On the Quantitative Effects of Unconventional Monetary Policies∗
Javier Garcı́a-Cicco
Central Bank of Chile
First Version: July, 2009This Version: August, 2010

Abstract
This paper quantitatively evaluates the effects of several unconventional monetary policies. In
particular, a new Keynesian model of a small-open economy is extended to include a liquidity
premium, deviations from the UIP condition, and a premium in the term structure of interest
rates; allowing the Central Bank to set, in addition to the policy rate, the stock of base money,
debt of different maturities and foreign assets. The model is calibrated to the case of Chile. We
find that policies affecting the liquidity channel can potentially have a big effect, but they dependon expectations about the future policy rate. On the other hand, alternatives working throughthe term premium have smaller effects, but they are less dependent on the expected path of thereference rate. We also study the possibility of undoing the unconventional policy as a possible exit
strategy, with results indicating that this alternative may induce a significant slowdown, particularly
if it is anticipated. Finally, we also consider the alternative of driving down and maintaining thepolicy rate to its lower bound. While this policy can also be greatly expansionary, credibility issues
regarding the promise of keeping the rate low for some time can severely undermine these effects.

∗
I would like to thank Carl Walsh, Pablo Garcia, Federico Mandelman, Juan Pablo Medina, Claudio Soto, Michael
Woodford and seminar participants at the Central Bank of Chile, the Atlanta Fed and the Central Bank of Argentina
for comments. Juan Sebastián Becerra provided excellent research assistance. The views and conclusions presented inthis paper are exclusively those of the author and do not necessarily reflect the position of the Central Bank of Chile orits Board members. E-mail: jgarcia-cicco@bcentral.cl

1Introduction

In normal times, many Central Banks use the nominal interest rate as the policy instrument to achieve
their inflation goals. However, when the policy rate reaches its lower bound the monetary authority
has to look for alternative (or unconventional) policies. What are the different tools that a Central
Bank can use in such a situation? What are the quantitative effects of these policies? Although there
is a significant branch of the literature that has evaluated the virtues of different options,1 the analysisof the quantitative impact of different unconventional policies is much less frequent.2 Moreover, while
until recently the experience of Japan (having a near-zero interest rate since the mid 1990’s) was
almost the only case of study, nowadays the aggressive response of many Central Banks aimed todampen the effects of the 2008-9 crisis –taking the interest rate to its lower bound– further emphasizes
the need for a quantitative assessment of different policy alternatives.
This paper is an attempt in this direction, presenting a quantitative exploration of different alternatives based on a dynamic and stochastic general equilibrium model. In particular, we extend a
simple new Keynesian model of a small open economy to include three frictions: a liquidity premiumthat introduces a gap between the policy rate and that of short-term debt, which depends on thevelocity of base money; a portfolio-balance effect for the determination of the nominal exchange rate
(implying deviations from the uncovered interest rate parity, UIP); and a premium in the term structure of interest rates that depends on the relative stock of short and long maturity of debt. Thesefrictions, albeit introduced in an ad-hoc fashion, allow the Central Bank to choose not only the interest
rate as its policy tool, but the composition of its balance sheet as well (the stock of money, short andlong term debt and foreign assets).
We use this model to evaluate the dynamics generated by different changes in the Central Bank’sbalance sheet, taking into account the expected path for the policy rate, alternative “exit strategies”,
the possibility of (exogenous) lack of credibility in the policy announcements, and the role of thezero lower bound. The model is calibrated to the case of Chile, which is one of the countries thathave driven the policy rate down to its lower bound during the 2008-9 crisis. In addition, from amethodological perspective, we present an algorithm to perform these type of exercises in the context
of linear rational expectations models, which is based on the idea of backward induction.
The results appears to indicate that polices working through the liquidity channel can potentiallyhave important expansionary effects; which are generated not by the direct effect that a monetary
expansion have in reducing this premium, but rather through the expected future inflation that theincrease in base money generates. However, these effects greatly depend on the perception about the
future path of the interest rate and its credibility. On the other hand, policies aimed to reduce theterm premium, while generating smaller expansions, are less dependent on the expected behavior of
the policy rate and on credibility issues. The channel associated with the deviations from UIP seems
to be of minor importance.
We also consider the possibility of undoing the original change in the balance sheet as a possible
exit strategy. This exercise shows that reversing the unconventionally policy can generate important
contractions, particularly if these changes are anticipated. Finally, we study the benefits of driving
the policy rate down to its lower bound and maintaining it at that value for some time, finding that
1

See, for instance, Eggertsson and Woodford (2003), Svensson (2001) and Bernanke and Reinhart (2004), among
many others.
2
One exception is the work of Coenen and Wieland (2003) who analyze several alternatives for the case of Japan.

