε̂) < 0 = T (ε∗ ), i.e., if and only if"# Rε1 − β̄δ αθ ε̂ H (ε − ε̂) dG(ε)β̄ 1 +< µ.
ε̂
Rε
Since T̂ (ε̂) = − (1 − αθ) (ε̂ − ε̄) + αθ ε̂ H (ε − ε̂) dG(ε) = 0, this last condition is equivalent toµ̂ < µ, where µ̂ is as defined in (10). The allocations and asset prices described in this stepcorrespond to those in the statement of the proposition for µ ∈ (µ̂, µ̄).
Step 2: Try to construct a stationary monetary equilibrium with asdt+1 > 0 for some d ∈ D
and asit+1 = 0 for all i ∈ I. The equilibrium conditions are (17), (44), (46), and (47), togetherwith
φs = β̄δ φ̄s

(53)
"

φs >

β̄δ
ε̄ + αθ1 − β̄δZ ε∗#(ε∗ − ε) dG(ε) , “ = ” if ãsit+1 > 0.

εL

50

(54)

ãsdt+1 ≥ 0, with “ > ” for some d ∈ D(55)ãsit+1 = 0, for all i ∈ I.

(56)

The conditions (44) and (53) are to be solved for ε∗ and φs . First use φ̄s = ε∗ + φs in (53) toobtain
φs =

β̄δ ∗
ε .
1 − β̄δ

Substitute (57) in (44) to obtain"#Rεαθ 1 − β̄δ ε∗H (ε − ε∗ ) dG(ε)
β̄
1+
,
1=
µ
ε∗

(57)

(58)

which is a single equation in ε∗ . Define
R (x) ≡

1 − β̄δ R εHx(ε − x) dG(ε) µ − β̄
−
x
β̄αθ

(59)

and notice that ε∗ solves (58) if and only if it satisfies R (ε∗ ) = 0. R is a continuous real-valued
function on [εL , εH ], with
1 − β̄δ (ε̄ − εL ) µ − β̄−R (εL ) =εLβ̄αθµ − β̄
R (εH ) = −
β̄αθ
and

Rε
[1 − G (x)] x + x H [1 − G (ε)] dε
R (x) = −
< 0.
1
x2
1−β̄δ
0

Hence, if R (εL ) > 0, or equivalently, if"#αθ 1 − β̄δ (ε̄ − εL )µ < β̄ 1 +≡ µo ,
εLthen there exists a unique ε∗ ∈ (εL , εH ) that satisfies R (ε∗ ) = 0 (and ε∗ ↓ εL as µ ↑ µo ).
Having solved for ε∗ , φs is obtained from (57). Given ε∗ and φs , the values of Z, φ̄s , φmt , andpt are obtained using (17) (with AsD = As − AsI = δAs ), (14), (15), and (16). Notice thatfor this case to be an equilibrium (54) must hold, or equivalently, using (57), it must be thatT̂ (ε∗ ) < 0, which in turn is equivalent to ε̂ < ε∗ . With (59), we know that ε̂ < ε∗ if and only
if R (ε∗ ) = 0 < R (ε̂), i.e., if and only if"#Rεαθ 1 − β̄δ ε̂ H (ε − ε̂) dG(ε),µ < β̄ 1 +ε̂51which using T̂ (ε̂) = 0 can be written as µ < µ̂. To summarize, the prices and allocations constructed in this step constitute a stationary monetary equilibrium provided µ ∈ (β̄, min (µ̂, µo )).
To conclude this step, we show that µ̂ < µ̄ < µo , which together with the previous step will
mean that there is no stationary monetary equilibrium for µ ≥ µ̄ (thus establishing part (ii )in the statement of the proposition). It is clear that µ̄ < µo , and we know that µ̂ < µ̄ from
Lemma 4. Therefore, the allocations and asset prices described in this step correspond to thosein the statement of the proposition for the case with µ ∈ (β̄, min (µ̂, µo )) = (β̄, µ̂).
Step 3: Try to construct a stationary monetary equilibrium with ãsdt+1 > 0 for some d ∈ D
and ãsit+1 > 0 for some i ∈ I. The equilibrium conditions are (17), (44), (45), (46), (47), and(53) withãsit+1 ≥ 0 and ãsdt+1 ≥ 0, with “ > ” for some i ∈ I or some d ∈ I.
Notice that ε∗ and φs are obtained as in Step 2. Now, however, (45) must also hold, which
together with (57) implies we must have T̂ (ε∗ ) = 0, or equivalently, ε∗ = ε̂. In other words,
this condition requires R (ε̂) = T̂ (ε̂), or equivalently, we must have µ = µ̂. As before, themarket-clearing condition (17) is used to obtain Z, while (14), (15), and (16) imply φ̄s , φmt , andpt , respectively. The allocations and asset prices described in this step correspond to those in
the statement of the proposition for the case with µ = µ̂.
Combined, Steps 1, 2, and 3 prove part (iv ) in the statement of the proposition. Part (v )(a)is immediate from (45) and (51), and part (v )(b) from (57) and (59).
Corollary 3 The marginal type, ε∗ , characterized in Proposition 2 is strictly decreasing in the∗rate of inflation, i.e., ∂ε∂µ < 0 both for µ ∈ (β̄, µ̂) and for µ ∈ (µ̂, µ̄).
Proof of Corollary 3. For µ ∈ β̄, µ̂ , implicitly differentiate R (ε∗ ) = 0 (with R given by(59)), and for µ ∈ (µ̂, µ̄), implicitly differentiate T (ε∗ ) = 0 (with T given by (51)) to obtainε∗− β̄αθ 1−β̄δ [1−G(εif β̄ < µ < µ̂∗ )]+µ−β̄
∗
(
)
∂ε
R εH
=
β̄αθ ε∗ [1−G(ε)]dε∂µ − n1+β̄αθh δG(ε∗ ) + 1−G(ε∗ ) io(µ−β̄ )2 if µ̂ < µ < µ̄.
1−β̄δ
µ−β̄
Clearly, ∂ε∗ /∂µ < 0 for µ ∈ (β̄, µ̂) and for µ ∈ (µ̂, µ̄).
Proof of Proposition 3. Recall that ∂ε∗ /∂µ < 0 (Corollary 3). (i ) From (13),
i ∂ε∗
∂φs
β̄δ h
=
I{β̄<µ≤µ̂} + I{µ̂<µ<µ̄} αθG (ε∗ )
< 0,
∂µ
∂µ
1 − β̄δ52

and from (19), ∂φs /∂ι = β̄∂φs /∂µ < 0. (ii ) Condition (14) implies ∂ φ̄s /∂µ = ∂ε∗ /∂µ +∂φs /∂µ < 0. (iii ) From (17) it is clear that ∂Z/∂ε∗ > 0, so ∂Z/∂µ = (∂Z/∂ε∗ )(∂ε∗ /∂µ) < 0.
m
From (15), ∂φmt /∂µ = (yt /At ) ∂Z/∂µ < 0.

Proof of Proposition 4. From condition (18),µ−β̄ ∗∗∂ε∗αθ [ε + β̄δ (ε̄ −ε ) I{µ̂<µ} ]=∂ (αθ)β̄αθ(1 − β̄δ) [1 − G (ε∗ )] + (µ − β̄) 1 + β̄δ [αθG (ε∗ ) − 1] I{µ̂<µ}

> 0.

(60)

(i ) From (13),∂φs∂ (αθ)

=





β̄δ
∂ε∗
>0
1−β̄δ ∂(αθ)
i
hR ∗
ε
β̄δ
∗ ) ∂ε∗
G
(ε)
dε
+
αθG
(ε
εL
∂(αθ) > 01−β̄δ

if β̄ < µ ≤ µ̂
if µ̂ < µ < µ̄.

(ii ) From (14), ∂ φ̄s /∂ (αθ) = ∂ε∗ /∂ (αθ) + ∂φs /∂ (αθ) > 0. (iii ) For µ ∈ (µ̂, µ̄), (17) impliesm∂Z/∂α = (∂Z/∂ε∗ ) (∂ε∗ /∂α) > 0 and therefore ∂φmt /∂α = (∂Z/∂α) (yt /At ) > 0.

Proof of Proposition 5. (i ) The result is immediate from the expression for AsD in Proposition
2. (ii ) From (10) and (11),∂ µ̂= β̄ 1 − β̄δ∂ (αθ)

(1 − αθ) ε̄
[1 − αθG (ε̂)] ε̂2Z ε̂ε̂ − ε̄G (ε) dε −
ε̂
εL


.

