HIAS-E-140

Deciphering the Neo-Fisherian Effect

Hafedh Bouakez (a), Takashi Kano (b)HEC Montr´eal
Hitotsubashi University

(a)
(b)

June 2024

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
2-1, Naka, Kunitachi, Tokyo 186-8601, Japan
tel:+81 42 580 8668 http://hias.hit-u.ac.jp/
HIAS discussion papers can be downloaded without charge from:https://hdl.handle.net/10086/27202
https://ideas.repec.org/s/hit/hiasdp.html

Deciphering the Neo-Fisherian Effect∗
Hafedh Bouakez†

Takashi Kano‡

June 2024

Abstract
The neo-Fisherian effect typically refers to the short-run increase in inflation associated witha permanent increase in the nominal interest rate. This positive comovement between the twovariables is commonly viewed — and empirically identified — as being conditional on permanentmonetary shocks, which are often interpreted as permanent shifts in the inflation target. Sucha view, however, implies that inflation and the nominal interest rate share a common stochastic
trend, a property that is hardly supported by the data, especially during episodes of stableinflation. Moreover, in countries that have adopted formal inflation targeting, changes in the
inflation target occur very infrequently, if at all, calling into question the interpretation of
inflation target shocks identified within standard time-series models based on quarterly data.
In this paper, we propose a novel empirical strategy to detect the neo-Fisherian effect, which
we apply to U.S. data. Our procedure relaxes the commonly used identifying restriction that
inflation and the nominal interest rate are cointegrated, and, more importantly, is agnostic about
the nature of the shock that gives rise to a neo-Fisherian effect. We find that the identified shock
has no permanent effect on the nominal interest rate or inflation, but moves them in the same
direction for a number of quarters. It also accounts for the bulk of their variability at any given
forecasting horizon, while explaining a non-negligible fraction of output fluctuations at businesscycle frequencies. Using Bayesian techniques, we show that the data favors the interpretationof the identified shock as a liquidity preference shock rather than an inflation target shock.
JEL classification: E12, E23, E31, E43, E52
Key words: Identification, Inflation, Liquidity preference, neo-Fisherian effect.

∗

We have benefited from useful discussions with Olivier Coibion, Kyle Jurado, and Nicolas Vincent. Financial
support from the HEC Montréal Foundation, the Japan Society for the Promotion of Science (grant # 23H00048),
and the Hitotsubashi Institute for Advanced Study is gratefully acknowledged.
†
Department of Applied Economics, HEC Montréal, 3000 chemin de la Côte-Sainte-Catherine, Montréal, Québec,
Canada H3T 2A7. E-mail: hafedh.bouakez@hec.ca.
‡
Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo, 186-8601, Japan; E-mail: tkano@econ.hit-u.ac.jp.

1

Introduction

The conventional view in monetary economics is that a rise in the nominal interest rates lowers
inflation and the output gap in the short run, whereas an interest-rate cut does the opposite. These
outcomes are consistent with the effects of transient monetary policy shocks in the standard New
Keynesian model. As forcefully shown by Cochrane (2016), however, this very same model implies
that a permanent increase in the nominal interest rate causes inflation to rise not only in the longrun (as required by the Fisher equation), but also in the short run — a prediction that turns out tobe remarkably robust across various modelling assumptions. This positive short-run comovement
between inflation and the nominal interest rate has come to be known as the neo-Fisherian effect.
While Cochrane (2016) considers the case of a pegged nominal interest rate, subsequent work by
Uribe (2022) shows that the neo-Fisherian effect continues to be present when monetary policy
follows an interest-rate rule that is subject to permanent shocks.1 Like much of the preceding
literature (e.g., Ireland, 2007; Cogley et al., 2010; Aruoba & Schorfheide, 2011), Uribe (2022)
interprets these shocks as permanent exogenous shifts in the inflation target.
The presence of permanent monetary shocks implies that inflation and the nominal interest rateshare a common stochastic trend, i.e., are cointegrated. This assumption underlies the empiricalstrategies proposed by Azevedo et al. (2022) and Uribe (2022) to test for the existence of a neoFisherian effect in the data. Alternatively, Lukmanova & Rabitsch (2023) attempt to directly
identify inflation target shocks by including various low-frequency measures of inflation in theirinformation set. While the assumption of cointegration between inflation and the nominal interest
rate may be difficult to reject in some countries during specific periods, inflation seems to havebecome stationary in several countries that have adopted a credible monetary policy regime. Table
1 illustrates this point by reporting unit-root test results for U.S. inflation for the period 1953Q3–
2023Q4 and the two sub-periods 1953Q3–1992Q1 and 1992Q2–2023Q4. Inflation has generally beenhigh and volatile in the U.S. before 1992 but low and stable in the post-1992 era.2 Overall, theresults provide decisive evidence that inflation contained a unit root before 1992 but has become
stationary after that date. Moreover, in the absence of announcements by the Federal Reserve
regarding changes in its explicit inflation target of 2 percent, adopted in January 2012, the very
notion of inflation target shocks after that date seems at least dubious (more on this below).
This paper offers novel insights into the neo-Fisherian effect, contributing to the extant literature
along two lines. First, we propose a new empirical strategy to detect and measure this effect. Our
1
Garı́n et al. (2018), Uribe (2022), and Lukmanova & Rabitsch (2023) show that monetary shocks can still giverise to a neo-Fisherian effect if they are highly persistent (though not permanent).
2
The selected break date of 1992Q1 was identified by Levin & Piger (2004) and Benati (2023) as marking a
structural break in the mean of U.S. inflation. However, our results are robust to alternative dates around theselected one.

1

Table 1: Unit-root tests for U.S. inflation.
Lag/Bandwidth
Length Criterion

Full Sample
1954Q3–2023Q4Sub-Samples
1954Q3–
1992Q2–
1992Q12023Q4

ADF

BIC
MBIC

−3.234∗∗−2.896∗∗−2.346−1.815−3.395∗∗−1.207

PP

Newey–West
Andrews−4.502∗∗∗−4.328∗∗∗−3.297∗∗−3.136∗∗−4.902∗∗∗−4.579∗∗∗

ERS

BIC
MBIC

−2.197∗∗−1.974∗∗−1.521−1.343−3.395∗∗∗−1.783∗

NP

BIC
MBIC

−2.180∗∗−1.978∗−1.513−1.360−3.093∗∗∗−1.936∗

Newey–West
Andrews

0.395∗
0.245

0.465∗∗
0.279

0.205
0.193

Test
H0 : Inflation has a Unit Root

H0 : Inflation is Stationary
KPSS

Notes: ADF refers the Augmented Dickey-Fuller test (Dickey & Fuller, 1979), PP refers to the Phillips-Perron Zt
test (Phillips & Perron, 1988), ERS refers to the Elliott-Rothenberg-Stock DF-GLS test (Elliott et al., 1996), NP
refers to the Ng-Perron M Zt test (Ng & Perron, 1995), and KPSS refers to the Kwiatkowski-Phillips-Schmidt-Shintest (Kwiatkowski et al., 1992). BIC (Schwarz information criterion) and MBIC (modified Schwarz information
criterion) are used to select the optimal number of lags of the dependent variable in ADF, ERS, and NP, where
the maximum number of lags is set according to the Schwert criterion. Newey-West (Newey & West, 1994)
and Andrews (Andrews, 1991) rules are used to determine the optimal bandwidth of the Bartlett kernel used toestimate robust standard errors in PP and KPSS. *** denotes rejection at the 1 percent level, ** denotes rejection
at the 5 percent level, and * denotes rejection at the 10 percent level.

approach does not require inflation and the nominal interest rate to share a common stochastic
trend, though it does not rule out this possibility. More importantly, it is agnostic about the natureof the shock that gives rise to a neo-Fisherian effect. Second, we provide theoretical arguments
and empirical evidence suggesting that the positive short-run comovement between inflation andthe nominal interest rate observed in the data is more likely to be driven by liquidity preference
shocks than by inflation target shocks.
Our empirical strategy consists in estimating a vector auto-regression (VAR) using data on
the nominal interest rate, inflation, and output, and identifying two orthogonal disturbances: astandard monetary policy shock and one that we label ‘neo-Fisherian shock’ without taking astand on its structural interpretation. To identify the monetary policy shock, we select the linear
combination of reduced-form residuals that has a transitory effect on the nominal interest rate
and that satisfies the restriction that a positive realization does not raise inflation or output on
impact. We then identify the neo-Fisherian shock using a Max Share approach, by selecting thelinear combination of reduced-form residuals that is orthogonal to the monetary policy shock andthat explains the largest fraction of the forecast-error variance of the nominal interest rate at along but finite horizon. The latter criterion is consistent with the theoretical prediction that theneo-Fisherian effect is associated with permanent (or highly persistent) changes in the nominal