1this policy can by itself be highly expansionary, although the issues regarding the credibility of theannouncement are also relevant in this case.
In terms of the related literature, several reasons explain the lack of studies quantifying the effectof unconventional policies. First, it is difficult to estimate the impact of these policies directly fromthe data. For, at least until recently, the lower bound episodes were almost exceptional, limiting thepower of direct econometric inference. The alternative is to use a model to tackle these issues, which is
the route taken in this paper. However, this is not a trivial task either. On one hand, allowing for theinterest rate to have a lower limit entails to consider non-linearities in the model, which are generally
difficult to handle computationally. On the other hand, even assuming that we can circumvent thecomputational issues, most models (particularly in the new-Keynesian tradition) are not rich enoughto allow for the monetary authority to use several different instruments. For instance, the role ofmoney and liquidity is generally absent in these model, as well as the role financial frictions.3 In
this work, we tackle the non-linearity by allowing the interest rate to be fixed at an arbitrary value(possibly zero) for some time as a way to capture deviations from the linear Taylor rule,4 and we
introduce frictions, although in an ad-hoc way, in order to consider different policy tools.5
Before turning to the analysis, a caveat is in order concerning the type of policies analyzed inthis paper. The term unconventional policy has been used somehow vaguely, referring to a numberof alternative tools that simply depart from the usual management of the monetary policy rate (e.g.,
in terms of a model, an unconventional policy could be any departure from the Taylor rule). One ofsuch an alternative implemented during the recent crisis was the purchase by Central Banks of risky
financial assets that were not part of the usual portfolio of these institutions, a type of policy thatis generally labeled as credit policy. The goal of these alternatives was to try to reduce a financial
distortion that have widen due to the drop in the price of assets used as collateral (e.g. mortgagebacked securities). Moreover, it is likely the case that these Central Banks would have implemented
this same type of strategies regardless of the policy rate reaching its lower bound. Although thesealternatives are relevant to discuss as well, we do not analyze them in this paper for at least two
reasons. First, these are not the only type of policies that have been evaluated and/or implemented;
for many countries did not have to deal with a situation of deep financial distress.6 This paper isthen more representative of the situation faced by countries like Chile, Sweden and Australia, among
others, that drove the policy rate to its lower limit and had to look for other alternatives to stimulate
the economy. Second, and related to a point previously highlighted, there is in the literature no clear
modeling strategy that would allow us to consider these issues; although as we mentioned significant
3
Some recent studies attempt to expand the basic framework in these lines; for instance, Curdia and Woodford (2009)
and Gertler and Karadi (2009).
4
Although we allow for non-linearities, we are still focusing on a local solution, as most studies in this topic do.
A different branch of the literature characterizes global solutions, showing that important differences may arise; mostremarkably, the possibility of converging to a liquidity trap (see, for instance, Benhabib et al., 2001, 2002). This
alternative is however no explored in this paper.
5
The fact that frictions are not micro-funded is clearly a drawback of the analysis. However, given that the literature
has not yet arrived to a consensus on which are the relevant frictions or how to model them, this analysis can provide
some insight on which are the relevant parts of the model (or “wedges”) that should be the focus of this research agenda.
6
For instance, Céspedes et al. (2009) compile a list of 56 policy announcements regarding unconventional policies,in the period from September 2008 to October 2009, for a group of 13 central banks the generally implement (explicit
or implicitly) inflation targets, finding that only 12 of these announcements were related with assets purchase programsor direct lending to (non-banks) financial firms, which were implemented by four central banks (Canada, Switzerland,
England and the U.S.).

2

progress is being made in this direction.
The rest of the paper is organized as follows. Section 2 presents the frictions introduced as well as
the alternative policy tools that the Central Bank has available. We also show there the estimation
exercise using Chilean data. In section 3 we present the quantitative results, where we analyze therole of different expected paths for the policy rate, lack of credibility, possible exit strategies and therole of the zero bound, as well some robustness exercises. Finally, section 4 concludes.

2

Model and Calibration

The structure of the model corresponds to a standard new Keynesian framework for a small open
economy with incomplete asset markets, modified to introduce ad-hoc frictions (or wedges in theusual equilibrium conditions) that allow the central bank to decide on the different elements in its
balance sheet, in addition to the policy rate. In particular, we consider a structure with households
having separable preferences in consumption and leisure and that can transfer resources over timeusing an internationally traded bond, money and bonds of different maturities issued by the central
bank. They consume a combination of domestic and foreign goods. There is also an infinite numberof monopolistic intermediate good firms that use labor to produce, using a constant returns-to-scaletechnology, and that are subject to a Calvo-type problem in setting prices, with full indexation to past
inflation. Finally, with the goal of simplifying the analysis, we abstract from capital as a productive
input (and therefore investment), as well as from fiscal policy.
In this section we first describe the optimality conditions associated with households assets decisions
in the standard model and then describe the modifications we introduce.7 Then, we characterize thecentral bank and the several policy alternatives we consider. Finally, we discuss the estimation of theparameters describing the frictions we introduce. The appendix contains a description of the otherequilibrium conditions of the model.

2.1

Assets and Monetary Policy in the Standard Model

We begin by describing the budget constrain and the optimality conditions in the frictionless model tosee how they reduce the central bank policy alternatives to only one instrument (generally, the interest
rate associated with short term bonds). Consider the following budget constrain faced by households,
Pt Ct +

D2,t−1
St Bt∗ D1,t D2,t Mt∗+++ M ≤ Wt Lt + St Bt−1
+ D1,t−1 ++ Mt−1 + Tt .
i∗t cpt
i1,t
i2,t
i1,t
it

Here, Ct denotes consumption, St is the nominal exchange rate, Pt is the price level, Wt stands fornominal wages, Lt are hours worked and Tt are lump-sum transfers from the central bank. In termsof assets, households have access to an internationally traded bond Bt∗ that has a price given 1/(i∗t cpt )(where i∗t is the world interest rate and cpt is the country premium that “closes” this small open
economy model), bonds issued by the central bank (Dk,t is a bond of maturity k with rate ik,t ),8 and
7
As we mentioned in the introduction, these modification are introduced ad-hoc, with no micro-foundations. Nevertheless, we discuss alternative mechanisms (introduced in the related literature) that may produce such departures fromthe standard model.
8
We only consider two period bonds here to ease the notation, while in the estimation we allow for maturities in linewith those available in our empirical case, Chile.