Notice that ∂ µ̂/∂ (αθ) approaches a positive value as αθ → 0 and a negative value as αθ → 1.
Also, µ̂ → β̄ both when αθ → 0 and when αθ → 1. Hence, µ > β̄ = limαθ→0 µ̂ = limαθ→1 µ̂ fora range of values of αθ close to 0 and a range of values of αθ close to 1. For those ranges ofvalues of αθ, AsD = 0. In between those ranges there must exist values of αθ such that µ < µ̂,
which implies AsD > 0.
Proof of Proposition 6. (i ) Differentiate (20) to get∂V∂ε∗= 2αG0 (ε∗ ) (As − δ ÃsD )
< 0,
∂µ
∂µ
where the inequality follows from Corollary 3. Also, from (19), ∂V/∂ι = β̄∂V/∂µ < 0. (ii )From (20),∂V∂ε∗= 2αG0 (ε∗ ) (As − δ ÃsD )
∂θ
∂θ

∗
∂V
∂ε
= 2 G (ε∗ ) + αG0 (ε∗ )(As − δ ÃsD ),
∂α
∂α
53

and both are positive since ∂ε∗ /∂ (αθ) > 0 (see (60)).

Proof of Proposition 7. (i ) For β̄ < µ ≤ µ̂, ∂P/∂µ = β̄δ/(1 − β̄δ) (∂ε∗ /∂µ) < 0, and forµ̂ < µ < µ̄, ∂P/∂µ = β̄δ/(1 − β̄δ) αθG (ε∗ ) (∂ε∗ /∂µ) < 0. In both cases, ∂P/∂ι = β̄∂P/∂µ <0. (ii ) For β̄ < µ ≤ µ̂, ∂P/∂ (αθ) = β̄δ/(1 − β̄δ) (∂ε∗ /∂ (αθ)) > 0, and for µ̂ < µ < µ̄,R ε∗∂P/∂µ = β̄δ/(1 − β̄δ) {αθG (ε∗ ) [∂ε∗ /∂ (αθ)] + εL G (ε) dε} > 0.

Proposition 8 Assume G (ε; σ) is a differentiable function of the parameter σ that indexes afamily of mean-preserving spreads, so that for any σ < σ 0 , G (·; σ 0 ) is a mean-preserving spreadof G (·; σ). Then in the stationary monetary equilibrium, ∂φs /∂σ > 0 and ∂ φ̄s /∂σ > 0.
Proof of Proposition 8. From the definition of the mean-preserving spread, for any ∆ > 0,Z x
[G (ε; σ + ∆) − G (ε; σ)] dε ≥ 0 for all x ∈ (εL , εH ) ,εLwith “=” if x ∈ {εL , εH }, and therefore
Z x
Z x[G (ε; σ + ∆) − G (ε; σ)]Gσ (ε; σ) dε ≥ 0 for all x ∈ (εL , εH ) ,dε =lim∆→0 εL∆εLwith “=” if x ∈ {εL , εH }, where Gσ (ε; σ) ≡ ∂Gσ (ε; σ) /∂σ. With this notation, the equilibrium
mapping (18) isT (x; σ) =

R εH
1−β̄δ
[1 − G (ε; σ)] dε
1−β̄δI{µ̂<µ} xβ̄δIx + 1−β̄δI{µ̂<µ}{µ̂<µ}

hµ − β̄i−,β̄αθε̄ + αθ εL G (ε; σ) dεRxand the equilibrium ε∗ satisfies T (ε∗ ; σ) = 0. By implicitly differentiating this condition, we
get
∂ε∗

=−

αθβ̄
1−β̄δI{µ̂<µ}β̄δδI{µ̂<µ} − 1−
µ−β̄

R ∗

ε
εL Gσ (ε; σ) dεi.
β̄δG (ε∗ ; σ) δI{µ̂<µ} + [1 − G (ε∗ ; σ)] 1−µ−β̄{µ̂<µ}

If µ ∈ β̄, µ̂ , then ∂ε∗ /∂σ > 0 since 1 − β̄δ / µ − β̄ − δI{µ̂<µ} = 1 − δ β̄ / µ − β̄ > 0. If∂σαθβ̄
1 + 1−β̄δIhµ ∈ (µ̂, µ̄), then ∂ε∗ /∂σ > 0 since β̄δ (1 − αθ) ε̄ + 1 − β̄δ (1 − αθ) εL<1δµ < δ µ̄ = 1 − 1 − β̄δβ̄δ ε̄ + 1 − β̄δ εL

54





implies − δ − 1 − β̄δ / µ − β̄ = (1 − δµ) / µ − β̄ > 0. Given that ∂ε∗ /∂σ > 0 for allµ ∈ β̄, µ̄ , (13) and (14) imply ∂φs /∂σ > 0 and ∂ φ̄s /∂σ > 0, respectively.
The following proposition shows there is a certain equivalence between α and G as fundamental determinants of trading activity.
Proposition 9 Consider Economy A with contact probability α and distribution of valuations
G on [εL , εH ] and Economy B with contact probability α̃ and distribution of valuations G̃ on[ε̃L , ε̃H ] (and all other primitives of Economy B are as in Economy A). Let ε∗ and ε̃∗ denote theequilibrium marginal type for Economy A and Economy B, respectively. Then for any α̃ > α,
there exists a G̃ such thatβ̄δI{µ̂<µ} 1 − αα̃

∗

ε̃ =

"



1 − β̄δ 1 − I{µ̂<µ} ε̄ + 1 −β̄δI{µ̂<µ} 1 − αα̃ #1 − β̄δ 1 − I{µ̂<µ} ε∗ ,and moreover, trade volume in Economy B is the same as in Economy A.
Proof of Proposition 9. In Economy A the marginal investor type, ε∗ , is characterized by(18), while in Economy B the marginal investor type is the ε̃∗ that solvesi R ε̃H h
1 − β̄δ α̃θ ε̃∗ 1 − G̃ (ε) dεµ − β̄
i
hR
−
= 0.

R ε̃∗
ε̃H
β̄
∗
G̃ (ε) dε I
1 − β̄δ ε̃ + β̄δεdG̃ (ε) + α̃θ
ε̃L

ε̃L

DefineG̃ (ε) = 0αα̃ G (ε − c) +

1 − α̃α

I{ε∗ <ε−c}1{µ̂<µ}for ε ≤ ε̃L
for ε̃L ≤ ε ≤ ε̃H
for ε̃H < ε(61)with ε̃L ≡ εL + c, ε̃H ≡ εH + c andβ̄δI{µ̂<µ}


α

c≡
1−
(ε̄ − ε∗ ) .
α̃
1 − β̄δ 1 − I{µ̂<µ}(62)With (61) and (62), the equilibrium mapping for Economy B becomes Rε 
1 − β̄δ α̃θ ε̃∗H−c 1 − α̃α G (z) − 1 − αα̃ I{ε∗ <z} dzµ − β̄
h
i
−
= 0.


R ε̃∗ −c  ααβ̄∗ <z} dz I{µ̂<µ}1 − β̄δ (ε̃∗ − c) + β̄δ ε̄ + α̃θ εL
G
(z)
+
1
−
I
{ε
α̃
α̃
If we replace ε̃∗ = ε∗ +c in this last expression, it reduces to (18), a condition that holds becauseε∗ is the equilibrium marginal type for Economy A. Hence, ε̃∗ = ε∗ + c with c given by (62) isthe equilibrium marginal type for Economy B. Notice thatα̃G̃ (ε̃∗ ) = α̃G̃ (ε∗ + c) = αG (ε∗ ) ,55

so (20) implies that trade volume in Economy B is the same as in Economy A.

56

B

Supplementary material

B.1

Heteroskedasticity-based estimator

In this section we explain the H-based estimator used in Section 6.2. Rigobon and Sack (2004)
show that the response of asset prices to changes in monetary policy can be identified basedon the increase in the variance of policy shocks that occurs on days of FOMC announcements.
They argue that this approach tends to be more reliable than the event-study approach basedon daily data because identification relies on a weaker set of conditions.
The idea behind the heteroskedasticity-based estimator of Rigobon and Sack (2004) is asfollows. Suppose the change in the policy rate, ∆it , and Yt (where Yt could be the stock market
return, RIt , or the turnover rate, TtI ) are jointly determined by∆it = κYt + $xt + t

(63)

Yt = ρ∆it + xt + ηt ,(64)

where t is a monetary policy shock and ηt is a shock to the asset price. To fix ideas, suppose
Yt = RIt . Then equation (63) represents the monetary policy reaction to asset returns andpossibly other variables represented by xt . Equation (64) represents the reaction of asset prices
to the policy rate and xt . The disturbances t and ηt are assumed to have no serial correlationand to be uncorrelated with each other and with xt . We are interested in estimating theparameter ρ. Let Σv denote the variance of some variable v. If (63) and (64) were the true modeland one were to run an OLS regression on an equation like (21), there would be a simultaneity
bias if κ 6= 0 and Ση > 0, and an omitted variable bias if $ 6= 0 and Σx > 0. Conditions (63) and11(64) can be solved for ∆it = 1−ρκ[t + κηt + (κ + $) xt ] and Yt = 1−ρκ[ρt + ηt + (1 + ρ$) xt ].