2

interest rate. Monte Carlo simulations based on artificial data generated from a New Keynesianmodel show that our identification strategy is remarkably successful in recovering the true shocks
both in large and small samples, regardless of whether the neo-Fisherian shock is permanent ornot.
Estimation results based on U.S. data indicate that the identified neo-Fisherian shock has no
permanent effect on the nominal interest rate or inflation, but moves them in the same direction
for a number of quarters. In other words, the shock generates a neo-Fisherian effect, thus justifying
its epithet. Positive realizations of the shock are also found to be expansionary in the short run,raising output for at least a year. Variance-decomposition results show that the neo-Fisherian shockaccounts for the bulk of fluctuations in the nominal interest rate and inflation at essentially any
frequency, while explaining only about 10 percent of output variability at business-cycle frequencies.
Nonetheless, this shock appears to have played a more significant role during the most recent
downturns, accounting for more than one-third of the decline in output growth during the GreatRecession and roughly 20 percent during the COVID-19 pandemic.
The second part of the paper delves into the interpretation of the identified shock. As statedabove, the common view in the empirical literature on neo-Fisherism is that the shock leading
to a short-run comovement of the nominal interest rate and inflation captures exogenous shifts in
the inflation target. Given that announcements about changing the inflation target are extremely
infrequent events that happen once in several decades (particularly in advanced economies), it
seems implausible that inflation target shocks can be extracted within VARs (or alternative timeseries models) estimated using quarterly data. To elucidate the nature of the neo-Fisherian shock
identified in the data, we start by showing that, in the context of a parsimonious New Keynesian
economy, the effects of an (positive) inflation target shock on the nominal interest rate, inflation,and output are akin to those of a (negative) liquidity preference shock, which affects households’
desire to hold safe and liquid assets. In fact, under certain parameter restrictions, the two shocks
are observationally equivalent for the dynamics of inflation and output. We then use Bayesian
methods to determine whether a model with liquidity preference shocks provides a better overall fitof U.S. data than does an otherwise identical model with inflation target shocks. We find that theformer has a much larger marginal likelihood, thus favoring the interpretation of the neo-Fisherian
shock as a liquidity preference shock.
Literature review

This paper contributes to the literature on neo-Fisherism. In addition to theaforementioned papers by Cochrane (2016), Garı́n et al. (2018), Uribe (2022), and Lukmanova &Rabitsch (2023), studies that have examined the conditions under which the neo-Fisherian effectarises in the class of New Keynesian models include those by Garcı́a-Schmidt & Woodford (2019),
Amano et al. (2016), and Schmitt-Grohé & Uribe (2022). Garcı́a-Schmidt & Woodford (2019) show3

that the neo-Fisherian effect ceases to exist when rational expectations are replaced by a learning
mechanism. Amano et al. (2016) show that the neo-Fisherian effect can emerge if monetary policy
deviates from the Taylor principle by committing to not responding too aggressively to off-target
inflation. This in turn requires the monetary authority to use inflation to manage the real valueof public debt. Finally, Schmitt-Grohé & Uribe (2022) extend the analysis of Uribe (2022) tothe context of an open economy. While these papers condition the neo-Fisherian effect on inflation
target shocks, we argue that, empirically, this effect is more likely to be driven by liquidity preference
shocks.
Our paper is also related to the vast literature on inflation dynamics. One strand of thisliterature emphasizes a highly persistent stochastic component with a positive long-run mean as
the fundamental characteristic of post-war U.S. inflation movements. Cogley & Sbordone (2008),
Coibion & Gorodnichenko (2011), and Ascari & Sbordone (2014) show that this positive long-run
mean of the inflation rate — often referred to as trend inflation — alters the shape of the loglinearized New Keynesian Phillips Curve, resulting in persistent and smooth deviations of inflationfrom its mean. Another strand attempts to account for the high persistence of U.S. inflationby allowing the long-run unconditional mean to vary over time. Specifically, Ireland (2007) andCogley et al. (2010) describe the Beveridge-Nelson permanent component of the inflation rate asa stochastic inflation target set by the central bank. Aruoba & Schorfheide (2011) interpret the
low-frequency movements of U.S. inflation in the 1970s and 1980s as evidence of a drifting target,
which they model as a stochastic process. Kano (2023) explores the role of persistent inflation target
shocks for inflation and exchange-rate dynamics in the context of a two-country New Keynesianmodel.
Our study also connects with the growing literature on the importance of liquidity preference
shocks for aggregate fluctuations. In Christiano et al. (2003), these shocks induce households toshift from deposits to currency. In Krishnamurthy & Vissing-Jorgensen (2012), they represent
stochastic changes in households’ preference for safe and liquid assets. Fisher (2015) shows that
this interpretation provides a micro-founded justification for the ad hoc risk premium shock introduced by Smets & Wouters (2007). Since positive realizations of liquidity preference shocks raisehouseholds’ desire to save, but only in risk-free bonds, they lower the natural rate of interest andlead a simultaneous fall in consumption, investment, and inflation. For this reason, these shocks
have sometimes been invoked to explain deep recessions, potentially characterized by liquidity traps(e.g., Bouakez et al., 2020; Cacciatore et al., 2021). Interestingly, based on an estimated modelof the U.S. economy using data from 1993Q1 to 2008Q3, Campbell et al. (2016) find that liquidity preference shocks account for the largest fraction of the variance of the federal funds rate atbusiness-cycle frequencies, thus comforting our interpretation of the source of the neo-Fisherian
effect found in the data.
4

Structure of the paper The rest of the paper is structured as follows. Section 2 describes
our empirical strategy to detect the neo-Fisherian effect, and performs Monte Carlo simulations
to gauge the reliability of our identification procedure. Section 3 describes the data and reports
our estimation results in terms of impulse responses, forecast-error variance decomposition, andhistorical decomposition. Section 4 investigates the economic interpretation of the shock underlying
the neo-Fisherian effect. It first discusses the near equivalence between inflation target and liquidity
preference shocks. Then, it uses Bayesian techniques to show that a model with liquidity preference
shocks provides a better overall fir of U.S. data than does a model with inflation target shocks.
Section 5 concludes.

2

Empirical Strategy

2.1

Identification

Let zt = [it , πt , yt ]0 be a vector of observables of length T , where it is the nominal interest rate, πt
is inflation, and yt is (the log of) aggregate output, and assume that zt has the following movingaverage representation (ignoring constant terms):
zt = B(L)ut ,where ut is a 3 × 1 vector of statistical innovations, whose variance-covariance matrix is denotedby Σ. Let t = [m,t , n,t , y,t ]0 be a 3 × 1 vector of structural innovations, where m,t is a monetary
policy shock, n,t is a shock that has a persistent/permanent effect on the nominal interest rate
— henceforth labeled neo-Fisherian shock — and y,t is a shock that need not be identified. The
variance-covariance matrix of t is I3 . If a linear mapping between the statistical innovations, ut ,and the structural shocks, t , exists, thenut = At ,where the impact matrix, A, must be such that AA0 = Σ.
Let Ã denote the Cholesky decomposition of Σ. Any impact matrix A = ÃD, where D is anorthonormal matrix, also satisfies the requirement AA0 = Σ. We identify the monetary policy
shock, m,t , by selecting the orthonormal matrix D that satisfies the following two requirements.
First, the shock does not have a permanent effect on the nominal interest rate. We implement thisrestriction by minimizing the sum of the squared impulse response functions of the nominal interest
rate beyond a given horizon, h. This criterion ensures that the dynamic effects of a monetary policy
shock on the nominal interest rate die out after h periods. Second, a positive realization of the shock
does not raise inflation or output on impact. These sign restrictions reflect the widely accepted5view about the short-run effects of monetary policy shocks.3
Let γj denote the jth column of D. Since the impulse vector to m,t is Ãγ1 (the first columnof ÃD), we only need to characterize γ1 . Denote by ri,γ1 (k), rπ,γ1 (k), and ry,γ1 (k) the impulse
responses of the nominal interest rate, inflation, and output, respectively, to the impulse vectorÃγ1 at horizon k. Our strategy to identify the monetary policy shock amounts to selecting thevector γ1 that solves the following minimization problem:
min
{γ1 }

∞
X

2
ri,γ
(k)
1

k=h

s.t.
0

γ1 γ1 = 1,
ri,γ1 (1) ≥ 0,rπ,γ1 (1) ≤ 0,ry,γ1 (1) ≤ 0.
The first constraint ensures that γ1 is a column vector of an orthonormal matrix. The remaining
constraints implement the sign restrictions on the responses of the nominal interest rate, inflation,and output.
Once m,t is identified, we identify n,t using a Max Share approach, by selecting the linear
combination of reduced-form residuals that is orthogonal to m,t and that explains the largest
fraction of the forecast-error variance of it at a long but finite horizon, H.4 This criterion reflects
the theoretical prediction that the neo-Fisherian effect is associated with permanent (or highly
persistent) changes in the nominal interest rate.
The k-step-ahead forecast error of vector zt is
zt+k − Et zt+k =k−1
X

Bl ÃDt+k−l ,l=0with B0 = I 3 . Letting Ωi,j (h) denote the share of the forecast-error variance of variable i attributable to structural shock j at horizon k (k = 1, 2...), we have0

Ωi,j (k) ≡

ei

P

0
0
0
k−1
l=0 Bl ÃDej ej D ÃBl
0

ei

P

0
k−1
l=0 Bl ΣBl



ei

3



ei

Pk−1

=

0

0

l=0 Bi,l Ãγj γj ÃBi,l,Pk−1
0
l=0 Bi,l ΣBi,lUribe (2022) adopts a similar sign-restriction approach to identify the (transitory) monetary policy shock.
The Max Share approach was first introduced by Francis et al. (2014) to identify technology shocks, and has
been subsequently used by several authors to identify TFP new shocks (e.g., Kurmann & Sims, 2021; Nam & Wang,
2019; Bouakez & Kemoe, 2023, etc.).
46

where0Bi,l = ei Bl ,γj = Dej ,and ei is a selection vector with 1 in the ith position and zero elsewhere.
Our approach to identify the neo-Fisherian shock therefore consists in selecting the vector γ2that solves

PH−1

max Ω1,2 (H) ≡
{γ2 }

0

0

τ =0 B1,τ Ãγ2 γ2 ÃB1,τ
PH−10
τ =0 B1,τ ΣB1,τ

s.t.
0

γ2 γ1 = 0,0γ2 γ2 = 1.
The first constraint ensures that n,t is orthogonal to m,t , and the second constraint ensures that
γ2 is a column vector of an orthonormal matrix. In practice, we choose the truncation horizons
h = 20 quarters and H = 80 quarters.