3money, Mt , which has an acquisition price equal to 1/iMt (we discuss the role of this rate below).
In the frictionless model, the first order conditions associated with asset decisions can be written,
after a log linear approximation, as9
λ̂t = Et {λ̂t+1 } + î1,t − Et {π̂t+1 },(1)î2,t = î1,t + Et {î1,t+1 },(2)î1,t = î∗t + cp
ˆ t + Et {ŝt+1 } − ŝt ,(3)

î1,t − îMt = −ϕm (m̂t − p̂t − ŷt ).

(4)

where λt is the marginal utility of consumption and the parameter ϕm will depend on the specificassumption about the role of money.10
In the usual treatment of this model, it is also assumed that iM
t = 1 and that the policy rate,MP
it , equals the short term rate i1,t . Therefore, in the standard model there is only one monetary
P , which is determined by a Taylor-type rule) and the different
policy instrument available (usually iMtelements in the central banks balance sheet are irrelevant for the determination of other asset prices:the demand for real balances (4) determines the stock of money for a given itM P , the expectations
hypothesis (2) determines interest rates at different horizons, and equation (3) (the uncovered interest
parity, UIP) pins down the exchange rate. To overcome this impossibility we redefine the monetary
policy rate and introduce wedges in these familiar conditions, in such a way that the stock of different
assets will show up in these conditions, as we next show.

2.2

Modifying the Standard Framework

The first departure from the standard framework is to modify the assumption iM
t = 1 and to consider
M
PM
this to be the policy rate (i.e. it= it ). The motivation behind this choice is that usually thetarget for the policy rate it is not that of a short term bond but, instead, that of a monetary market,
such as the overnight or the repo rate. These are the type of markets that central banks actually use
to introduce money into the economy trough open market operations and, although in those markets
treasuries and/or central bank bonds are generally exchanged for money, the interest rate charged forthese operations do not need to (and generally they actually do not) coincide with that of these assets,
for the later may generally include a liquidity premium.11
We are not the first, however, to use this type of assumption. Reynard and Schabert (2009)
introduce a framework explicitly modeling the market for money where only short term bonds are
eligible for money acquisitions. In their framework monetary policy targets the rate prevailing inthat market, which is generally different from that of other assets in the economy. Sims (2009) also
consider a model with two policy rates, although his interpretation for the rate iM
t is rather different;
9

We use small-caps hatted variables to denote log-deviations from the steady state.
Here we are implicitly assuming that, in the absence of price stickiness, money is neutral; as it is generally assumed
by the “cashless limit” modeling strategy. Additionally, it is worth mentioning that in a fully micro-funded model(particularly in one of a small open economy) consumption should appear in equation (4) instead of output. However,
because in our estimation we use monthly data to overcome a short available sample we use output because we do nothave available a monthly consumption series but we do have a monthly series for aggregate economic activity. Finally,
we are implicitly assuming that money demand has a unitary elasticity with respect to output.
11
Such a premium does not need to be high (as it doesn’t seems to be in the data) for this mechanism to work, but aslong as it exist this mechanism will generate the results presented below.
10

4

for he considers this to be the interest rate paid on reserves as a way to capture the policy recently
implemented in the U.S. by the Fed.
This assumption introduces an additional degree of freedom for monetary policy, which can now be
implemented by choosing two instruments, in our case iMt and the stock of money Mt , leaving the shortterm rate i1,t to be determined by equation (4). From a technical point of view, this assumption has also
be considered in some textbook treatments such as Woodford (2003, Ch. 2). In particular, Woodfordshows (in the context of a flexible price model) that a monetary policy specified by exogenous paths for
12 This results is relevant because, although weboth iMt and Mt will deliver a determinate equilibrium.
will consider a familiar Taylor-type rule for iMt satisfying the Taylor principle as the “normal times”
policy rule (see below), we can also assume that iM
t is fixed at a given value forever without facing
indeterminacy issues; a feature that we will exploit to consider unconventional monetary policies.
From a qualitative point of view, given this friction, a permanent increase in the stock of Mt will
have, ceteris paribus, two effects. On one hand, the reduction of i1,t will increase desired consumption.
On the other hand, the rise in Mt will increase the price level which, in the presence of price rigidities,
will generate expectations of future inflation, reducing the real interest rate and further expanding
consumption demand.
The second difference with the standard framework is that we also consider deviations from theuncovered interest rate parity (UIP) such that,ŝt − Et {ŝt+1 } − î∗t − cp
ˆ t + î1,t = ϕs (∆b̂∗BC
),
t

(5)

where ∆ denotes the first difference operator and Bt∗BC denote the holdings of of foreign assets by thecentral bank (described below). In this way, changes in the central banks holdings of foreign assets
will have a direct effect (ceteris paribus) on the nominal exchange rate.13
Finally, we modify equations (1) and (2) to consider a term premium in the yield curve structurethat affect intertemporal allocations. In particularˆ t,
λ̂t = Et {λ̂t+1 } + î1,t − Et {π̂t+1 } + ∆tp
where



ˆ t ≡ îk,t − î1,t − Et {î1,t+1 } = ϕk dˆ2,t − dˆ1,t .
tp(6)Therefore, different combinations of long and short term debt will have first order effects in consumption decisions.14 In particular, either increases in D1,t or drops in D2,t will shift consumption demand
upwards.15
12

See Proposition 2.11. This result is also present in the model of Reynard and Schabert (2009), which additionallyfeatures price rigidities.
13
A similar deviation from UIP gap can be obtained in a model with costly adjustment of the international portfolio.
See, for instance, Sierra (2008).
14
ˆ t entering the Euler equation with a coefficient different thanIn principle, we could consider the possibility of ∆tp
one. However, to estimate such a coefficient will be difficult, for there is no monthly data on consumption. Nevertheless,in the robustness section we check the robustness with respect to the parameter ϕk which can be seen also as a robustness
check in terms of modifying this assumption.
15
A similar modification of the Euler equation and the expectation hypothesis can be obtained, for instance, in linewith Andres et al. (2004), by assuming imperfect assets substitution between long and short term bonds.