Divide the data sample into two subsamples: one consisting of FOMC policy announcement
days and another consisting of the trading days immediately before the policy announcementdays. In what follows we refer to these subsamples as S1 and S0 , respectively. Let Ωk denotethe covariance matrix of ∆it and RIt for t ∈ Sk , for k ∈ {0, 1}. Then k
1
Ω11 Ωk12Ωk =,kk(1 − ρκ)2 Ω21 Ω22
where Ωk11 ≡ Σk + κ2 Σkη + (κ + $)2 Σkx , Ωk12 = Ωk21 ≡ ρΣk + κΣkη + (κ + $) (1 + ρ$) Σkx , Ωk22 ≡ρ2 Σk + Σkη + (1 + ρ$)2 Σkx , and Σkx denotes the variance of variable x in subsample Sk , for57k ∈ {0, 1}. Provided Σ1x = Σ0x and Σ1η = Σ0η ,Σ1 − Σ0Ω −Ω = (1 − ρκ)2
1

0



1 ρ
ρ ρ2


.

Hence, if Σ1 − Σ0 > 0, then ρ can be identified from the difference in the covariance matrices of
the two subsamples. This suggests a natural way to estimate ρ. Replace Ω1 and Ω0 with theirsample estimates, denoted Ω̂1 and Ω̂0 . Define Ω̂ ≡ Ω̂1 − Ω̂0 and use Ω̂ij to denote the (i, j)element of Ω̂. Then ρ can be estimated by Ω̂12 /Ω̂11 ≡ ρ̂. Rigobon and Sack (2004) show that
this estimate can be obtained by regressing RIt on ∆it over the combined sample S0 ∪ S1 using
a standard instrumental variables regression.
The standard deviation of ∆it is 3.53 basis points (bps) in subsample S0 and 6.84 bps
in subsample S1 . The standard deviation of RIt is 49.67 bps in subsample S0 and 64.22 bps
in subsample S1 . The correlation between ∆it and RIt is 0.19 in subsample S0 and −0.4 in
subsample S1 . Stock returns are more volatile on the days of monetary policy announcements
than on other days, which is consistent with policy actions inducing some reaction in thestock market. The relatively large negative correlation between the policy rate and stock
returns for announcement days contrasts with the much smaller and positive correlation for
non-announcement days, suggesting that the negative effect of surprise increases in the nominal
rate on stock prices that has been documented in the empirical literature (e.g., Bernanke andKuttner, 2005, Rigobon and Sack, 2004).

B.2

High-frequency IV estimator

In this section we consider a version of the event-study estimator that, instead of daily changes
in the interest rate, uses intraday high-frequency tick-by-tick interest rate data to isolate thechange in the interest rate that takes place over a narrow window around each policy announcement. We refer to this as the high-frequency instrumental variable estimator (or “HFIV”
estimator, for short).
Specifically, the HFIV estimator is obtained by estimating (21), where instead of directly
using the daily change in the 3-month Eurodollar future rate, we instrument for it using the daily
imputed change in the 30-day federal funds futures rate from the level it has 20 minutes after the
FOMC announcement and the level it has 10 minutes before the FOMC announcement.34 By
34

By “daily imputed” we mean that in order to interpret the change in the federal funds futures rate as the
surprise component of the change in the daily policy rate, it is adjusted for the fact that the federal funds futures

58

focusing on changes in a proxy for the policy rate in a very narrow 30-minute window around thetime of the policy announcement, the resulting HFIV estimator addresses the omitted variablebias and the concern that the Eurodollar futures rate may itself respond to market conditions
on policy announcement days.
The data for the high-frequency interest change are constructed as follows. For each announcement day t ∈ S1 , we define zt ≡ it,m∗t +20 −it,m∗t −10 , where it,m denotes the (daily imputed)
30-day federal funds futures rate on minute m of day t, and for any t ∈ S1 , m∗t denotes the timeof day (measured in minutes) when the FOMC announcement was made.35 We then estimate bededin (21) using the following two-stage least squares (2SLS) procedure. Define ∆iedt ≡ it − it−1 ,where iedt denotes the rate implied (for day t) by the 3-month Eurodollar futures contract withclosest expiration date at or after day t. First, run the regression ∆iedt = κ0 + κzt + ηt on
sample S1 (where ηt is an error term) to obtain the OLS estimates of κ0 and κ, namely κ̂0 andκ̂. Second, construct the fitted values ẑt ≡ κ̂0 + κ̂zt and run the event-study regression (21)setting ∆it = ẑt .
The sample period is again 1994-2001, but the data from Gorodnichenko and Weber (2016)that we use contain intraday high-frequency interest rates for 64 policy announcement days
(we used data for 73 announcement days in the analysis of Section 6.2). The columns labeled“HFIV” in Table 1 report the HFIV estimates. The stock return is expressed in percentage
terms. The first row reports estimates of the reaction of equity returns to monetary policy
shocks. The point estimate for b in (21) is −8.57. This means that a 1 bp increase in thepolicy rate causes a reduction of 8.57 bps in the stock market return on the day of the policy
announcement. The estimated effects of monetary policy announcements on the turnover rate
are reported in the second row. A 1 bp increase in the policy rate causes a change in the level ofthe marketwide turnover rate on the day of the policy announcement equal to −.000043. The
daily marketwide turnover rate for the sample period is .0037 (i.e., on average, stocks turn over
.94 times during a typical year composed of 252 trading days), which means that an increasein the policy rate of 1 bp causes a reduction in the marketwide turnover rate on the day of thepolicy announcement of about 1.16 percent of its typical level.
contracts settle on the effective federal funds rate averaged over the month covered by the contract. See Section
B.4.4 for details.
35
We use the data set constructed by Gorodnichenko and Weber (2016) with tick-by-tick data of the federal
funds futures trading on the CME Globex electronic trading platform (as opposed to the open-outcry market).
The variable we denote as zt is the same variable that Gorodnichenko and Weber denote as vt . Their data are
available at http://faculty.chicagobooth.edu/michael.weber/research/data/replication dataset gw.xlsx.

59

For each of the 20 portfolios analyzed in Section 6.3, the two columns in Table 4 labeled“HFIV” report the HFIV estimates of the responses (on the day of the policy announcement)of the return and turnover of the portfolio to a 1 pp increase in the policy rate.
All the estimates in the column labeled “Return” are negative (all but one are significant
at the 1 percent level) as predicted by the theory. Also as predicted by the quantitative theory,the magnitude of the estimates tends to increase with the turnover liquidity of the portfolio.
The HFIV estimates for the response of the turnover rates are all negative and for the mostpart statistically significant at the 1% level, as predicted by the theory. Also as predicted by
the theory, the magnitude of the estimates tends to increase with the turnover liquidity of theportfolio.

B.3

More on disaggregative announcement-day effects

In Section 6.3 and Section B.2, we sorted stocks into 20 portfolios according to the level ofturnover of each individual stock and found that changes in the nominal rate affect stocks withdifferent turnover liquidity differently, with more liquid stocks responding more than less liquid
stocks. In this section, we complement that analysis by using an alternative procedure to sortstocks into portfolios. Specifically, in this section we sort stocks according to the sensitivityof their individual return to changes in an aggregate (marketwide) measure of turnover. This
alternative criterion is useful for two reasons. First, it will allow us to control for some differences
across stocks, such as the conventional risk factors used in empirical asset-pricing models.
Second, this sorting criterion emphasizes the responsiveness of the individual stock return tochanges in an aggregate measure of turnover, which is another manifestation of the transmission
mechanism that operates in the theory. To construct the portfolios, we proceed as follows.
For each individual stock s in our sample, we use daily time-series data to run
Rst = αs + β0s TtI +K
X