2.2

Monte Carlo simulations

Before applying our identification strategy to actual data, we evaluate its reliability both in large
and small samples using Monte Carlo simulations.
2.2.1

Data generating process

Consider a simple New Keynesian economy without capital in which prices are set à la Calvo.
Non-optimizing firms index their prices to past inflation. The economy features a monetary policy shock (mt ), an inflation target shock (τt ), and a technology shock (at ). Log-linearizing themodel’s equilibrium conditions around the deterministic steady state yields the following equations(a detailed description of the model is provided in the Appendix):

Etyt+1 − (it − Etπt+1 − ln β −1),πt = (1 + β)−1 [πt−1 + β Et πt+1 + λ (yt − at )],
yt =

(1)
(2)

it = (1 − %) ln β −1 + %it−1 + (1 − %) [φπ πt + φy (yt − ytf )] − [(1 − %) φπ − 1] τt − %τt−1 + mt ,(3)where yt is output, it is the nominal interest rate, πt is the inflation rate, and ytf = at is theflexible-price (or natural) level of output. The variables yt and ytf are expressed as percentage
deviations from their steady-state values, while it , πt , mt , τt , and at are expressed in levels. The
model parameters are defined as follows: 0 < β < 1 is the discount factor, ϕ > 0 is the inverse> 0, with 0 < θ < 1 being the Calvoof the Frisch elasticity of labor supply, λ ≡ (1−θ)(1−βθ)(1+ϕ)θ
probability of not changing prices, 0 ≤ % < 1 is the interest-smoothing parameter, and φπ > 1 and7

φy ≥ 0 are the coefficients attached to, respectively, inflation and the output gap in the interest-rate
rule.
The forcing variables follow the autoregressive processes given by
mt = ρm mt−1 + m,t ,m,t ∼ N (0, σm ),τt = ρτ τt−1 + τ,t ,τ,t ∼ N (0, στ ),at = ρa at−1 + a,t ,a,t ∼ N (0, σa ),where 0 ≤ ρm , ρa < 1, and 0 ≤ ρτ ≤ 1 to allow for the possibility of permanent inflation target
shocks.5
To generate artificial data from the model, we assign the following standard values to its parameters: β = 0.99, ϕ = 1, θ = 0.75, % = 0.8, φπ = 1.5, and φy = 0.125. We also set the standard
deviations of the three shocks, σm , στ , and σa , to 0.01. Since TFP shocks are usually found tobe highly persistent, we set ρa = 0.95. Finally, to allow for a sharp distinction between transitoryand persistent/permanent shifts in monetary policy, we set ρm = 0 and ρτ ∈ {0.95, 1}. Considering both highly persistent and permanent inflation target shocks will help determine whether ouridentification procedure is robust regardless of whether or not inflation and the nominal interest
rate share a common stochastic trend.
The theoretical impulse responses are depicted in Figure 1. The top panels of the figure showthat a positive realization of m,t raises the nominal interest rate while lowering inflation and output
in the short run, and that these effects dissipate after about 10 quarters. These patterns rationalize
the identifying restrictions imposed to pin down the monetary policy shock, discussed in Section
2. Notice that, despite its transitory effects, the shock still explains roughly 10 percent of theforecast-error variance of the nominal interest rate at long horizons, as reported in Table 2.
The dynamic responses to a positive inflation target shock are shown in the middle panelsof Figure 1, for ρτ = 0.95 and ρτ = 1. In both cases, the shock gives rise to a neo-Fisherian
effect: positive short-run comovement of inflation and the nominal interest rate. When the shockis persistent, the two variables remain higher than average for a prolonged period of time before
eventually returning to their initial levels. When the shock is permanent, on the other hand, theyboth converge to permanently higher levels. Table 2 shows that the shock’s contribution to theforecast-error variance of the nominal interest rate at the 80-quarter horizon amounts to 90 percent
when ρτ = 0.95 and reaches 99 percent when ρτ = 1. The observation that the inflation target
shock is the dominant force driving the conditional variance of the nominal interest rate at longhorizons underlies our Max Share identification strategy, discussed in Section 2. Interestingly, the5When ρτ = 1, inflation and the nominal interest rate do not have steady-state values, as their unconditional
means do not exist. In this case, we solve the model by assuming that the initial values of the inflation target is equal
to 0.

8

inflation target shock is expansionary, as output rises for about 6 quarters after the shock.
Monetary Policy Shock
Nominal Interest Rate

Inflation

0.5

0

0

0

-0.5

10

15

20

25

-1
-2

-1
5

Output1

Percent

Percentage Points

Percentage Points

1

30

-3
5

10

15

20

25

30

5

10

15

20

25

30

20

25

30

20

25

30

Inflation Target Shock
Inflation

1
Persistent Shock
Permanent Shock

0.5

0

1

0.5

0
5

10

15

20

25

Output130

0.5

Percent

Percentage Points

Percentage Points

Nominal Interest Rate

0

-0.5
5

10

15

20

25

30

5

10

15

Technology Shock
Nominal Interest Rate

Inflation

0

-0.1

-0.2

-0.3

-0.1

-0.2

0.5

-0.3
5

10

15

20

25

30

Output1

Percent

Percentage Points

Percentage Points

0

0
5

10

15

20

25

30

5

10

15

Figure 1: Theoretical impulse responses.
Notes: The figure shows the theoretical impulse responses to monetary policy, inflation target, and technology shocks. The
responses of the nominal interest rate and inflation are expressed in percentage-point deviations from their pre-shock levels.
The response of output is expressed as percentage deviation from its pre-shock level. In all cases, the size of the shock is 1
percent. For the inflation target shock, the solid lines represent the case of a persistent shock (ρτ = 0.95), while the dashedlines represent the case of a permanent shock (ρτ = 1).

Finally, the bottom panels of Figure 1 show that a positive technology shock raises output while
lowering inflation and the nominal interest rate. This negative comovement between output andinflation violates the sign restrictions we impose to identify the monetary policy shock, thus ensuringthat the latter is not confounded by changes in technology. On the other hand, the technology
shock leads to positive comovement between the nominal interest rate and inflation. However, theresponse of the nominal interest rate is smaller (in absolute value) at any given horizon than thattriggered by an inflation target shock, both when ρτ = 0.95 and when ρτ = 1. The share of theforecast-error variance of the nominal interest rate accounted for by the technology shock does
not exceed 1 percent, even at long horizons. This observation greatly alleviates the concern thattechnological disturbances confound the identification of the neo-Fisherian shock.
9

Table 2: Theoretical (model-based) variance decomposition.

Nominal Interest Rateρτ = 0.95
ρτ = 1

0.468
0.447

Horizonk = 8 k = 16 k = 32
Monetary Policy Shock0.259 0.162
0.110
0.092
0.198 0.098
0.047
0.023

Inflationρτ = 0.95
ρτ = 1

0.760
0.656

0.557
0.326

Outputρτ = 0.95
ρτ = 1

0.920
0.896

0.734
0.710

Nominal Interest Rateρτ = 0.95
ρτ = 1

k=1

k=4

0.175
0.082

0.088
0.009

0.147
0.042

0.142
0.017
0.485
0.476

0.532
0.553

0.624
0.538
0.495
0.607
0.525
0.484
Inflation Target Shock0.735 0.829
0.880
0.898
0.797 0.897
0.949
0.974

ρτ = 0.95
ρτ = 1

0.229
0.335

0.433
0.667

0.744
0.836

0.847
0.957

0.853
0.983

Outputρτ = 0.95
ρτ = 1

0.020
0.042

0.068
0.099

0.047
0.067

Nominal Interest Rateρτ = 0.95
ρτ = 1

0.000
0.001

0.006
0.004

0.061
0.052
0.048
0.086
0.074
0.068
Technology Shock
0.009
0.010
0.010
0.005
0.004
0.003

Inflationρτ = 0.95
ρτ = 1

0.011
0.009

0.010
0.007

0.007
0.005

0.006
0.003

0.006
0.001

0.081
0.001

Outputρτ = 0.95
ρτ = 1

0.060
0.062

0.198
0.191

0.315
0.307

0.410
0.401

0.457
0.448

0.467
0.458

Inflation

0.249
0.159

k = 80

0.819
0.915

0.901
0.990

0.011
0.001

Notes: The table reports the theoretical fractions of the k-step ahead forecast-error variance of eachvariable attributed to each of the structural shocks. The fractions may not add up to 1 due to rounding.