5

2.3

Monetary Policy

In each period t, the Central Bank (CB) decides on four types of assets: base money (Mt ), foreign
CB and D CB ).16 Additionally,
reserves (Bt∗CB ), and domestic debt of both short and long maturity (D1,t2,tP and lump-sum transfers to households (T ).
it choses the monetary policy rate iMtt
The central bank resource constrain in any given period is given by,17
CB
CB
CB
D2,t−1
D1,t
D2,t
St BtCB∗
Mt
CB
CB∗
+
D
+
+
M
=
S
B
+
+
+ M + Tt .
t−1
t t−1
1,t−1
∗
it cpt
i1,t
i1,t
i2,t
it

It is further assumed that the central bank uses transfers in such a way to rebate any capital gain/loss
to households,18 implying that the bank’s total net worth is kept constant in nominal terms, i.e.
∗CB
CB
CB
CB
CB
(St Bt∗CB − St−1 Bt−1
) = (Mt − Mt−1 ) + (D1,t− D1,t−1) + (D2,t− D2,t−1).

(7)

In normal times, policy is conducted through a familiar Taylor rule,PPîM= ρi îMtt−1 + (1 − ρi )(απ π̂t + αy ŷt ).

(8)

On the other hand, the unconventional policies that we will consider have two components: first, thepolicy rate is fixed at some value (either its steady state level or zero) for some time and, second, theCB changes the position of an asset in its balance sheet, compensated by a change in another asset
such that the bank’s total net worth is kept constant in nominal terms (i.e. equation (7) holds).

2.4

Estimating the Frictions

Given the frictions previously introduced, different changes in the CB balance sheet may have different
expansionary effects. To quantify these effects we need to assign values to the parameter describing
these frictions (ϕm , ϕs and ϕm ) which we will obtain by estimating equations (4) to (6) using monthly
Chilean data from 01-2003 to 05-2009. Here we just report the results regarding these parameters,
while the Appendix includes details about the estimation and discusses some data limitation issues.19
In term of equation (4), we obtain that the relationship between the difference of the short-term
P , and velocity is,20
rate, i1,t , and the policy rate, iMtP= −0.0014(m̂t − p̂t − ŷt ).
î1,t − îMt(0.0007)

For equation (5), the deviations from the standard UIP is related with the stock of central bank’s
16
That a CB can issue its own debt is not a commonly observed characteristic, but it is the case for Chile. MoreCB
CB
generally, thinking of a CB that does not share this characteristic, we can interpret D1,t
and Dk,t
as the negative of
the CB holdings of treasury bonds.
17
In the absence of the frictions previously introduced this equation is irrelevant in equilibrium.
18
Reynard and Schabert (2009), in a closed economy model, make a similar assumption. In principle, this is not the
only possibility that one can consider. But allowing for a change in the CB net worth would require to set a rule for theevolution of this variable, and it is not clear how this rule should be specified (nor there is guidance in the literature onthese issues). This exercise, on the other hand, is simpler and will allow us to obtain a cleaner intuition.
19
The calibration of the other parameters in the model is also described in the Appendix.
20
Newey-West robust standard errors in parenthesis.

6foreign assets Bt∗CB as,ŝt − Et {ŝt+1 } − î∗t − cpt + î1,t = 0.2047(∆b̂∗CB).
t(0.036)

Finally, in terms of equation (6) relating deviation from the expectations hypothesis with theˆCBdifference of the stock of bonds at different maturities, dˆCBk,t − d1,t , we obtained,îk,t −

k−1
X
j=0



ˆCB .
Et {î1,t+j } = 0.017 dˆCB
−
d
1,t
k,t
(0.001)

Finally, it is relevant to highlight that these equations are not meant to represent a full theory
explaining these gaps observed in the data (e.g. not all deviations from UIP are due to exchange
rate interventions). Actually, the share of the variance of the left-hand-side variables explained by the
proposed regressors is generally low. Nevertheless, the fact that the coefficients displayed the correct
sign and appear to be statistically different from zero suggests that some explanatory power can be
attributed to the proposed regressors.

3

The Effect of Unconventional Policies

We proceed with the numerical exercises in four steps. First, we study the effects of a given change in
P remains fixed in its steady state value forever. Although thisthe CB balance sheet assuming that iMt
is clearly not a realistic assumption, it will be useful to grasp the intuition behind the dynamics implied
P fixed atby the different alternatives. Second, we analyze what happens if the CB promises to keep iMt
its steady state value only for a limited period of time, using the Taylor rule afterwards. Additionally,
we allow people to assign an exogenous probability to the event that the CB returns to the Taylor
rule before it was promised. Third, we consider the possibility of undoing the unconventional policy(either by a surprise or with an anticipated announcement) as a possible exit strategy. Finally, weevaluate the virtues of driving the policy rate down to zero and maintaining it there for some time as
another policy alternative.
To keep the analysis as clean as possible we will not assume that the economy is hit by a negative
shock that motives the use of an unconventional policy, neither we study what is the optimal response
to such a shock (as analyzed, for instance, by Eggertsson and Woodford, 2003, in the context of amuch simpler model).21 Our goal is somehow simpler: to understand what is the effect of a given
policy regardless of the shock that is hitting the economy. While the issue of the optimal response
to a recessionary shock is clearly a relevant one, we will proceed in this way so we can better understand the different channels through which the unconventional policy alternatives affect the economy.
Additionally, if we were to use a shock to represent, for instance, the 2008 crisis, it is not clear either
which type of shocks should be considered.
21
However, the algorithm presented in the appendix can be accommodated to consider the type of shocks analyzed in
Eggertsson and Woodford (2003).