βjs fj,t + εst ,

(65)

j=1

where εst is an error term, Rst is the daily stock return (between day t and day t − 1), TtI is
the aggregate (marketwide) turnover rate on day t, and {fj,t }K
j=1 are K pricing factors. We
set K = 3, with f1,t = M KTt , f2,t = HM Lt , and f3,t = SM Bt , where M KTt is a broadmeasure of the market excess return, HM Lt is the return of a portfolio of stocks with highbook-to-market value minus the return of a portfolio of stocks with low book-to-market value,60and SM Bt is the return of a portfolio of small-cap stocks minus the return of a portfolio oflarge-cap stocks. That is, M KTt is the typical CAPM factor, while HM Lt and SM Bt are
the long-short spreads constructed by sorting stocks according to book-to-market value andmarket capitalization, respectively, as in the Fama and French (1993) three-factor model.36 Let
tk denote the day of the k th policy announcement (we use 73 policy announcement days fromour sample period 1994-2001). For each stock s, regression (65) is run 73 times, once for eachpolicy announcement day, each time using the sample of all trading days between day tk−1sand day tk . Thus, for each stock s we obtain 292 estimates, {{βjs (k)}3j=0 }73
k=1 , where βj (k)denotes the estimate for the beta corresponding to factor j for stock s, estimated on the sample
consisting of all trading days between the policy announcement days tk−1 and tk . For eachpolicy announcement day, tk , we sort all NYSE stocks into 20 portfolios by assigning stocks
with β0s (k) ranked between the [5 (i − 1)]th percentile and (5i)th percentile to the ith portfolio,
for i = 1, ..., 20. For each portfolio i ∈ {1, ..., 20} constructed in this manner, we compute thei , and run the event-study
daily return, Rit , and the daily change in the turnover rate, Tti − Tt−1
i , as inregression (21) portfolio-by-portfolio, first with Yti = Rit and then with Yti = Tti − Tt−1

Section 6.3.
For each of the 20 portfolios, Table 5 reports the E-based estimates of the responses (on the
day of the policy announcement) of the return and turnover of the portfolio to a 1 pp increasein the policy rate. Estimates are negative, as predicted by the theory. Also as predictedby the theory, the magnitude of the estimates tends to be larger for portfolios with higher
indices. From these estimates we learn that stocks whose returns are more sensitive to aggregate
measures of aggregate market turnover tend to experience larger declines in returns in response
to unexpected increases in the nominal rate. This finding is in line with the turnover-liquidity
channel of monetary policy.
36

In order to construct the Fama-French factors HM Lt and SM Bt , stocks are sorted into six portfolios obtained from the intersections of two portfolios formed on size (as measured by market capitalization and labeled “Small” and “Big”) and three portfolios formed on the ratio of book value to market value (labeled “Value,” “Neutral,” and “Growth”). Then SM Bt = (1/3) RSG
+ RSN
+ RSV
−
t
t
t



(1/3) RBG
+ RBN
+ RBV
and HM Lt = (1/2) RSV+ RBV− (1/2) RSG+ RBG, where RBG
denotes
t
t
t
t
t
t
t
t
the return on portfolio “Big-Growth,” “RSVt ” denotes the return on portfolio “Small-Value,” and so on.
For a detailed description of the breakpoints used to define the six portfolios, see Kenneth French’s website, http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data Library/six portfolios. The CAPM factor,
M KTt , is a broad measure of the market excess return, specifically, the value-weighted return of all CRSP firms
incorporated in the United States and listed on the NYSE, AMEX, or NASDAQ that have a CRSP share codeof 10 or 11 at the beginning of month t, good shares and price data at the beginning of t, and good return data
for t minus the one-month Treasury bill rate (from Ibbotson Associates). The data for the three Fama-French
factors were obtained from Wharton Research Data Services (WRDS).

61

Notice that by sorting portfolios on the β0 ’s estimated from (65), we are controlling for thethree standard Fama-French factors. To explore how the portfolios sorted in this manner varyin terms of the three standard Fama-French factors, we construct the series of monthly return
20
for each of the 20 portfolios for the period 1994-2001, { Rit i=1 }, and run (65) to estimate the3vector of betas, {{βji }20
i=1 }j=0 . The estimated betas corresponding to each portfolio are displayed
20
in Figure 8.37 Notice that there is no correlation between the turnover-liquidity betas, β0i i=1 ,
20
and the CAPM betas, β1i i=1 . To get a sense of whether the different cross-portfolio responsesof returns to policy shocks documented in Table 5 can be accounted for by the standard CAPM,
consider the following back-of-the-envelope calculation. Let b denote the effect of a 1 bp increasein the policy rate on the marketwide stock return on the day of the policy announcement (e.g.,
the E-based estimate obtained from running (21) with YtI = RIt ). Then according to the basic
20
CAPM model, the effect on portfolio i ∈ {1, ..., 20} would be b̃i ≡ β1i × b, where β1i i=1 is thevector of betas estimated on monthly data for each of the 20 portfolios sorted on β0i (plotted in i 20i 20
Figure 8). Figure 9 plots {(i, b̃i )}20
i=1 and {(i, b )}i=1 , where b i=1 corresponds to the E-based
estimates for the effect of monetary policy on returns reported in Table 5.

B.4

VAR estimation

B.4.1

Identification

We conjecture that the data, {Yt } with Yt ∈ Rn , correspond to an equilibrium that can beapproximated by a structural vector autoregression (SVAR),
KYt =

J
X

Cj Yt−j + εt ,

(66)

j=1

where K and Cj are n × n matrices, J ≥ 1 is an integer that denotes the maximum number oflags, and εt ∈ Rn is a vector of structural shocks, with E (εt ) = 0, E (εt ε0t ) = I, and E (εt ε0s ) = 0
for s 6= t, where 0 is a conformable matrix of zeroes and I denotes the n-dimensional identity.
If K is invertible, (66) can be represented by the reduced-form VAR
Yt =

J
X

Bj Yt−j + ut ,

(67)

j=1

where Bj = K −1 Cj and
ut = K −1 εt37

1The vector {β0i }20
i=1 shown in the figure has been normalized by dividing it by β0 .

62

(68)

is an error term withΞ ≡ E ut u0t = K −1 K −10 .

(69)

The reduced-form VAR (67) can be estimated to obtain the matrices {Bj }Jj=1 , and theresiduals {ut } from the estimation can be used to calculate Ξ. From (68), we know that the
disturbances of the reduced-form VAR (67) are linear combinations of the structural shocks,
εt , so in order to use (67) and the estimates {Bj }Jj=1 to compute the impulse responses to thestructural shocks, it is necessary to find the n2 elements of the matrix K −1 . However, given
the known covariance matrix Ξ, (69) only provides n(n + 1)/2 independent equations involving
the elements of K −1 , so n (n − 1) /2 additional independent conditions would be necessary tofind all elements of K −1 . This is the well-known identification problem of the SVAR (66).
Only three specific elements of K −1 are relevant for our analysis. To find them, we use an
identification scheme that relies on external instruments.38
0
The VAR we estimate consists of three variables, i.e., Yt = it , RIt , TtI , where it , RIt , and
TtI are the measures of the policy rate, the stock return, and turnover described in Sections 6.1

T 0 , u = ui , uR , uT 0 , andand 6.2. Denote εt = εit , εRtt , εtt tt
kii kiR kiTi
R kT  .
kR
K −1 =  kRR
kTi kTR kTTThen ut = K −1 εt can be written as
 T 
 R 
 i   i 
ki
ki
ki
ut
i  εi +  k R  εR +  k T  εT .


 uR
k
=
t
t
t
t
R
R
R
kTT
kTR
kTi
uTt

(70)

Since we are only interested in the impulse responses for the monetary shock, εit , it suffices
to find the first column of K −1 . The identification problem we face, of course, stems fromT
the fact that the structural shocks, εit , εRt , εt , are unobservable and some of the elements ofK −1 are unknown (three elements are unknown in this 3 × 3 case). Suppose we had data on
 i
εt . Then we could run the regression uit = κii εit + ηt to estimate κii , where ηt is an error i R R

T T
i
T T
term. From (70) we have ηt = kiR εR= 0 (sincet + ki εt , so E εt ηt = E εt ki εt + ki εt
we are assuming E(εt ε0t ) = I), and thus the estimate of κii could be used to identify kii (up toa constant) via the population regression of uit onto εit . Since εit is unobservable, one natural38

The identification methodology has been used by Mertens and Ravn (2013), Stock and Watson (2012),
Gertler and Karadi (2015), Hamilton (2003), and Kilian (2008a, 2008b), among others.

63alternative is to find a proxy (instrumental) variable for it. Suppose there is a variable zt such
that



E zt ε R= E zt εTt = 0 < E zt εit ≡ v for all t.
tTheniΛ ≡ E(zt ut ) = K −1 E(zt εt ) = kii , kR
, kTi

0

v.

(71)

Since Λ = (Λ1 , Λ2 , Λ3 )0 is a known (3×1) vector, we can identify the coefficients of interest,i , kikii , kR
T up to the sign of the scalar v. To see this, notice (71) implies
vkii = Λ1

(72)

i
vkR
= Λ2

(73)

vkTi = Λ3(74)v 2 = E(zt ut )0 Ξ−1 E(zt ut ).