2.2.2

Results

Using model (1)–(3) as a data-generating process, we simulate 2000 sequences {it , πt , yt }Tt=1 oflength T. We consider both a large sample (T = 10, 000) and a small sample (T = 275). The
latter roughly corresponds to the number of observations used in our empirical analysis, discussed
in the next section. In each case, the resulting sample is obtained after discarding the first 100
observations to ensure that the results do not depend on initial conditions. As stated above, theartificial series are generated both under the assumptions that the inflation target shock is persistent(ρτ = 0.95) and permanent (ρτ = 1).
For each Monte Carlo replication, we estimate a VAR(3) using the synthetic series it , πt , andyt , and identify the monetary policy and inflation target shocks using the identification proceduredescribed in Section 2. The results are shown in Figures 2 and 3 for T = 10, 000 and in Figures
4 and 5 for T = 275. In each figure, the dashed lines represent the true (model-based) responses,
the solid lines represent the estimated median responses across the 2000 replications, the dark and10

light shaded areas represent, respectively, the 84 and 95 percent confidence bands. Table 3 reports
the estimated median forecast-error variance decomposition.
Consider first the results for T = 10, 000, which illustrate the asymptotic properties of ourestimates. The top panels of Figures 2 and 3 show that our identification procedure yields unbiased
estimates of the effects of the monetary policy shock, regardless of whether the data are driven bypersistent or permanent inflation target shocks. The estimated effects of the inflation target shockon the nominal interest rate and inflation are also unbiased, but that on output exhibits verysmall downward bias when the shock is persistent. In all cases, the estimated confidence bands
are extremely narrow, indicating very little variability of the estimates. In addition, Table 3 showsthat the estimated forecast-error variance decomposition is nearly identical to its model-based
counterpart. These results suggest that, asymptotically, our identification procedure is remarkably
successful at recovering the true structural shocks, including the neo-Fisherian one.
Next, consider the results for T = 275. Our procedure again performs very well in extracting
the true monetary policy shock and in estimating its effects with precision, as shown in the top
panels of Figures 4 and 5. The estimated effects of the inflation target shock on the nominal interest
rate and inflation are biased downward at long horizons when the true shock is permanent (see thebottom panels of Figure 5), but in all cases, the theoretical responses lie within the estimated 95
percent confidence intervals. The estimated variance decomposition is also considerably similar to
its theoretical counterpart, irrespective of whether the true inflation target shock is persistent orpermanent.
Based on these results, we deem our identification procedure reliable not only in large samples
but also in small samples of the size typically used in times-series analysis with aggregate data.
Robustness To check the robustness of our identification method, we also generate artificial
data under the following alternative assumptions about the model economy: (i) non-optimizing
firms index their prices to the past level of the inflation target (rather than to past inflation), (ii)technology shocks are permanent (rather than persistent), and (iii) monetary policy shocks are
mildly autocorrelated, with ρm = 0.5 (rather than being i.i.d.). In all cases, our identification
strategy performs remarkably well in recovering the true shocks. To conserve space, the results are
not reported but are available upon request.

11

Monetary Policy Shock
Nominal Interest Rate

0.6
0.4
0.2
0

-0.4

-1
20

25

-0.5
-1
-1.5

-0.6

-0.4
15

0

-0.2

-0.8

10

0.5

0

-0.2

5

Output1

0.2

Percentage Points

0.8

Percentage Points

Inflation

95% Confidence Band
84% Confidence Band
Estimated Response
True Response

Percent

1

30

-2
-2.5
-3
5

10

15

20

25

30

5

10

15

20

25

30

20

25

30

Persistent Inflation Target Shock
Nominal Interest Rate
95% Confidence Band
84% Confidence Band
Estimated Response
True Response

Output1

0.8

Percentage Points

Percentage Points

0.8

Inflation

1

0.6

0.4

0.2

0.5
0.6

Percent

1

0.4

0
0.2

0

0
5

10

15

20

25

30

-0.5
5

10

15

20

25

30

5

10

15

Figure 2: Simulation results: Impulse responses estimated using artificial data with a persistent
inflation target shock (large sample).
Notes: The figure shows the impulse responses to monetary policy and inflation target shocks estimated within a three-equation
VAR using artificial data generated from a simple New Keynesian model. The series used in estimation are those of the nominalinterest rate, inflation, and output, each including 10,000 observations. The responses of the nominal interest rate and inflation
are expressed in percentage-point deviations from their pre-shock levels. The response of output is expressed as percentage
deviation from its pre-shock level. The dark and light shaded areas represent, respectively, the 84 and 95 percent confidence
bands based on 2000 draws. The solid lines are the median impulse responses. The dashed lines are the theoretical (true)
responses.

12Monetary Policy Shock
Nominal Interest Rate

0.6
0.4
0.2
0

-0.4

-1
20

25

-0.5
-1
-1.5

-0.6

-0.4
15

0

-0.2

-0.8

10

0.5

0

-0.2

5

Output1

0.2

Percentage Points

0.8

Percentage Points

Inflation

95% Confidence Band
84% Confidence Band
Estimated Response
True Response

Percent

1

30

-2
-2.5
-3
5

10

15

20

25

30

5

10

15

20

25

30

20

25

30

Permanent Inflation Target Shock
Inflation

1.2

1.2

1

1

0.8
0.6
0.4
95% Confidence Band
84% Confidence Band
Estimated Response
True Response

0.2

0.5
0.8
0.6

0

0.4
0.2

0

0
5

10

15

20

25

Output1

Percent

Percentage Points

Percentage Points

Nominal Interest Rate

30

-0.5
5

10

15

20

25

30

5

10

15

Figure 3: Simulation results: Impulse responses estimated using artificial data with a permanent
inflation target shock (large sample).
Notes: The figure shows the impulse responses to monetary policy and inflation target shocks estimated within a three-equation
VAR using artificial data generated from a simple New Keynesian model. The series used in estimation are those of the nominalinterest rate, inflation, and output, each including 10,000 observations. The responses of the nominal interest rate and inflation
are expressed in percentage-point deviations from their pre-shock levels. The response of output is expressed as percentage
deviation from its pre-shock level. The dark and light shaded areas represent, respectively, the 84 and 95 percent confidence
bands based on 2000 draws. The solid lines are the median impulse responses. The dashed lines are the theoretical (true)
responses.

13Monetary Policy Shock
Nominal Interest Rate

0.6
0.4
0.2
0

-0.4

-1
20

25

-0.5
-1
-1.5

-0.6

-0.4
15

0

-0.2

-0.8

10

0.5

0

-0.2

5

Output1

0.2

Percentage Points

0.8

Percentage Points

Inflation

95% Confidence Band
84% Confidence Band
Estimated Response
True Response

Percent

1

30

-2
-2.5
-3
5

10

15

20

25

30

5

10

15

20

25

30

20

25

30

Persistent Inflation Target Shock
Nominal Interest Rate
95% Confidence Band
84% Confidence Band
Estimated Response
True Response1

0.8

Percentage Points

Percentage Points

0.8

OutputInflation

1

0.6

0.4

0.6

Percent

1

0.4

0.5

0

0.2

0.2

0

0

-0.5
5

10

15

20

25

30

5

10

15

20

25

30

5

10

15

Figure 4: Simulation results: Impulse responses estimated using artificial data with a persistent
inflation target shock (small sample).
Notes: The figure shows the impulse responses to monetary policy and inflation target shocks estimated within a three-equation
VAR using artificial data generated from a simple New Keynesian model. The series used in estimation are those of the nominalinterest rate, inflation, and output, each including 275 observations. The responses of the nominal interest rate and inflation
are expressed in percentage-point deviations from their pre-shock levels. The response of output is expressed as percentage
deviation from its pre-shock level. The dark and light shaded areas represent, respectively, the 84 and 95 percent confidence
bands based on 2000 draws. The solid lines are the median impulse responses. The dashed lines are the theoretical (true)
responses.

14

Monetary Policy Shock
Nominal Interest Rate

0.6
0.4
0.2
0

-0.4

-1
20

25

-0.5
-1
-1.5

-0.6

-0.4
15

0

-0.2

-0.8

10

0.5

0

-0.2

5

Output1

0.2

Percentage Points

0.8

Percentage Points

Inflation

95% Confidence Band
84% Confidence Band
Estimated Response
True Response

Percent

1

30

-2
-2.5
-3
5

10

15

20

25

30

5

10

15

20

25

30

20

25

30

Permanent Inflation Target Shock

1.2

1.2

1

1

0.6
0.4
95% Confidence Band
84% Confidence Band
Estimated Response
True Response1

0.8

Percent

0.8

0.2

OutputInflation

Percentage Points

Percentage Points

Nominal Interest Rate

0.6

0.5

0

0.4
0.2

-0.5

0

0
5

10

15

20

25

30

5

10

15

20

25

30

5

10

15

Figure 5: Simulation results: Impulse responses estimated using artificial data with a permanent
inflation target shock (small sample).
Notes: The figure shows the impulse responses to monetary policy and inflation target shocks estimated within a three-equation
VAR using artificial data generated from a simple New Keynesian model. The series used in estimation are those of the nominalinterest rate, inflation, and output, each including 275 observations. The responses of the nominal interest rate and inflation
are expressed in percentage-point deviations from their pre-shock levels. The response of output is expressed as percentage
deviation from its pre-shock level. The dark and light shaded areas represent, respectively, the 84 and 95 percent confidence
bands based on 2000 draws. The solid lines are the median impulse responses. The dashed lines are the theoretical (true)
responses.