7

3.1

P
Fixing iM
Forevert

P at its steady state value forever and to permanently change
The exercise we consider first is to fix iMt
two components of the CB balance sheet in such a way that equation (7) is satisfied. In particular,
we assume that the change in the balance sheet is equivalent to 10% of nominal GDP in steady state.
Figure 1 shows the effect of a purchase of foreign assets financed with an increase in the stock ofmoney. This policy generates an increase of 6.1% on inflation and a rise on GDP of 3.7% on impact,
and these possitive effects last for several quarters. The fact that the response of inflation lies always
above zero indicates, in line with the discussion before, that the price level experiences a permanentchange in the long run.22 This change does not fully materialize in the first quarter due to the pricerigidities, generating, in particular, an increase in expected inflation. This rise in expectations reduces
the ex-ante real interest rate, rt ≡ i1,t − Et πt+1 , which has an expansionary effect. In principle, thisrate is also reduced by the drop in i1,t . However, as can be seen in the figure, i1,t decreases by littlecompared with the increase in Et πt+1 .
This alternative also has an effect through the exchange rate channel in equation (5). To see how
important this mechanism is, the dashed-dotted line in figure 1 displays the same response assuming
ϕs = 0. As can be seen, the responses are quite similar compared to the benchmark case, which further
emphasizes the role of the increase in expected inflation as the relevant channel.
Monetary expansions can also be used to purchase either type of CB debt. In this case, the channel
in equation (5) will play no role, but we may have an additional expansionary effect if long debt isacquired, for it reduces the term premium in equation (6). Figure 2 displays the dynamics under bothalternatives. We can observe effects on impact on GDP and inflation similar to the previous exercise,
CB . There is a somehow smaller response on
with a slightly bigger effect in the case of a purchase of Dk,t
the real exchange rate in this case (the impact response in the previous case was a real depreciationof near 5.3%, while in this case it is around 4.7%), but in general the dynamics are similar indicatingthat this alternative also creates an expansionary effect trough the rise of inflation expectations.
In terms of policies that do not involve monetary expansions, Figure 3 plots the responses to apurchase of long debt financed by new short debt, a policy that has a direct effect through the term
premium channel in equation (6). This alternative generates a milder expansion of GDP and inflation
on impact, 0.3% and 0.05% respectively. While the nominal exchange rate depreciates in this case,there is a real appreciation on impact which partially explains this smaller response. Moreover, given
the shape of the inflation path, the real interest rate actually rise and therefore the expectations
channel is not present in this case.
Finally, figure 4 shows the effect of a purchase of foreign assets financed with either long or shortnew debt. In this case, both the friction affecting the UIP condition, equation (5), and the yield curve,
equation (6), are in place. However, we can see that the effect on output is only positive when the
CB , which appears to indicate that the second channel is relatively
operation involves an increase in D1,tmore important. Additionally, while the increase in output is smaller compared with the case in figure
3, the response of inflation (at least on impact) brought about by the nominal depreciation is bigger
in this case, although this change is reversed in the second quarter.
Overall, these exercises highlight that policies working through the liquidity channel may havesignificant effects, while other alternatives can also have expansionary effect but of a smaller magnitude.
22

Actually, given that equation (4) depends on velocity, and given that the stock of money increases by 10% of GDP,the integral of the response of inflation should be 10%, which can be verified numerically.

8However, the maintained assumption so far was that the monetary policy rate is expected to remain
fixed forever. In what follows, we relax this important assumption.

3.2

P
A Temporary Fix of iMt

We consider the following exercise. At time t0 , the CB announces that the policy rate is going tostay fixed until period T − 1, and from T onwards it will be determined according to the Taylor Rulein equation (8). Additionally, it announces an unconventional policy, which also takes place at t0
but that will never be reversed (an alternative that we consider in the next subsection). Everybody
believes that, beginning on period T , the Taylor rule will be in place and that it will stay for anyt ≥ T . However, for t ∈ [t0 , T − 2], while the interest rate is fixed, people assign an exogenous
probability sequence pt to the CB breaking its promise and setting the nominal interest rate according
to the Taylor rule staring on t + 1. Moreover, if this is the case, they believe that the Taylor rule
will stay forever. Finally, regardless of people not trusting the CB, we assume that (ex-post) it fulfills
its promises and actually waits until T to use the Taylor rule. The Appendix presents an algorithm,
based on the idea of backward induction applied to linear rational expectation models, that can beused to compute the equilibrium trajectories under these assumptions. The algorithm also constrains
the equilibrium path to satisfy the non-negativity constrain on the policy rate.
We begin with the case a purchase of foreign assets financed by a monetary expansion. Figure 5
shows the dynamics for different T ’s, assuming full credibility (i.e. pt = 0). As can be seen, allowing the
Taylor rule to return after some time significantly dampens the expansionary effects of this alternative.
For instance, if the policy rate is fixed for eight quarters, the impact response of output is near 0.9%,
dropping to close to 0.2% for T = 2, while these figures for inflation are, respectively, close to 2%
and 0.7%. These are significantly smaller than the rise of 3.7% on GDP and 6.1% of inflation that is
P stays fixed ad infinitum.
generated if iMt
In part, this smaller effect is due to the qualitatively different response of inflation: while before
it used to lie completely above zero, now it increases on impact, decreases to a negative value in thesecond quarter and then it becomes positive and converges to zero from above.23 This different pathfor inflation transaltes into a less expansionary path for the real interest rate, which now actually
increases on impact and then becomes negative, converging to zero from below. Although the infinite
sum of the real interest rate (i.e. the relevant measure for the determination of consumption according
to the IS curve) is still negative (close to -0.6% when T = 8), it is significantly smaller that before(almost -4%) which explains why the response of output has dampen in this case.
However, this is only a part of the explanation, for the shape of the inflation path changes in thiscase because inflation is more affected by the dynamics of the nominal exchange rate than before,
which explains why it decreases below zero in the second quarter.24 In particular, if ϕs = 0 the
response of inflation lies completely above zero. However, it is still the case that the convergence ofthe price level to its new steady state is much slower.
The difference in terms of the speed of convergence of the price level is clearly due to the expected
23