(75)

with

Since the sign of v is unknown, we could look for restrictions that do not involve v, and in thisi , k i , i.e., combiningcase these conditions only provide two additional restrictions on kii , kRT(72) with (73), and (72) with (74), yields
i
kR
Λ2
=
i
Λ1
ki

(76)

kTi
Λ3
=
.
i
Λ1
ki

(77)

i and k i are identified. From (72), k i is also identified but up to the sign of v.
Thus, kRi
Tii
Notice that if we run a 2SLS regression of uRt on ut using zt as an instrument for ut , thenthe estimate of the slope coefficient on this regression is Λ2 /Λ1 . Similarly, Λ3 /Λ1 corresponds
to the instrumental variable estimate of the slope coefficient of a regression of uTt on uit usingzt as an instrument for uit .
In our application, as an instrument for the structural monetary policy shock, εit , we use the(daily imputed) change in the 30-day federal funds futures from the level it has 10 minutes before
the FOMC announcement and the level it has 20 minutes after the FOMC announcement.39
39
By “daily imputed” we mean that in order to interpret the change in the federal funds futures rate as the
surprise component of the change in the daily policy rate, it is adjusted for the fact that the federal funds futurescontracts settle on the effective federal funds rate averaged over the month covered by the contract. See Section
B.4.4 for details.

64That is, we restrict our sample to t ∈ S1 and set {zt } =it,m∗t +20 − it,m∗t −10 , where it,mdenotes the (daily imputed) 30-day federal funds futures rate on minute m of day t, and for anyt ∈ S1 , m∗t denotes the time of day (measured in minutes) when the FOMC announcement was
made.40 All this leads to the following procedure, used by Mertens and Ravn (2013), Stock andWatson (2012), and Gertler and Karadi (2015), to identify the coefficients needed to estimate
the empirical impulse responses to a monetary policy shock:Step 1: Estimate the reduced-form VAR by least squares over the whole sample of all trading
days to obtain the coefficients {Bj }Jj=1 and the residuals {ut }.
Step 2: Run the regression uit = κ0 + κi zt + ηt on sample S1 to obtain the OLS estimates ofκ0 and κi , namely κ̂0 and κ̂i , and construct the fitted values ûit = κ̂0 + κ̂i zt .
R i
T
T i
Step 3: Run the regressions uR
t = κ0 + κ ût + ηt and ut = κ0 + κ ût + ηt on sample S1 toobtain the OLS estimates of κR and κT , namely κ̂R and κ̂T . Since κ̂R = Λ2 /Λ1 andi /k i and κ̂T = k i /k i .
κ̂T = Λ3 /Λ1 , (76) and (77) imply κ̂R = kRiiTFor the purpose of getting impulse responses with respect to the shock εit , the scale and signof kii are irrelevant since the shock εit is typically normalized to have any desired impact on agiven variable.41 For example, in our impulse responses we normalize the shock εit so that it
induces a 1 pp increase in the level of the policy rate it on impact. To see this, consider (70)Tiwith εRt = εt = 0. Then for any ki , the shock that induces an x pp increase in the level of thepolicy rate on impact (e.g., at t = 0) is εi0 = (x/100)/kii = (x/100)/(Λ1 /v).
B.4.2

Confidence intervals for impulse responses

The 95 percent confidence intervals for the impulse response coefficients estimated from the data
are computed using a recursive wild bootstrap using 10,000 replications, as in Gonçalves and40

We use the data set constructed by Gorodnichenko and Weber (2016) with tick-by-tick data of the federal
funds futures trading on the CME Globex electronic trading platform (as opposed to the open-outcry market).
The variable we call zt is the same variable that Gorodnichenko and Weber denote as vt . Their data are
available at http://faculty.chicagobooth.edu/michael.weber/research/data/replication dataset gw.xlsx. We havealso performed the estimations using a different instrument for the high-frequency external identification scheme,
namely the 3-month Eurodollar rate (on the nearest futures contract to expire after the FOMC announcement)from the level it has 10 minutes before the FOMC announcement and the level it has 20 minutes after the FOMC
announcement. That is, we restrict our sample to t ∈ S1 and set {zt } = {ied− ied}, where iedt,m denotes
t,m∗
t,m∗
t +20
t −10
the 3-month Eurodollar futures rate on minute m of day t, and for any t ∈ S1 , m∗t denotes the time of day(measured in minutes) when the FOMC announcement was made. The results were essentially the same.
41
Alternatively, (72) and (75) can be combined to get kii = Λ1 /v, which is then identified up to the sign of v.

65

Kilian (2004) and Mertens and Ravn (2013). The procedure is as follows. Given the estimates
of the reduced-form VAR, {B̂j }Jj=1 , and the residual, {ût }, we generate bootstrap draws, Ytb ,P
recursively, by Ytb = Jj=1 B̂j Yt−j + ebt ût , where ebt is the realization of a scalar random variabletaking values of −1 or 1, each with probability 1/2. Our identification procedure also requires usto generate bootstrap draws for the proxy variable, ztb , so following Mertens and Ravn (2013),
we generate random draws for the proxy variable via ztb = ebt zt . We then use the bootstrap
samples Ytb and ztb to reestimate the VAR coefficients and compute the associated impulse
responses (applying the covariance restrictions implied by the bootstrapped instrument ztb ).
This gives one bootstrap estimate of the impulse response coefficients. The confidence intervals
are the percentile intervals of the distribution of 10,000 bootstrap estimates for the impulse
response coefficients.
B.4.3

Lag length

The Akaike information criterion (AIC) suggests a VAR specification with 10 lags, while
Schwarz’s Bayesian information criterion (SBIC) and the Hannan and Quinn information criterion (HQIC) suggest a specification with 5 lags. In order to choose between these alternatives,
we check which specification delivers a better estimate of the true impulse responses generated
by the quantitative theory presented in Section 7. Specifically, the baseline parameters andthe dividend process are calibrated as in Section 7.2. For the policy rate we adopt the AR(1)process estimated from the data (see footnote 29 in Section 7.2 for details). We then proceed
as follows:
1. Compute the equilibrium functions characterized by (23)-(26) for the calibrated model.
2. Simulate 1,000 samples of the dividend and the policy rate, each sample of length equal
to our data sample (1,996 days, which is the number of trading days between January 1,
1994, and December 31, 2001).42
3. Compute the equilibrium of the model 1,000 times (one for each realization of the simulated dividend and policy rate paths) to produce 1,000 synthetic data samples (each42
The policy rate is simulated using the AR(1) process we estimated for our proxy for the policy rate (seefootnote 29 in Section 7.2). The initial condition for the policy rate is the mean value of the estimated AR(1)process. The initial condition for the dividend is normalized to 1.

66

consisting of 1,996 days) for the policy rate and the (arithmetic) average (across the 20
stock classes) of the stock return and the turnover rate.43
4. For each synthetic data set, estimate our baseline VAR specification with 10 lags andan alternative specification with 5 lags. For each specification, calculate the impulse
responses to a 1 bp increase in the policy rate.44
5. Calculate the average (median) impulse responses over the 1,000 synthetic samples andcalculate the 95 percent confidence intervals using the distribution of estimates over the
1,000 synthetic samples. Do this for the specification with 10 lags and for the one with 5
lags.
6. Use the equilibrium conditions of the model to compute the true theoretical impulse
responses to a 1 bp increase in the policy rate that follows the estimated AR(1) process.45
For each variable, Figure 7 reports the average (median) impulse response and the 95 percent
confidence interval corresponding to the VAR specifications with 5 and 10 lags, along with thetrue theoretical impulse response implied by the equilibrium conditions. The impulse responses
from the VAR specification with 10 lags and those from the VAR specification with 5 lags
approximate the true theoretical responses about as well.
B.4.4

Changes in federal funds future rate and unexpected policy rate changes
 0 T
Fix a month, s, and let the intervals {[t, t + 1]}Tt=1 denote the T days of the month. Let fs,t
t=1
denote the market prices of the federal funds futures contract at the end of day t of month s.
The superscript “0” indicates that the contract corresponds to the current month, s.46 Let{rt }Tt=1 be the (average) daily fededral funds rate calculated at the end of day t. Finally, forj = 1, ..., T − t, let Et rt+j denote the expectation of the spot federal funds rate on day t + j
43

Measurement noise is added to the equilibrium path of the turnover rate, as explained in footnote 31. Linear
interpolation is used to compute the equilibrium variables for realizations of the policy rate that lie outside theset of states of the Markov chain that is used to compute the equilibrium functions.
44
We use the same policy dates on the synthetic data as in the actual data. We also follow the same identification procedure used in the empirical section and described in Section B.4.1. To replicate this instrumental-variable
identification procedure in the context of the model simulations, as instrument (zt ) we simply use with the change
in the policy rate (which is indeed exogenous in the model).
45
Here again, linear interpolation is used to compute the equilibrium variables for realizations of the policy
rate that are not states of the Markov chain used to compute the equilibrium functions.
46
5
Contracts can range from 1 to 5 months. For example, fs,twould be the price of the 5-month forward on
day t of month s.