15

Table 3: Simulation results: Fractions of forecast-error variance attributed to monetary policy andinflation target shocks estimated using artificial data.

Nominal Interest Rate

Theoretical
Estimated (Large Sample)
Estimated (Small Sample)

0.468
0.466
0.439

Horizonk = 8 k = 16 k = 32
Monetary Policy Shock0.259 0.162
0.110
0.092
0.257 0.161
0.110
0.092
0.249 0.167
0.120
0.102

Inflation

Theoretical
Estimated (Large Sample)
Estimated (Small Sample)

0.760
0.761
0.732

0.557
0.434
0.412

OutputTheoretical
Estimated (Large Sample)
Estimated (Small Sample)

0.920
0.921
0.884

0.734
0.734
0.705

Nominal Interest Rate

Theoretical
Estimated (Large Sample)
Estimated (Small Sample)

Inflation

OutputPanel A: ρτ = 0.95

k=1

k=4

0.249
0.250
0.244

0.175
0.175
0.181

k = 80
0.088
0.088
0.097

0.147
0.148
0.156

0.142
0.143
0.151
0.485
0.486
0.489

0.532
0.531
0.524

0.624
0.538
0.495
0.624
0.538
0.495
0.608
0.531
0.497
Inflation Target Shock0.735 0.829
0.880
0.898
0.741 0.838
0.889
0.907
0.718 0.810
0.861
0.872

0.901
0.910
0.870

Theoretical
Estimated (Large Sample)
Estimated (Small Sample)

0.229
0.235
0.216

0.433
0.564
0.537

0.744
0.748
0.723

0.819
0.822
0.788

0.847
0.849
0.806

0.853
0.854
0.804

Theoretical
Estimated (Large Sample)
Estimated (Small Sample)

0.020
0.009
0.020

0.068
0.046
0.062

0.061
0.040
0.066

0.052
0.038
0.079

0.048
0.037
0.089

0.047
0.037
0.095

Nominal Interest Rate

Theoretical
Estimated (Large Sample)
Estimated (Small Sample)

0.447
0.445
0.433

0.198
0.197
0.200

Monetary Policy Shock
0.098
0.047
0.023
0.098
0.047
0.023
0.108
0.059
0.036

0.009
0.009
0.024

Inflation

Theoretical
Estimated (Large Sample)
Estimated (Small Sample)

0.656
0.657
0.628

0.326
0.326
0.312

OutputTheoretical
Estimated (Large Sample)
Estimated (Small Sample)

0.896
0.897
0.865

0.710
0.711
0.685

Nominal Interest Rate

Theoretical
Estimated (Large Sample)
Estimated (Small Sample)

Inflation

OutputPanel B: ρτ = 1

0.159
0.160
0.162

0.082
0.083
0.094

0.042
0.042
0.058

0.017
0.017
0.040
0.476
0.477
0.481

0.553
0.554
0.516

0.607
0.525
0.484
0.609
0.527
0.486
0.595
0.529
0.497
Inflation Target Shock
0.797 0.897
0.949
0.974
0.799 0.898
0.950
0.975
0.747 0.856
0.918
0.946

0.990
0.990
0.960

Theoretical
Estimated (Large Sample)
Estimated (Small Sample)

0.335
0.335
0.313

0.667
0.667
0.630

0.836
0.836
0.796

0.915
0.915
0.876

0.957
0.957
0.920

0.983
0.982
0.942

Theoretical
Estimated (Large Sample)
Estimated (Small Sample)

0.042
0.039
0.034

0.099
0.092
0.092

0.086
0.079
0.095

0.074
0.069
0.102

0.068
0.064
0.112

0.067
0.063
0.130

Note: The table reports the median fraction (across 2000 bootstrap replications) of the k-step ahead forecast-error varianceof each variable due to monetary policy and inflation target shocks.

16

3

Empirical Evidence

In this section, we use our empirical strategy to estimate the dynamic effects of neo-Fisherian shocks
and their relative contribution to aggregate fluctuations in the U.S.

3.1

Data

We use quarterly U.S. data spanning the period 1954Q3–2023Q4. The nominal interest rate ismeasured by the federal funds rate, expressed in percent per year. Inflation is measured by thegrowth rate of the GDP deflator, expressed in percent per year. Output is measured by annualized
real GDP (chained 2017 dollars, seasonally adjusted) divided by the civilian non-institutional population 16 years of age and older, and expressed in logarithm. All the series are retrieved from the
FRED database.

3.2

Results

Below, we discuss our empirical findings based on our agnostic identification strategy. The results
are based on a VAR(3) but are robust to higher lag orders.
3.2.1

Impulse responses

Figure 6 shows the impulse responses to the identified monetary policy and neo-Fisherian shocks.
The dark and light shaded areas represent, respectively, the 84 and 95 percent bias-correctedbootstrap confidence intervals, computed using Kilian (1998)’s procedure with 2000 replications.
The solid lines represent the median responses.
Consistently with the imposed sign restrictions, a positive realization of the monetary policy
shock raises the nominal interest rate but lowers inflation and output on impact (see the top panelsof Figure 6). These two variables remain below their pre-shock levels for a prolonged period oftime. Thus, although the sign-restriction approach — by construction — prevents the occurrenceof a price puzzle on impact, the estimated response of inflation does not exhibit such a puzzle atsubsequent horizons.
The impulse responses to the neo-Fisherian shock are depicted in the bottom panels of Figure 6.
A positive realization of the shock leads to a persistent and hump-shaped increase in the nominalinterest rate, which, however, ultimately returns to its initial level. Inflation also rises persistently
in response to the shock, though its response is not as persistent as that of the nominal interest rateand lacks the hump-shaped pattern. These observations suggest that the data do not support theexistence of a shock that raises permanently both the nominal interest rate and inflation. Finally,
the identified neo-Fisherian shock is expansionary in the short run, as it raises aggregate output
for at least a year.

17Monetary Policy Shock
Nominal Interest Rate
0.2

Percentage Points

Percentage Points

0.8

OutputInflation

95% Confidence Band
84% Confidence Band
Estimated Response

0.6
0.4
0.2

0.2

0

0

-0.2

-0.2

Percent

1

-0.4
-0.6

-0.4
-0.6

-0.8

-0.8

-1

-1

-1.2

-1.2

0
-0.2

-1.4
5

10

15

20

25

30

-1.4
5

10

15

20

25

30

5

10

15

20

25

30

20

25

30

Neo-Fisherian Shock
Nominal Interest Rate

1
1

0.8

Percentage Points

Percentage Points

1

0.6
0.4
0.2

0.6
0.4

-0.2

-0.2
15

20

25

30

0

-0.5

0.2
0

10

0.5

0.8

0

5

OutputInflation

1.2

95% Confidence Band
84% Confidence Band
Estimated Response
True Response

Percent

1.2

-1

5

10

15

20

25

30

5

10

15

Figure 6: Estimated impulse responses.
Notes: The figure shows the impulse responses to monetary policy and neo-Fisherian shocks estimated within a three-equation
VAR using U.S. data. The series used in estimation are those of the nominal interest rate, inflation, and output. The responsesof the nominal interest rate and inflation are expressed in percentage-point deviations from their pre-shock levels. The responseof output is expressed as percentage deviation from its pre-shock level. The dark and light shaded areas represent, respectively,
the 84 and 95 percent bias-corrected bootstrap confidence intervals, computed using Kilian (1998)’s procedure with 2000
replications. The solid lines represent the median responses.