The cumulative response of inflation (i.e. the price level) is the same in the long run in both cases (equal to 10%P
as we dicussed before). However, while in the case of iM
fixed forever the convergence was after just a few periods, itttakes now more than 3000 quarters to reach this value.
24
Actually, as can be seen in Figure 1, the response of inflation also displayed a small negative bump in the second
quarter even with the policy rate fixed forever. However, the response in inflation in that case was so large that this had
a minor impact.

9

rise in the policy rate. In particular, although this increase is small (less than 3 basis point when
T = 2), the path of the interest rate also lies above zero for a significant period of time (until the
price level converges to its new steady state value).25 Of course, this description cannot be interprated
causally, for both inflation and interest rates are jointly determined in equilibrium. However, it isclear that the anticipation of the return to the Taylor rule significantly limits the expansionary effectthat this policy alternative may have.
We consider next the effect of lack of credibility. Figure 6 shows the same unconventional policy,
fixing T = 6, and setting pt = pt for different values of p (this implies that, as times goes by and people
see that the CB has actually kept its promise, the assigned probability of an anticipated return to the
Taylor rule decreases). This lack of credibility also dampens the expansionary effect that this policy
may have. In particular, for p = 0 the impact increase in GDP is close to 0.5%, but with p = 0.9 itdrops to almost 0.3%. The response of inflacion is also milder the higher this probability is, although
the difference are relatively smaller. Overall, the patterns are anyway similar to those described forthe case of pt = 0.26
We can also consider how are the results affected if the unconventional policy implemented isa purchase of long term debt financed with new short term debt. Figure 7 show the dynamics fordifferent T ’s, keeping pt = 0. Compared with the case of a monetary expansion, here the responses of
output and inflation are not that sensitive to changes in T . Actually, the effects seem to be slightly
P fixed forever, and the responses of both variables lie now completely above
bigger than the case of iMtzero. This different shapes particularly generates a drop in real interest rate, and although the policy
rate rises once the Taylor rule returns, the increase is quite small and the real rate remains below zerofor several period.27 Finally, we analyze the effect of this policy when people assigns a positive but
decreasing probability to the early return of the Taylor rule, a case described in 8. It appears that the
dynamics are not significantly affected either when we allow for pt to vary.

3.3

Undoing the Unconventional Policy

Part of the current discussion about unconventional policies refers to possible “exit strategies”, which
is clearly a broad term. In principle, one can consider to return to the usual Taylor rule being, byitself, an exit strategy. However, the discussion appears to focus on undoing the unconventional policy
initially implemented. For instance, one might be worried about the effects that the excess of liquidity
introduced. As an example, we described before how a policy that entails increases in M can generatesignificant persistence in the inflation process once the Taylor rule is reintroduced. In what follows,
we present several exercises that help to sharpen this discussion.28
A first alternative is to consider the same policies as before (i.e. the CB announces an unconvenP will be fixed until T − 1, an the Taylor rule will be used for t > T ), andtional policy and that iMt
to additionally assume that the CB also announces that in period T − k (with k ∈ [0, T − t0 + 1])
25
Taking together, while the infinite sum of the real interest rate is more negative if ϕs = 0 (close to -1.2% when
T = 8), it is still significantly smaller than what it was in the case of the polict rate fixed forever.
26
Other policies that entail an increase in M also show similar results as those obtained in this case.
27
This actually suggest that, for this type of policy, the responses vary non-monotonically with changes in T . In thePcase of a monetary expansion, the bigger the T the bigger and closer to the case of iM
fixed forever the responses were.
t
In this case, on the contrary, there seems to be a value for T (in the numerical exercises it is close to 25) such that the
responses are bigger at to that point, becoming closer to the case in Figure 3 afterwards.
28
To simplify the exposition, we will just show the case of a purchase of foreign currency financed by a monetary
expansion. The other alternatives yield qualitatively similar results.

10the unconventional policy will be completely reversed (assuming, for the moment, full credibility).29
For instance, if at t0 the CB buys foreign assets using base money, at T − k it sells the same amountof dollars in exchange for currency. This alternative is considered in figure 9, which displays theequilibrium paths for different values of k. To clarify the exercise, the figure also displays the paths
for Mt and Bt∗BC . It is clear that the anticipation of the reversal further dampens the expansion,actually generating a recession in most cases. In particular, the smaller the k, the more negative theresponses of output an inflation are. This is because the policy reversal is contractionary and theeffect is partially mitigated if the nominal rate remains fixed for more time.
These responses can be rationalized by realizing that the mechanism generating the expansion witha policy that expands M (i.e. the expected rise in the price level) disappears when the reversal isanticipated. Figure 10 actually shows that gradually undoing the unconventional policy is even worst
when the change is anticipated, while figure 11 illustrates the if people believes that the change canoccur earlier the economy contracts even further.
We also explore the possibility of undoing the unconventional policy unexpectedly. Figure 12
displays the dynamics for different values of k, for the case of full reversal at T − k. Evidently,
the impact response is the same we described in the previous subsection, and when the policy isundone we have a contraction, with output dropping slightly below zero at that time and inflation
also becoming negative. What actually mitigates the negative effect on output is the fact that theshock is unanticipated: although inflation drops on impact at T − k, it is expected to be positive in
the next period (partly due to the dynamics of the nominal exchange rate), making the real interest
rate negative and generating an increase in consumption demand. However, it is relevant to highlightthat when the Taylor rule returns, the policy rates displays a minor drop (of less than a basis point)
because both output and inflation converge to zero from below slowly. Finally, the case of a gradualreversal starting at T − k is considered in Figure 13. This reversal produces a bigger drop in output
and inflation. However, inflation now converges from above zero and, consistently, the policy rate rises
after period T .