67

conditional on the information available at the end of day t. Then, since federal funds futurescontracts settle on the average daily rate of the month, we have


XT
1 Xt
0
Et ri , for t = 1, ..., T.
ri +fs,t =
i=t+1
i=1
THence, for t = 1, ..., T ,00fs,t− fs,t−1
=

1
1 XT
1
1 XT
Et ri −
Et−1 ri ,rt − Et−1 rt +
i=t+1
i=t+1
T
T
T
T

0 ≡ f1
where fs,0s−1,T . Assume the federal funds rate changes at most once during the month,and suppose it is known that the announcement takes place at the beginning of day t ≥ 1.47
Then
Et ri = rt for i = t, ..., T
Et−1 ri = Et−1 rt for i = t + 1, ...T.
Thus, the change in the forward rate at the time of the announcement, i.e., t = 1, ..., T , is00fs,t
− fs,t−1=

T +1−t(rt − Et−1 rt ) ,T(78)

where rt − Et−1 rt is the surprise change in the federal funds rate on day t (the day of thepolicy announcement). If we know the daily change in the forward rate at the time of the
0 −f0announcement, fs,ts,t−1 , then from (78) we can recover the unexpected change in the federal

funds rate on the day of the FOMC announcement, t, as follows:
rt+1 − Et rt+1 =T
00
fs,t+1− fs,tfor t = 0, ..., T − 1.
T −t

(79)

This condition is the same as condition (7) in Kuttner (2001), which is the convention used by
the event-study literature to map the change in the 30-day federal funds futures rate on the dayof the FOMC policy announcement into the surprise change in the daily policy rate on the dayof the announcement. In terms of the notation for our high-frequency instrument introduced

0
0
0
0 is
− fs,tin Section B.4.1, we set zt = TT−t fs,t+1− fs,t≡ it,m∗t +20 − it,m∗t −10 , where fs,t+1
measured (using high-frequency data) as the change in the 30-day federal funds futures rateover a 30-minute window around the FOMC announcement that takes place on day t.
47
If rt were the actual target federal funds rate, then the assumption that it changes at most once in the monthwould be exactly true for most of our sample; see, e.g., footnote 16 in Gorodnichenko and Weber (2016). In
general this has to be regarded as an approximation, since on any given day the effective federal funds rate, rt ,can and does deviate somewhat from the announced federal funds rate target rate (see Afonso and Lagos, 2014).

68B.4.5

Portfolio-based VAR analysis

In Section 6.4 we estimated a VAR consisting of three variables, i.e., it , RIt , TtI , where it is a
proxy for the policy rate (the 3-month Eurodollar futures rate), and RIt , and TtI are the daily
measures of the average NYSE stock return and turnover described in Sections 6.1 and 6.2.
In this section we follow the same methodology described in Section 6.4, but instead estimate
a separate VAR for each of the individual liquidity-based portfolios constructed as describedin Section 6.3. That is, for each portfolio i = 1, ..., 20, we estimate a VAR consisting of threevariables it , Rit , Tti . The results from this exercise confirm and extend the findings of Section
6.3 and Section 6.4. To illustrate, for three liquidity-based portfolios, Figure 10 reports theimpulse responses of the daily portfolio return and turnover rate to a 1 bp unexpected increasein the policy rate for a forecast horizon of 30 days. As in Section 6.3, we again find that on theannouncement day, the negative responses of returns and turnover to an unexpected increasein the nominal rate tend to be larger in magnitude for portfolios with higher turnover liquidity.
However, here these responses appear to be estimated much more precisely than by the E-basedand H-based estimates reported in Table 2.48 Interestingly, Figure 10 also confirms that, as in
the quantitative theory, the dynamic responses of returns and turnover for more liquid portfolios
tend to be not only larger but also more persistent than for less liquid portfolios.

B.5

Equilibrium conditions for the model with M states and N assets


N +3
m
denotes, where Ams
Let the state of this economy be xt = (Amstt )s∈N , At , yt , µt ∈ R+
the amount of money that investors have available to trade asset s at the beginning of period
t.49 The laws of motion for the state variables Am
t , yt , and µt are exogenous (as described
ms (x ), where the function Ψms is determined in equilibrium. Assetabove) while Amstt+1 = Ψprices in the recursive equilibrium are time-invariant functions of the state, i.e., φst = φs (xt ),
m
s
s
s∗
s∗
φ̄st = φ̄s (xt ), φmt = φ (xt ), pt = p (xt ), and εt = ε (xt ). Lemma 1 still characterizes theequilibrium post-trade portfolios of dealers and investors in the OTC market. The investor’s
48

Aside from the fact that the VAR specification is more flexible than (21), our VAR estimation also relies on
the HFIV identification scheme. In fact, notice that even for the simple specification (21), Table 4 and Table 6
show that in general, the HFIV identification strategy by itself already delivers estimates that are more precise
and more statistically significant than the E-based and H-based estimates.
49
As it will become clear, the state can be reduced to xt = (Amt , yt , µt ) in the class of equilibria we consider.

69

value functions are
 X m
s
s
s
I
W I (ams[φ (xt ) ams
t , at )s∈N , −kt ; xt =t + φ (xt ) at ] − kt + W̄ (xt )
s∈N

V

I

s
(ams
t , at )s∈N , ε; xt



=

X

s
s
I
{φm (xt ) ams
t + [εyt + φ (xt )] at } + W̄ (xt )

s∈N

X

1
αs θ [ε − εs∗ (xt )] yt I{εs∗ (xt )<ε} sams
p (xt ) t
s∈N

s
s∗
s
+ α θ [ε (xt ) − ε] yt I{ε<εs∗ (xt )} at ,
+

where
I



X

−
φm (xt ) ãms
+ φs (xt ) ãst+1
max
t+1
2N
s
(ãms
t+1 ,ãt+1 )s∈N ∈R+
s∈N
Z




I
ms
s
+ βE
V
ãt+1 , at+1 s∈N , ε; xt+1 dG(ε) xt

W̄ (xt ) ≡ T (xt ) +ssswith T (xt ) ≡ (µt − 1) φm (xt ) Amt and at+1 ≡ δãt+1 + (1 − δ) A .

s
Lemma 5 Let ãms
t+1 , ãt+1 s∈N denote the portfolios chosen by an investor in the second subperiod of period t of a recursive equilibrium. These portfolios must satisfy the following first-ordernecessary and sufficient conditions:
"
Z

#
1
xt
[ε − ε (xt+1 )] yt+1 dG(ε) sφ (xt ) ≥ βE φ (xt+1 ) + α θp (xt+1 )εs∗ (xt+1 )"#Z εs∗ (xt+1 )ssss∗
φ (xt ) ≥ βδE ε̄yt+1 + φ (xt+1 ) + α θ[ε (xt+1 ) − ε] yt+1 dG(ε) xt ,
m

m

s

εH

s∗

(80)
(81)

εL
s
where (80) holds with “=” if ãms
t+1 > 0 and (81) holds with “=” if ãt+1 > 0.

Proof. The proof is similar to the proof of Lemma 2, so it is omitted.
Definition 2 A recursive equilibrium for the economy in which only investors can hold equity
overnight is a collection of functions
 m
φ (·) , {φs (·) , ps (·) , εs∗ (·) , Ψms (·)}s∈N70

that satisfy#
1xt
φm (xt ) ≥ βE φm (xt+1 ) + αs θ[ε − εs∗ (xt+1 )] yt+1 dG(ε) sp (xt+1 )s∗
ε (xt+1 )#"Z εs∗ (xt+1 )ssss∗
φ (xt ) = βδE ε̄yt+1 + φ (xt+1 ) + α θ[ε (xt+1 ) − ε] yt+1 dG(ε) xt
"

Z εH

εL

[1 − G (εs∗ (xt ))] Amstps (xt ) =G (εs∗ (xt )) As
ps (xt ) φm (xt ) − φs (xt )εs∗ (xt ) =yt
ms
ms
At+1 = Ψ (xt )XmAms
t+1 = At+1 ,s∈Nwhere the first condition holds with equality if Ams
t+1 > 0.
We conjecture that the recursive equilibrium has the property that real prices are linear
functions of the aggregate dividend, i.e., suppose xt = (Amt , yt , µi ). Thenφs (xt ) = φsi yt(82)

φ̄s (xt ) ≡ ps (xt ) φm (xt ) = φ̄si yt(83)

φm (xt ) Am
t = Z i yt

(84)

sφm (xt ) Ams
t = Z i yt(85)εs∗ (xt ) ≡φ̄s (xt ) − φs (xt )yt

and the equilibrium conditions reduce to (23)-(26).