3.2.2

Variance decomposition

By assumption, our identification strategy requires that the neo-Fisherian shock accounts for thebulk of the forecast-error variance of the nominal interest rate at the 80-quarter horizon. But how
large is the contribution of this shock to the conditional variance of not only the interest rate, butalso inflation and output at short horizons and at business-cycle frequencies?
Table 4 reports the median fractions (across 2000 bootstrap replications) of the forecast-error
variance attributed to monetary policy and neo-Fisherian shocks at various horizons. The neoFisherian shock turns out to be important in explaining movements in the nominal interest at any
given horizon, with a contribution that amounts to roughly 40 percent at the one-quarter horizon

18

but that keeps increasing with the horizon. The monetary policy shock explains essentially theremainder of the forecast-error variance of the nominal interest rate, but its contribution dwindles
as the horizon increases. The neo-Fisherian shock also proves to be the main driver of movements
in inflation, accounting for more than 57 percent of its conditional variance at the one-quarter
horizon and roughly 68 percent at business-cycle frequencies. In contrast, the contribution of themonetary policy shock to the conditional variance of inflation never exceeds 10 percent at any given
horizon. Finally, the neo-Fisherian shock does not seems to play a major role in driving fluctuations
in aggregate output, though it still accounts for more than 10 percent of its conditional varianceat business-cycle frequencies. Overall, these results indicate that the neo-Fisherian shock is theprincipal source of fluctuations in nominal variables, while being responsible for a non-negligible
fraction of output variability.
Table 4: Fractions of forecast-error variance attributed to monetary policy and neo-Fisherian
shocks.
k=1
Nominal Interest Rate
Inflation
Output

0.503
0.102
0.090

Nominal Interest Rate
Inflation
Output

0.397
0.567
0.234

Horizonk = 8 k = 16 k = 32
Monetary Policy Shock0.292 0.206
0.143
0.114
0.065 0.058
0.058
0.059
0.105 0.107
0.088
0.060
Neo-Fisherian Shock0.604 0.722
0.797
0.818
0.668 0.685
0.684
0.680
0.198 0.132
0.096
0.106

k=4

k = 80
0.104
0.057
0.036
0.808
0.676
0.118

Note: The table reports the median fractions (across 2000 bootstrap replications) of the
k-step ahead forecast-error variance of each variable due to monetary policy and neoFisherian shocks.

3.2.3

Historical decomposition

Further insights into the relative importance of the neo-Fisherian shock in accounting for fluctuations in the nominal interest rate, inflation, and output can be gained by inspecting their historicaldecomposition. Figure 7 depicts the median time paths (across 2000 bootstrap replications) of thesevariables simulated from the estimated VAR under the assumption that the neo-Fisherian shock is
the only stochastic disturbance driving the data. For ease of visualization, output is expressed in
growth-rate terms, and the actual series are superimposed on the simulated ones. Table 5 reports
the correlations and variance ratios between the actual and simulated series.
Figure 7 shows that the simulated series of the nominal interest rate and especially of inflation
track very closely their actual counterparts. The correlation between the simulated and actual series
amounts to 0.84 for the nominal interest rate and 0.97 for inflation. In each case, the simulated

19

series is roughly 80 percent as volatile as the actual one. These observations confirm that theneo-Fisherian shock has been the dominant source of fluctuations in these variables.
Nominal interest Rate
20

NBER Recessions
ActualSimulated

15
10
5
0
1960

1970

1980

1990

2000

2010

2020

2000

2010

2020

2000

2010

2020

Inflation
10
5
0
1960

1970

1980

1990

Output Growth
0.05
0
-0.05
-0.1
1960

1970

1980

1990

Figure 7: Historical decomposition.
Notes: The figure shows the actual series (thick black lines) and the ones simulated from the VAR assuming that the neoFisherian shock is the only disturbance generating the data (thin blue lines). Each simulated data point is the median across2000 bootstrap replications. The shaded areas indicate the dates of the U.S. recessions identified by the NBER.

On the other hand, the simulated series of output growth is weakly correlated with — and muchsmoother than — the actual one, suggesting that the neo-Fisherian shock has played a limited rolein accounting for output fluctuations on average. Nonetheless, this shock appears to have played amore significant role during the most recent downturns, accounting for more than one-third of thedecline in output growth during the Great Recession and roughly 20 percent during the COVID-19
pandemic. Intuitively, in normal times, where monetary authorities operate away from the effective
lower bound (ELB) on the policy rate, the positive comovement between inflation and the nominalinterest rate induced by the neo-Fisherian shock leads to modest variation in the real interest rate
and thus in economic activity. In contrast, during severe recessions that drive the policy rate toits ELB, as was the case during the Great Recession and the COVID-19 pandemic, the partial
adjustment of the nominal interest rate prompts a more substantial change in the real interest rate.

20

Table 5: Correlations and variance ratios between the simulated and actual series.
Nominal Interest Rate
Inflation
Output Growth

Correlation
0.840
0.969
0.253

Variance Ratio
0.775
0.792
0.289

Notes: The table reports the median (across 2000 bootstrap replications) correlations and variance ratios between the simulated andactual series. The former are constructed from the VAR under theassumption that the neo-Fisherian shock is the only disturbance generating the data.

4

What is being Identified?

The consensual view in the literature on neo-Fisherism is that the shock leading to a positive
comovement of the nominal interest rate and inflation captures exogenous shifts in monetary policythat are highly persistent or even permanent. Often, such policy shifts are construed as changes
in the inflation target (e.g., Garı́n et al., 2018; Uribe, 2022). Given that announcements aboutchanging the inflation target are extremely infrequent events that happen once in several decades(particularly in advanced economies), it seems implausible that inflation target shocks can beextracted within VARs (or alternative time-series models) estimated using quarterly data. In thissection, we offer an alternative interpretation of the neo-Fisherian shock identified in the data.

4.1

Inflation target shocks vs liquidity preference shocks

Let us amend the utility function in the benchmark economy by assuming that households’ preferences depend on the real value of risk-free bonds, as in Krishnamurthy & Vissing-Jorgensen (2012).
More specifically, the representative household maximizes

E0

∞
X
t=0

"

β

t

Nt 1+ϕ+ ζt Vln(Ct ) − ψ
1+ϕ



Bt
Pt

#

,

(4)

where Bt is a stock of one-period riskless nominal bonds that mature in period t, V (·) is increasing0

and concave in BPtt and is such that V (0) = 0 and V (0) = 1, and ζt is an exogenous shock thatevolves according to the following process:ζ,t ∼ N (0, σζ ),ζt = ρζ ζt−1 + ζ,t ,where 0 < ρζ < 1. Positive realizations of ζ,t therefore imply that the representative household
values the liquidity and safety of risk-free assets. Accordingly, we shall refer to ζt as a liquidity
preference shock. In this model, the Euler equation becomes1
1
Bt= β(1 + it )Et+ ζt V 0.
Ct(1 + πt+1 )Ct+1
Pt




21





(5)

Equation (5) implies that an increase in ζt lowers the marginal disutility of saving in the risk-free
bond



1
0
Ct − ζt V



Bt
Pt



, thus increasing the incentive to save. As a result, the natural rate ofinterest falls. With flexible prices, this does not affect consumption (and output) in equilibrium.
With sticky prices, however, and to the extent that monetary policy is sub-optimal, the nominalinterest rate does not fall sufficiently to replicate the flexible-price allocation, leading to a fallin consumption and output. This contraction in turn lowers inflation (see Bouakez et al., 2020).
Conversely, a negative realization of the liquidity preference shock raises the nominal interest rate,
output, and inflation, just as does an exogenous increase in the inflation target.
In order to show the similarity between the effects of liquidity preference and inflation target
shocks, let us abstract from monetary policy and technology shocks and from interest-rate smoothing (by setting % = 0), and assume for simplicity that steady-state output is normalized to 1.6 The
log-linearized model therefore becomes

Etyt+1 − (it − Etπt+1) − ζt,πt = (1 + β)−1 (πt−1 + β Et πt+1 + λyt ),yt =it = φπ πt + φy yt − (φπ − 1) τt .

(6)
(7)
(8)

It is straightforward to see that
dyt+j
dζt
dπt+j
dζt
dit+j
dζt

dyt+j
,
j = 0, 1...
dτt
dπt+j
= − (φπ − 1),
j = 0, 1...
dτt


dit+j
= − (φπ − 1)
+ φπ − 1 ,dτt= − (φπ − 1)

j = 0, 1...

Thus, to the extent that τt and ζt are equally persistent, the responses of output and inflation
to a positive realization of τ,t are proportional (thus identical in shape) to their responses to anegative realization of ζ,t . In the special case where ρζ = ρτ and σζ = (φπ − 1) στ , one can easily
see that a model with liquidity preference shocks would be observationally equivalent to a modelwith inflation target shocks with respect to the dynamics of output and inflation. On the other
hand, while the liquidity preference shock always gives rise to a short-run positive comovement
between the nominal interest and inflation rates (i.e., a neo-Fisherian effect), the inflation target
shock does so only when it is sufficiently persistent. Figure 8 illustrates these predictions for ρτ = ρζ
∈ {0, 0.95}.
In the more general case where % > 0, the dynamic responses to a liquidity preference shock are
no longer affine functions of those implied by an inflation target shock. However, as is depicted in
6

At the steady state, Y =

1
ψ

ν−1
ν

1
 1+ϕν. Setting ψ to ν−1ensures that Y = 1.