3.4

P
Driving iM
to the Lower Boundt

So far we have assumed the policy rate remaining fixed at its steady state level for some time, as away to complement the effects of changes in the CB balance sheet. However, we can alternatively
consider the possibility of the CB driving the policy rate all the way down to its lower bound, whichis also in line with the recent experience. For instance, Céspedes et al. (2009) document that, in agroup of 20 countries with (implicit or explicit) inflation targets, 13 of them drove the policy rate
down to a lower bound at some point after September 2008 and have maintained it there for an
extended period of time. Additionally, while maintaining the rate at its lower bound can be seen as
a complementary action to enhance the effects of movements in the balance sheet, driving the policy
rate to zero and leaving it there for some time also implies a departure from the usual Taylor rule.
Therefore, we can consider this action as an unconventional policy by itself, regardless of its potential
role in complementing movements in the balance sheet.
Figure 14 shows the responses associated with a decrease (from its steady state value) of the policy
rate down to zero, leaving it at that value for alternative horizons, and assuming that afterwards
29

As shown in the Appendix, the algorithm can be generalized to consider this alternative.

11

this rate is determined by the Taylor rule.30 As can be seen, this policy alternative have important
expansionary effects: for instance, if the rate drops to zero and remains at this value for one quarter,
output increases by near 5% while inflation rises by almost 7%. In addition, we can see that the
key channel for this effect is the reduction in the ex-ante real interest rate. To put these numbers
in context, in order to obtain a similar response (on impact) for both inflation and output with an
expansionary monetary shock to the Taylor rule, we would require a disturbance generating a drop on
impact in the policy rate to -17% (in annual terms).
It is relevant to mention at this point an important caveat, particularly in trying to interpret thesenumbers in the context of recent experiences. If we were to hit this economy with a negative shockin the spirit of current events, the final response will not be equivalent to simply add to the figure
presented the impulse response generated by the negative shock; for the presence of the zero bound
imposes non linearities in the solution. As we mentioned before, we are reluctant to consider such ashock because it is not clear which one should be considered, nor it is obvious how to calibrate its size.
Nevertheless, it is still important to highlight that initial conditions might not only affect the size but
the dynamics of the responses as well.
As we did before, we can also analyze the role of (exogenous) credibility in the announcementof the policy rate staying at zero for some time. Figure 15 displays the response for the case inwhich it is announced that the Taylor rule will return after six quarters, but with people assigninga probability pt = pt to the anticipated return of the Taylor rule in t + 1. In line with the results
obtained previously, the effects of lack of credibility can be pervasive in this case as well; significantly
dampening the expansionary effects brought about by the zero bound.
Finally, we could also consider the role of driving the policy rate down to zero in complementing
unconventional policies that alter the CB balance sheet. However, given that the expansionary effects
brought about by the drop in the interest rate are of an order of magnitude larger than what we haveobtained in subsection 3.2, it can be verified that adding movements in the balance sheet generate
only minor differences compared to the results in figures 14 and 15.

3.5

Robustness

As a final exercise, we check the robustness of the results with respect to some parameter values. In
particular, we explore how the analysis presented in subsection 3.2 is affected by alternative values
for the parameter describing the estimated frictions (plausible according to their estimated confidence
bands), and also in terms of the coefficients in the Taylor rule that is assumed to follow after theP.
temporary fix in iMt
Figure 16 shows the case of a purchase of foreign assets financed by a monetary expansion when
the policy rate is fixed for 6 periods, presenting the case of the original value for ϕm (-0.0014), as wellas two other alternatives representing plus an minus one standard error of the estimated parameter(-0.0007 and -0.0021). As can be seen, these differences in the coefficient affecting the liquidity channel
significantly change the impact of this policy, particularly in terms of the responses of output and thereal exchange rate and, to a less extent, in term of inflation and nominal depreciation.
On the other hand, figure 17 presents the alternative of a purchase of long debt financed with new
short debt under different values of the parameter ϕk (0.017, the point estimate, as well as 0.019 and30

Notice that because the figure plots log-deviations from steady state, the lower bound in the graph corresponds toa value close to -1.6% (given that the steady state of this rate is, in annual, terms is 6.5%).