= φ̄si − φsi ≡ εs∗i(86)Once (23)-(26) have been solved for

s
s
s
s
m
{εs∗
i , φi , Zi , Zi }i∈M,s∈N , for any given xt = (At , yt , µi ), φ (xt ) is obtained from (82), φ̄ (xt )sm
ms from (85).
from (83) (with φ̄si = εs∗i + φi ), φ (xt ) from (84), and At

B.6

Extended sample: 1994-2007

We have focused our empirical analysis on the sample period 1994-2001. We chose to end thesample period in 2001 because our theory abstracts from credit, and credit conditions in the
U.S. financial market appear to have eased dramatically in the six years leading up to the2007 financial crisis.50 In this section, we report empirical results for the longer sample period,50

For example, the private-label securitization market grew in issuance from under $500 billion to over $2
trillion in 2006, the year before the crisis, see e.g., Gorton and Metrick (2012).

71

1994-2007. The basic results on the announcement-day effects of monetary policy surprises on
returns and turnover are reported in Table 6 (this table is the analogue of Table 4).

72

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77

78

φts

0

1

2

3

4

5

6

7

0

2

4

6

8

10

φts

0

×10 5

0

×10 5

20

10

20

α = 0.18
α = 0.8310µ = β̄
µ = µ̂

40

50

t (years)

60

70

30

40

60

70

80

80

90

90

100

100

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

1

2

3

4

5

6

7

8

9

10

0

×10 6

0

×10 7

20

10

20

α = 0.50
α = 0.7810µ ≈ β̄
µ > β̄

30

30

50

t (years)

60

40

t (years)

50

60

Real balances

40

Real balances

70

70

80

80

90

90

100

100

0

1

2

3

4

5

6

7

8

0

0.5

1

1.5

2

2.5

0

×10 -50

×10 -3Figure 1: Effects of monetary policy and OTC frictions on asset prices.

t (years)

50

Equity prices (ex dividend, daily)

30

Equity prices (ex dividend, daily)

φtmAtm
φtmAtm

12

1/φm
t
1/φtm

20

10

20

α = 0.50
α = 0.78

10µ < γ̄
µ = γ̄
µ > γ̄

30

30

50

t (years)

60

40

t (years)

50

60

Nominal price level40

Nominal price level

70

70

80

80

90

90

100

100

79

H-basedEstimate Std dev
-6.18
1.87
-.0045.0017

HFIV
Estimate Std dev
-8.57
1.69
-.0043.0009

Table 1: Empirical response of stock returns and turnover to monetary policy. All estimates are significant at the 1
percent level.

ReturnTurnoverE-basedEstimate Std dev
-3.77
1.02
-.0025.0007

80

Turnover
.11
.19
.25
.31
.37
.42
.47
.53
.58
.65
.71
.78
.86
.95
1.06
1.19
1.36
1.61
2.02
3.11

H-basedReturnTurnoverEstimate Std dev
Estimate Std dev
***
-3.40
1.12
-.0008
.0005
***
**
-3.77
1.45
-.0014.0006-3.02*
1.78-.0012*
.0006
-2.69
1.84
.0005
.0011
**
**
-5.21
2.43
-.0020.0009-4.58**
1.81-.0039***
.0013
**
***
-6.11
2.80
-.0031
.0011
**
**
-4.81
2.12
-.0024
.0012
**
**
-5.00
2.45
-.0034.0016-7.25***
2.50-.0065**
.0030
***
***
-6.20
2.33
-.0039
.0014
***
***
-5.98
2.30
-.0059
.0020
***
***
-6.62
2.40
-.0050.0019-7.71***
2.43-.0061**
.0025
***
**
-7.61
2.50
-.0068
.0027
***
**
-9.10
2.92
-.0066.0028-10.50***3.20-.0136 ***
.0044
***
-8.82
3.01
-.0001
.0002
***
***
-10.57
3.62-.0153.0050*****-12.014.10-.0172.0078

Table 2: Empirical responses of stock returns and turnover to monetary policy across NYSE liquidity portfolios. ***denotes significance at the 1 percent level, ** significance at the 5 percent level, * significance at the 10 percent level.

Portfolio
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

E-basedReturnTurnoverEstimate Std dev
Estimate Std dev
***
-2.03
.73
-.0003
.0002
**
**
-1.83
.85
-.0008.0004-1.57*.91-.0007
.0005
-1.05
.87
.0002
.0007
**
**
-2.54
1.03
-.0011.0005-2.67***
1.01-.0021**
.0009
***
**
-3.33
1.06
-.0016
.0007
**
-2.55
1.12
-.0011
.0007
**
***
-2.65
1.20
-.0023.0008-4.33***
1.19-.0027***.0009-3.88***
1.22-.0023***
.0009
***
***
-3.76
1.26
-.0030
.0009
***
*
-3.98
1.15
-.0028.0015-4.73***
1.31-.0034***
.0011
***
***
-4.69
1.30
-.0035
.0014
***
***
-5.37
1.56
-.0037.0012-6.02***
1.44-.0078***
.0019
***
-5.21
1.67
-.0001
.0002
***
***
-5.93
1.81
-.0083
.0021
***
***
-6.27
2.09
-.0088
.0030

81
.0084

.0086

.0085

yes

(25.76)(25.36)(3.08)-121.14(2.26)(2.26)

17.54-109.43

25.36

25.93

(.091)

-2.46

(III)

.0314

yes

(25.28)-100.43(2.29)

22.37(.097)

-2.37

(IV)

.0316

yes

yes

(25.68)-111.09(3.16)

13.93(.098)

-2.44

(V)

.0132

(465.31)(392.48).0132-1921.29(.042)(.041)-1696.88

.947

yes

(28.71)-415.17(7.55)

42.58(.100)

-3.37

(VII)

.947

(28.22)-403.98(5.71)

45.29(.099)

-3.36

(VI)

.0363

(389.33)-1378.21(.042)

1.00

yes(28.06)-398.96(5.72)

39.09(.110)-3.62(VIII).0364(466.21)-1418.23(.043)

1.00

yes

yes

(28.55)-410.15(7.71)

33.13(.110)

-3.63

(IX)

Table 3: Effects of monetary policy on stock returns of individual stocks. Each column reports the coefficients from aseparate pooled OLS regression based on (3). Number of observations: 117,487. Standard errors in parentheses. All
estimates are significant at the 1 percent level.

R2(Tts )2(∆it )2Dt

Ds

Tts × ∆it

Tts

-2.40
(.090)-2.52(.086)

∆it

(II)

(I)

Variable82

Basis points

15

20

25

30

0

20

40

Days

60

Policy rate

80

100

120

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

0

0

20

5

40

10

Days

60

Stock return

Days

15

Stock return

80

20

100

25

120

30

-5

-4

-3

-2

-1

0

1

-5

-4

-3

-2

-1

0

1

0

×10

0

×10

-5

-5

20

5

40

10

Days

60

Daily turnover rate

Days

15

Daily turnover rate

80

20

100

25

Figure 2: Empirical impulse responses to a 1 basis point increase in the policy rate. Solid lines are point estimates. Brokenlines are 95 percent confidence intervals.

0.4

0.6

0.8

1

1.2

1.4

1.6

Days

-9

10

0.8

5

-8

0

-7

0.9

0.85

-5

-4

-3

-2

-1

0

1

-6

Basis points

0.95

1

1.05

1.1

1.15

1.2

1.25

Policy rate

Basis points
Basis points

1.3

Times per day
Times per day

120

30

83
0
-7

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

-6

-4

-3

-2

-1

0

Basis Points (per 1 bp increase in policy rate)-51Model Distribution
Data Estimate
5-95 percentiles

Estimates of Response of Stock Return to Policy Rate

2

Figure 3: Estimates of announcement-day response of stock return to policy rate.

Frequency

84
-10

-8

-6

-4

-2

0

0

2

4

Model
Data
6

8

12

Portfolio

10

14

16

18

20

Response of Stock Return to Policy Rate: Model and Data

Figure 4: Cross-sectional announcement-day responses of stock returns to policy rate.

Basis Points (per 1 bp increase in policy rate)

85
-6

-5

-4

-3

-2

-1

0

1

0

2

4

Model
Data
6

8

12

Portfolio

10

14

16

18

20

Response of Turnover Rate to Policy Rate: Model and Data

Figure 5: Cross-sectional announcement-day normalized responses of turnover rates to policy rate.