22

=Nominal Interest Rate

0.5

-0.1
-0.2
-0.3

0.4
0.04

Percent

0

0.03

-0.1

0
10

15

20

25

30

-0.2
5

10

15

=
Nominal Interest Rate

0.8
0.6
0.4

25

30

20

25

30

20

25

30

1

0.6
0.4
0.2

0.4

0

0
20

15

Output

1.2

0.6

0
15

10

= 0.95

0.8

0.2

10

5

0.8

0.2

5

30

1

Percentage Points

Percentage Points

1

25

Inflation

1.2

Inflation Target ShockLiquidity Preference Shock20

Percent

1.2

0.2

0

-0.5
5

0.3

0.1

0.02
0.01

-0.4

Output0.6

0.05

Percentage Points

Percentage Points

0.1

Inflation

0.06

Inflation Target ShockLiquidity Preference Shock

=0

-0.2
-0.4
5

10

15

20

25

30

5

10

15

Figure 8: Impulse responses to inflation target and liquidity preference shocks without interest-ratesmoothing (% = 0).
Notes: The figure shows the impulse responses to a positive inflation target shock and to a negative liquidity preference shockwith % = 0. The two shocks are i.i.d (ρτ = ρζ = 0) in the top panels and are equally persistent (ρτ = ρζ = 0.95) in the bottom
panels. The size of the inflation target shock is equal to 1 percent and that of the liquidity preference shock is equal to (φπ − 1)
percent, where φπ = 1.5. The responses of the nominal interest rate and inflation are expressed in percentage-point deviations
from their pre-shock levels. The response of output is expressed as percentage deviation from its pre-shock level.

Figure 9, where we assume that % = 0.8 and ρζ = ρτ = 0.95, the two shocks still lead to a positive
comovement of the nominal interest rate, inflation, and output, consistently with the empiricalfindings. Interestingly, the response of the nominal interest rate to a negative liquidity preference
shock exhibits the same hump-shaped pattern observed in the data, unlike the response triggeredby an inflation target shock, which is rather monotonic.
These arguments, along with the observation that there have not been explicit announcements
about persistent/permanent changes in the U.S. inflation target since its implementation, suggestthat the neo-Fisherian effect measured in the data is more likely to be due to persistent liquidity
preference shocks rather than to long-lasting exogenous shifts in U.S. monetary policy. The next
section provides formal support to this conjecture.
23

Nominal Interest Rate

1.2

Inflation

1.2

2.5

1

0.8

0.6

0.4

0.2

2
0.8

Percent

Percentage Points

Percentage Points

1

0.6

10

15

20

25

30

1
0.5
0

0
5

1.5

0.4

0.2

0

Output3

Inflation Target ShockLiquidity Preference Shock

-0.5
5

10

15

20

25

30

5

10

15

20

25

30

Figure 9: Impulse responses to inflation target and liquidity preference shocks with interest-ratesmoothing (% = 0.8).
Notes: The figure shows the impulse responses to a positive inflation target shock and to a negative liquidity preference shockwith % = 0.8. The two shocks are equally persistent (ρζ = ρτ = 0.95). The size of the inflation target shock is equal to 1
percent and that of the liquidity preference shock is equal to (φπ − 1) percent, where φπ = 1.5. The responses of the nominalinterest rate and inflation are expressed in percentage-point deviations from their pre-shock levels. The response of output isexpressed as percentage deviation from its pre-shock level.

4.2

Which story best fits the data? A Bayesian perspective

In this section, we use Bayesian methods to determine whether a model with liquidity preference
shocks (henceforth referred to as the LP model) provides a better overall fit of U.S. data than does
an otherwise identical model with inflation target shocks (henceforth referred to as the IT model).
In the Appendix, we provide a detailed description of the two models, and derive the corresponding
three log-linear equations characterizing the dynamics of output, inflation, and the nominal interest
rate. These are given by equations (A.8)–(A.10) for the IT model and equations (A.11)–(A.13) for
the LP model. Notice that the two models are equally parsimonious and share the same deepparameters.
Both models are estimated using Bayesian methods. To do so, we use quarterly data on thenominal interest rate, inflation, and output growth. From the solution of each model, we derive
the state-space representation and apply the Kalman filter to construct the likelihood function
L(z T |Θ), where z T is the vector of observables and Θ is the vector of structural parameters (seethe Appendix for details). Let p(Θ) denote a prior probability of the structural parameters. Then,by Bayes’ law, p(Θ|z T ) ∝ p(Θ)L(z T |Θ), where p(Θ|z T ) is the posterior distribution. We estimate
the latter using the Random-Walk Metropolis-Hastings (RW-MH) algorithm with 1 million draws.
Table 6 summarizes our Bayesian analysis of the two models. The second, third, and fourthcolumns of the table report the prior distributions of the structural parameters listed in the firstcolumn. Our prior configuration is quite standard and follows closely that chosen by Uribe (2022).

24

In particular, we assume that ρm follows a Beta distribution with mean 0.3 and a standard deviationof 0.2, and ρτ and ρζ follow a Beta distribution with mean 0.7 and a standard deviation of 0.2.
Table 6: Prior and posterior distributions of the structural parameters.
Prior Distribution
Parameter
Distribution
β
BETA
η
GAMMA
θ
BETA
%
BETA
φπ
GAMMA
φy
GAMMA
ρa
BETA
ρm
BETA
ρτ
BETA
ρζ
BETA
σa
INVGAMMAσmINVGAMMAστINVGAMMAσζINVGAMMA
Log Marginal Likelihood

Mean
0.990
1.000
0.500
0.700
1.500
0.125
0.700
0.300
0.700
0.700
0.010
0.010
0.010
0.010

S.D.
0.005
0.200
0.200
0.200
0.250
0.100
0.200
0.200
0.200
0.200
0.010
0.010
0.010
0.010

Mean
0.991
1.054
0.545
0.863
2.424
0.056
0.999
0.421
0.988
–
0.012
0.002
0.002
–

Posterior Distribution
IT ModelLP Model
S.D.
HPD
Mean
S.D.
HPD
0.004
[0.983 0.998]
0.990
0.004
[0.981 0.998]
0.033[0.992 1.109]
1.041
0.031
[0.983 1.095]
0.020[0.503 0.579]
0.458
0.017
[0.428 0.489]
0.015[0.833 0.887]
0.583
0.057
[0.501 0.689]
0.034[2.365 2.482]
2.332
0.073
[2.238 2.495]
0.024[0.007 0.103]
0.125
0.029
[0.073 0.186]
0.000[0.996 1.000]
0.996
0.003
[0.988 1.000]
0.048[0.337 0.483]
0.436
0.079
[0.286 0.539]
0.007[0.976 0.999]
–
–
–
–
–
0.929
0.017
[0.898 0.964]
0.004[0.011 0.013]
0.011
0.003
[0.010 0.012]
0.004[0.002 0.003]
0.004
0.003
[0.003 0.004]
0.004[0.002 0.003]
–
–
–
–
–
0.002
0.003
[0.001 0.002]-75.011-13.351

Notes: The posterior distributions are constructed using 1 million Monte Carlo Markov Chain (MCMC) draws from the
RW-MH posterior sampler with 100,000 burn-in draws. HPD denotes the 95 percent Bayesian highest probability density
interval. The log marginal likelihoods are estimated from the MCMC draws using Geweke (1999)’s harmonic mean estimator.

The fifth, sixth, and seventh columns of Table 6 display the posterior means, posterior standard
deviations, and 95 percent highest probability density (HPD) intervals of the structural parameters of the IT model, while the eighth, ninth, and tenth columns report those of the LP model,
respectively. The two models lead to almost identical posterior inferences regarding the common
structural parameters, the only exceptions being that the IT model infers a larger Calvo probability and a larger degree of interest-rate smoothing than does the LP model. Importantly, theinflation target shock is estimated to be nearly permanent, unlike the liquidity preference shock;the posterior means of ρτ and ρζ are 0.988 and 0.929, and the corresponding upper bounds of the
95 percent HPD intervals are 0.999 and 0.964, respectively.
The last raw of Table 6 reports the log marginal likelihoods of the two models, which are
estimated using Geweke (1999)’s harmonic mean estimator. They are equal to −75.011 and −13.351
for the IT and LP models, respectively. According to Kass & Raftery (1995), this difference in themarginal likelihoods implies very strong evidence in favor of the LP model against the IT model.

5Conclusion

Existing empirical studies estimate the neo-Fisherian effect under the assumption that the underlying shock is a permanent shit in monetary policy, often interpreted as a lasting exogenous change
in the inflation target. In this paper, we have proposed an alternative empirical strategy to detect25and measure such an effect, which is agnostic about the persistence and nature of the shock thatgenerates it. Monte Carlo simulations based on artificial data confirm that our approach is robustly reliable. Applying our methodology to U.S. data, we find strong evidence of a neo-Fisherian
effect. The underlying shock, however, is found to persistent but not permanent, causing inflationand the nominal interest rate to eventually return to their pre-shock levels. The identified shock
proves to be the main driver of these two variables at any given frequency, while accounting for a
non-negligible fraction of output variability at business-cycle frequencies.
We have then provided theoretical and empirical arguments that advocate for the interpretationof the identified shock as a liquidity preference shock rather than an inflation target shock. While
the two disturbances imply similar dynamics of the nominal interest rate, inflation, and output,
Bayesian analysis indicates that a model with liquidity preference shocks is better supported by
the data than an otherwise identical model with inflation target shocks.
Given the importance of liquidity preference shocks in accounting for interest-rate variability, itis plausible to believe that they also play a significant role in explaining exchange-rate movements.
Extending our analysis to the context of an open economy would help shed light on this conjecture.
We leave this inquiry for future research.