Normalized Response(as proportion of average response)

86

Basis points

0

0

20

Data
Model

5

Data
Model

40

10

15

Days

60

Policy rate

Days

80

20

100

25

120

30

-10

-8

-6

-4

-2

0

2

4

-10

-8

-6

-4

-2

0

2

4

0

0

20

5

40

10

Days

60

Stock return

Days

15

Stock return

80

20

100

25

120

30

-5

-4

-3

-2

-1

0

1

-5

-4

-3

-2

-1

0

1

-5

0

×10 -5

0

×10

20

5

40

10

Days

60

Daily turnover rate

Days

15

Daily turnover rate

80

20

100

25

Figure 6: Impulse responses to a 1 basis point increase in the policy rate. Solid lines are point estimates. Broken lines
are 95 percent confidence intervals.

0.4

0.6

0.8

1

1.2

1.4

1.6

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

Basis points

Policy rate

Basis points
Basis points

1.3

Times per day
Times per day

120

30

87

1.1

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

0

5

10

True model
VAR with 10 lags
VAR with 5 lags

Days

15

Policy rate

20

25

30

Basis points
-7

-6

-5

-4

-3

-2

-1

0

1

2

3

0

5

10

Days

15

Stock return

20

25

30

-1.9

-1.8

-1.7

-1.6

-1.5

-1.4

-1.3

-1.2

-1.1

-1

-0.9

0

×10 -6

5

10

Days

15

Daily turnover rate

20

25

Figure 7: Impulse responses to a 1 basis point increase in the policy rate: true theoretical responses versus responses
estimated from model generated data. Solid lines are point estimates. Broken lines are 95 percent confidence intervals.

Basis points

1.15

Times per day

30

88

Turnover
.11
.19
.25
.31
.37
.42
.47
.53
.58
.65
.71
.78
.86
.95
1.06
1.19
1.36
1.61
2.02
3.11
.94

E-basedReturnTurnover***-2.03-.0003**-1.83-.0008**-1.57*-.0007-1.05.0002-2.54**-.0011**-2.67***-.0021**-3.33***-.0016****-2.55-.0011**-2.65-.0023***-4.33***-.0027******-3.88-.0023***-3.76***-.0030***-3.98***-.0028*-4.73***-.0034******-4.69-.0035***-5.37***-.0037***-6.02***-.0078******-5.21-.0001***-5.93-.0083***-6.27***-.0088***-3.77***-.0025***

H-basedReturnTurnover***-3.40-.0008***-3.77-.0014**-3.02*-.0012*
-2.69
.0005
**
-5.21
-.0020 **-4.58**-.0039***-6.11**-.0031*****-4.81-.0024**-5.00**-.0034**-7.25***-.0065*****-6.20-.0039***-5.98***-.0059***-6.62***-.0050 ***-7.71***-.0061*****-7.61-.0068**-9.10***-.0066**-10.50***-.0136 ***
***
-8.82
-.0001***-10.57-.0153***-12.01***-.0172**-6.18***-.0045**

HFIV
ReturnTurnover***-5.80-.0008**-5.95***-.0011**-4.54***-.0014*****-4.29-.0031***-5.89***-.0017***-3.43**-.0016-6.38***-.0022******-6.14-.0024***-8.02***-.0025***-8.19***-.0029******-6.63-.0036***-8.84***-.0048***-11.15***-.0036***-9.13***-.00341****-9.35-.0052***-12.66***-.0047***-12.15***-.0078******-13.90-.0098***-13.37***-.0098***-15.70***-.0125***-8.57***-.0043***

Table 4: Empirical responses of stock returns and turnover to monetary policy across NYSE liquidity portfolios. ***denotes significance at the 1 percent level, ** significance at the 5 percent level, * significance at the 10 percent level.

Portfolio
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
NYSE

89

TurnoverEstimate Std dev-.0033.0021***-.0052.0019-.0052***.0013***-.0048.0015-.0036**.0014-.0040***.0013-.0032**.0014-.0037***.0014-.0028*.0015-.0028*.0013-.0053***.0016-.0006.0015-.0034**.0013*-.0025.0014-.0047***.0014-.0032*.0017-.0044***.0015-.0055***.0017-.0080***.0019***-.0082.0023

Table 5: Empirical responses of stock returns and turnover to monetary policy across portfolios sorted on return sensitivity
to aggregate turnover (E-based estimates). *** denotes significance at the 1 percent level, ** significance at the 5 percent
level, * significance at the 10 percent level.

Portfolio
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

ReturnEstimate Std dev
-.85
1.89
-2.11
1.41
-1.22
1.23
***
-3.38
1.19
-2.69**
1.20-2.68**
1.10-2.64***
.99
**
-2.39
1.06
-3.59***
1.02-3.17***
1.03-3.92***
1.09-4.71***
1.05-4.41***
1.17
***
-6.12
1.28
-6.53***
1.43-6.63***
1.50-7.25***
1.57-6.66***
1.78-10.16***2.42***-13.17
3.02

90
-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0

2

4

6

8

10

12

Portfolio

14

16

18

20

Exposure to market turnover
Exposure to market risk
Exposure to SMB factor
Exposure of HML factor

Exposure to risk factors

Figure 8: Betas for the 20 portfolios sorted on sensitivity of return to aggregate turnover.

β is

91
-20

-15

-10

-5

0

5

0

2

4

6

8

12

Portfolio

10

Actual response
Response according to CAPM
14

16

Response of stock return to policy rate

18

20

Figure 9: E-based estimates of responses of announcement-day stock returns to a 1 basis point surprise increase in thepolicy rate: CAPM vs. response based on portfolio analysis (for portfolios sorted on sensitivity of return to aggregate
turnover).

Basis points (per 1 bp increase in policy rate)

92

Percentage points

0

0

×10

0

-5

5

5

10

10

15

Days

15

Daily turnover rate

Days

20

20

25

25

30

30

-3

-2.5

-2

-1.5

-1

-0.5

0

-7

-6

-5

-4

-3

-2

-1

0

1

2

0

×10

0

-3

5

5

10

10

Days

15

Daily turnover rate

Days

15

Stock return

20

20

25

25

30

30

-25

-20

-15

-10

-5

0

5

-0.035-0.03-0.025-0.02-0.015-0.01-0.0050

0.005

Percentage points
0

0

5

5

10

10

Days

15

Daily turnover rate

Days

15

Stock return

20

20

25

25

Figure 10: Impulse responses to a 1 percentage point increase in the policy rate for selected liquidity-based portfolios.
The left panels correspond to portfolio 1, the middle panels to portfolio 10, and the right panels to portfolio 20. Solidlines are point estimates. Broken lines are 95 percent confidence intervals.

-20

-15

-10

-5

0

5

-2.5

-2

-1.5

-1

-0.5

Times per day

Stock return

Percentage points
Times per day

0.5

Times per day

30

30

93

Turnover
.18
.35
.45
.54
.62
.69
.76
.84
.91
.97
1.06
1.15
1.26
1.37
1.49
1.66
1.85
2.13
2.57
3.63
1.23

H-basedReturnTurnover***-9.75-.0015*****-12.13-.0017***-9.46-.0015***-11.40-.0012***-14.28-.0025-12.79***-.0009***-15.21-.0014***-21.28-.0019-14.78***-.0009***-16.57-.0040***-14.48-.0025***-17.27-.0047-18.10***-.0038***-18.36-.0026***-18.97-.0080*-22.26***-.0050-21.64***-.0042***-23.26-.0076***-23.74-.0115-27.55***-.0061***-16.79-.0037

HFIV
ReturnTurnover***-6.25-.0009******-6.84-.0012**-5.69***-.0022**-5.49***-.0029***-7.23***-.0019-5.16***-.0018***-7.33-.0029*-7.244***-.0026*****-8.79-.0030*-9.08***-.0036*-8.02***-.0034***-9.46-.0049******-11.40-.0042***-9.84***-.0049**-10.00***-.0059***-13.25***-.0069******-13.03-.0093***-14.31***-.0096***-13.85***-.0123******-16.40-.0189***-9.43***-.0053**

Table 6: Empirical responses of stock returns and turnover to monetary policy across NYSE liquidity portfolios (19942007 sample). *** denotes significance at the 1 percent level, ** significance at the 5 percent level, * significance at the 10
percent level.

Portfolio
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
NYSE

E-basedReturnTurnover***-3.85-.0004-4.26***-.0006***-3.60-.0008***-3.22-.0002***-4.83-.0010-3.65***-.0009***-4.88-.0008***-4.34-.0011-5.10***-.0013***-5.60-.0016***-5.12-.0016***-5.73-.0022****-6.87-.0020***-5.95-.0026***-6.48-.0039**-7.60***-.0035****-7.46-.0035***-8.35***-.0041*-8.33***-.0061*****-9.28-.0062***-5.73-.0022*