26

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29

A

Appendix

Model with inflation target shocks (IT model)
The representative household in this economy has the following expected lifetime utility:

E0

∞
X

"

β

t=0

t

#

Nt 1+ϕln(Ct ) − ψ,
1+ϕ

where Ct is a consumption basket, Nt is hours worked, 0 < β < 1 is the subjective discount factor,
ψ > 0 is a utility shifter, and ϕ > 0 is the Frisch elasticity of labor supply.
The consumption basket consists of a Dixit–Stiglitz aggregator of a continuum of final goods,
each of which is produced by a monopolistically competitive firm:Z 1

Ct (ι)

Ct =

ν−1
ν

ν
 ν−1

dι

0

where Ct (ι) is the consumption demand for a particular final good indexed by ι ∈ [0, 1], and ν > 1
represents the price elasticity of demand.
The static cost minimization problem of the representative household yields the demand function
for each final good. Given the price of final good ι, Pt (ι), the demand function is

Ct (ι) =Pt (ι)
Pt

−ν

Ct ,

where Pt is the aggregate price level, which satisfies
Z 1

Pt =

1−ν

Pt (ι)

1
 1−ν

dι

.

0

The household maximizes its expected lifetime utility subject to the following budget constraint:
Bt + Pt Ct ≤ (1 + it−1 )Bt−1 + Wt Nt + Dt ,where Bt , it , Wt , and Dt denote nominal bond holdings, the nominal interest rate, the nominal
wage, and nominal dividends from monopolistically competitive firms, respectively. The first-ordernecessary conditions associated with this maximization problem are11= β(1 + it )Et,
Ct(1 + πt+1 )Ct+1

(A.1)

and
ψNtϕ =

Wt
,
Ct

t
t
− 1 is the inflation rate between t − 1 and t, and Wt ≡ W
where πt ≡ PPt−1
Pt is the real wage.

30

(A.2)

Firm ι produces its final good using the production technology
Yt (ι) = eat Nt (ι),
where at denotes labor productivity (or technology), which is governed by the following process:a,t ∼ N (0, σa ),at = ρa at−1 + a,t ,where 0 ≤ ρa < 1. Given the production function above, the economy’s average real marginal costis
M Ct =

Wt
.
eat

(A.3)

Final-good-producing firm set prices à la Calvo: in each period, a given firm resets its optimal
price with probability 0 ≤ 1 − θ < 1. Firms that cannot optimally reset prices fully index theircurrent prices to past inflation. Let Pt denote the optimal reset price at time t. A firm that getsto reset its price at time t solves the following problem:
max EtPt∞
X

θi Qt,t+i {Pt − MC t+i } Ct+i|i ,

i=0where Qt,t+i ≡ β i (Ct /Ct+i )(Pt /Pt+i ) is the stochastic discount factor for nominal payoffs, Ct+i|i ≡


Pt
Pt+i

−ν

Ct+i , and MC t+i is the the economy’s average nominal marginal cost. The FONC for the

optimal price is∞ν Et i=1 θi Qt,t+i MC t+i Ct+i|i
Pt =ν−1
Et P∞i=1 θiQt,t+iCt+i|i .

P

(A.4)

Given the optimal price, the price index, Pt , follows the law of motion:
Pt1−ν = (1 − θ)Pt1−ν + θ[Pt−1 (1 + πt−1 )]1−ν .

(A.5)

As in Uribe (2022), we assume that monetary policy is determined by a generalized Taylor rulethat allows for stochastic variations in the inflation target, τt . More specifically, the central banksets the short-term nominal interest rate according to

1 + it1 + τt

=

1 + it−1
1 + τt−1

%



1

β



1 + πt1 + τt

 φπ

Yt
Ytf

!φy 1−%


emt ,

(A.6)

where 0 ≤ % < 1, φπ > 1, φy ≥ 0, Yt /Ytf is the output gap, defined as the difference between actual
output and its flexible-price counterpart, Ytf , and mt is a monetary policy shock. The processes

31governing the evolution of mt and τt are given by
mt = ρm mt−1 + m,t ,m,t ∼ N (0, σm ),τ,t ∼ N (0, στ ),τt = ρτ τt−1 + τ,t ,where 0 ≤ ρm < 1, and 0 ≤ ρτ ≤ 1.
Let Nt ≡R1

0 Nt (ι)dι denote aggregate labor and ∆t ≡R 1  Pt (ι) −ν0Ptdι. Then, we can write∆t Yt = At Nt .

(A.7)

Consider a deterministic steady state in which real variables are constant, and let variables
without a time subscript denote steady-state values and lower-case variables denote percentage
deviations of their upper-case counterparts from their steady-state values (e.g., yt = YtY−Y ≈ ln Yt −ln Y ). Imposing the resource constraint Yt = Ct , using (A.2), (A.3), and (A.7) to substitute for
M Ct in (A.4), and log-linearizing the equilibrium conditions around the deterministic steady state,
the dynamics of output, inflation, and the nominal interest rate can be summarized by the following
equations:

Etyt+1 − (it − Etπt+1 − ln β −1),πt = (1 + β)−1 [πt−1 + β Et πt+1 + λ (yt − at )],
yt =

(A.8)
(A.9)

it = (1 − %) ln β −1 + %it−1 + (1 − %) [φπ πt + φy (yt − ytf )] − [(1 − %) φπ − 1] τt − %τt−1 +(A.10)
mt ,> 0 and ytf = at .
where λ ≡ (1−θ)(1−βθ)(1+ϕ)θ

Model with liquidity preference shocks (LP model)
In the LP model, we abstract from stochastic variations in the inflation target by assuming τt =
τ = 0 for all t. Moreover, we assume that the expected lifetime utility function is given by

E0

∞
X
t=0

"

β

t

Nt 1+ϕln(Ct ) − ψ+ ζt V
1+ϕ



Bt
Pt

#

,
0

where V (·) is increasing and concave in BPtt and is such that V (0) = 0 and V (0) = 1, and ζt is aliquidity preference shock that evolves according to the following process:ζ,t ∼ N (0, σζ ),ζt = ρζ ζt−1 + ζ,t ,where 0 < ρζ < 1.
In this case, the Euler equation becomes1
1
Bt= β(1 + it )Et+ ζt V 0,
Ct(1 + πt+1 )Ct+1
Pt




32





while the rest of the FONCs are identical to those of the IT model. The two models also share thesame steady state.
The log-linearized LP model is given by

Etyt+1 − (it − Etπt+1 − ln β −1) − ζt,πt = (1 + β)−1 [πt−1 + β Et πt+1 + λ (yt − at )],yt =(A.11)(A.12)

it = (1 − %) ln β −1 + %it−1 + (1 − %) [φπ πt + φy (yt − ytf )] + mt .

(A.13)

Construction of the likelihood function
Let M = {IT, LP} denote the linear rational expectations model under consideration and xt denotea 9 × 1 unobserved state column vector defined as(xt ≡[yt , πt , it , Et yt+1 , Et πt+1 , mt , τt , at , πt−1 ]0[yt , πt , it , Et yt+1 , Et πt+1 , mt , ζt , at , πt−1 ]0

if M = IT.
if M = LP

Furthermore, let t and ηt denote 3 × 1 random vectors consisting of the structural shocks andrational expectations errors, respectively:
(

t ≡
and

[m,t , τ,t , a,t ]0[m,t , ζ,t , a,t ]0

if M = IT,
if M = LPηt ≡ [y − Et−1 yt , πt − Et−1 πt ]0 .

The vector of structural shocks, t , is assumed to be normally distributed, with a zero mean and a
diagonal variance-covariance matrix Σ: t ∼ i.i.d.N (0, Σ) with diag(Σ) ≡ [σm , στ , σa ]0 .
For each model, the equilibrium conditions can be written as
G0 (Θ)xt = G1 (Θ)xt−1 + Q(Θ)ηt + R(Θ)t ,where G0 , G1 , Q, and R are coefficient matrices and Θ is the vector of structural parameters.
Applying Sims (2002)’s QZ algorithm to the system above yields a unique solution as the following
stationary transition equation of the state vector:
xt = F (Θ)xt−1 + Φ(Θ)t ,(A.14)

where F and Φ are conformable coefficient matrices.
Let zt denote the information set that consists of the nominal interest rate, inflation, and thegrowth rate of output:
zt ≡ [it , πt , ∆yt ]0 .
Assuming that all the shocks follow stationary processes, it is straightforward to show that the33demeaned information set z̃t ≡ zt − Ezt is linearly related to the unobservable state vector xt viaz̃t = Ξxt ,(A.15)

where Ξ is a conformable coefficient matrix. Equations (A.14) and (A.15) jointly constitute astate-space representation of each of the two models.
Given the data set z T ≡ {zt }Tt=0 , we construct the likelihood L(z T |Θ) of each model by applying
the Kalman filter to the state-space representation (A.14)–(A.15). Let p(Θ) denote a prior probability of the structural parameters. Then, by Bayes’ law, p(Θ|z T ) ∝ p(Θ)L(z T |Θ), where p(Θ|z T )is the posterior distribution. The latter is simulated using the Random-Walk Metropolis-Hastings
algorithm.

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