12

0.015, i.e. plus and minus two standard errors respectively). In this case, changes in this parameterwithin its confidence range do not significantly affect the original responses.
We also check the robustness regarding the parameters describing the Taylor that will be in place
after the temporary fix in the policy rate. Because agents anticipate the contractionary policy thatwill follow once the Taylor rule returns, the expansion generated by the unconventional policy will
be smaller the more aggressive the response is with respect to inflation and output deviations, andthe less important the smoothing part of the rule is. Accordingly, figure 18 displays the result of apurchase of foreign assets financed by a monetary expansion when the policy rate is fixed for 6 periods,
considering the original Taylor rule as well as three alternative coefficients: half the response to theprevious interest rate, and double the reaction to either inflation or output. As can be seen, it appears
that these alternative rules affect mainly the response of output and, to a less extent those of inflationand the real exchange rate.
Similarly, figure 19 shows the impact that considering the same alternative rules has on the evaluation of purchasing long debt financed with new short debt, also fixing the policy rate for 6 periods.
In this case, the different specifications of the rule gave little impact on the responses of output andinflation, while affecting more the responses of the real exchange exchange rate and the ex-ante realinterest rate.
Finally, figure 20 evaluates the role of the Taylor rules parameters on we consider, as in section 3.4,
the experiment of driving the policy rate to zero for some time. For this case, what appears to generatesignificant difference is the smoothing coefficient, particularly for output and the real exchange rate.

4

Conclusions

This paper uses a new Keynesian model of a small-open economy to provide a quantitative evaluationof different policy alternatives that a CB can use when the nominal interest rate is fixed. In particular,
we have analyzed the effects of different changes in the CB balance-sheet, such as movements in theholdings of foreign assets, base money and debt at different maturities. To allow for the possibility of
the CB changing these assets, the standard model was extended to include several frictions; namely,a liquidity premium, deviations from the UIP condition, and a premium in the term structure ofinterest rates. Additionally, we introduced an algorithm that allows to take into account issues likethe anticipation of future polices, the (exogenous) lack of credibility in the announcements and therole of the zero bound.
The results show that polices working through the liquidity channel can potentially have important
expansionary effects, although its impact greatly depends on the perception about the future path of
the interest rate and its credibility. On the other hand, policies aimed to reduce the term premium,
while generating smaller expansions, are less dependent on the expected behavior of the policy rate andon credibility issues. Additionally, it seems that reversing the unconventionally policy as a possible exit
strategy might not be beneficial, particularly if these changes are anticipated. Finally, we described
how driving the policy rate down to its lower bound and maintaining it at that value for some time
can be highly expansionary as well, although the issues regarding the credibility of the announcement
are also relevant in this case.
It is useful to compare these results with those in Eggertsson and Woodford (2003). Using a simpler
model (in particular, a closed economy), their analysis suggests that the channel that is relevant to13

escape from a zero-bound situation is to affect the expectations about future inflation. Moreover,they show that the optimal policy in such a situation is a price level target. In our case, although weare not characterizing the optimal policy, we have described that the potentially big effect generated
by policies affecting the liquidity premium are due to the increase in the future price level brought
about by the permanent increase in base money, resembling the channel emphasized by these authors.
Additionally, they also highlight that the effectiveness of this policy depends on the credibility of theannouncement.31
It is also worth to mention that, although in our set up is money what helps to reduce the liquidity
premium, there is nothing exclusively special about money. Actually, in more general settings whereother assets can affect this premium (for instance, it might be the case that short run debt provides
liquidity as well), a permanent increase in such an asset can also generate the same effect, and its
importance will also be subject to the same caveats about the expected path of interest rates andcredibility that we have analyzed.
Additionally, it is worth mentioning that unconventional monetary strategies might not necessarilybe the only (neither the most effective) policy alternative available to use in a zero bound situation.
In particular, there are several fiscal tools that can be used as well. For instance, Christiano et al.
(2009) describe that the government-expenditure multiplier is actually quite large when the nominalinterest rate is at its lower bound. Additionally, Eggertsson (2006) highlights that government debt
can be used as a commitment device to ameliorate the credibility issues associated with the optimalmonetary policy. Our goal, however, was to analyze the virtues of these monetary alternatives, andwe left the interaction with fiscal tools for future research.32
Finally, an important extension that is left for future research is to rank the different alternativefrom the point of view of optimal policy. However, as we have argued, for this kind of exercises iswhere having a more precise microfoundation of the different channels is most relevant, as well as it
is to choose which shocks (and of which size) are the relevant to consider for this exercise.

References
Andres, J., J. D. López-Salido, and E. Nelson (2004), “Tobin’s imperfect asset substitution in optimizing general equilibrium.” Journal of Money, Credit and Banking, Vol. 36, No. 4.
Benhabib, J., S. Schmitt-Grohe, and M. Uribe (2001), “Monetary policy and multiple equilibria.”
American Economic Review, 91, 167–186.
Benhabib, J., S. Schmitt-Grohe, and M. Uribe (2002), “Avoiding liquidity traps.” The Journal of
Political Economy, 110, 535–563.
Bernanke, B. and V. Reinhart (2004), “Conducting monetary policy at very low short-term interest
rates.” American Economic Review, 94, 85–90.
31

Eggertsson (2006) formally discusses the issues of credibility arising from this optimal policy.
Moreover, in terms of the country we choose for the application, Chile has a fiscal rule in place since 2001, limiting
the fiscal deficits according to the revenues from copper exports. This characteristic places a bound to the type of fiscal
tools than can be implemented; which further emphasizes the importance of analyzing the monetary alternatives.
32

14

Bodenstein, M., C. Erceg, and L. Guerrieri (2009), “The effects of foreign shocks when interest rates
are at zero.” International Finance Discussion Papers, Board of Governors of the Federal Reserve
System.
Céspedes, L. F., R. Chang, and J. Garcı́a-Cicco (2009), “Heterodox central banking.” mimeo, Central
Bank of Chile.
Christiano, L., M. Eichenbaum, and S. Rebelo (2009), “When is the government spending multiplierlarge?” mimeo, Northwestern University.
Coenen, G. and V. Wieland (2003), “The zero-interest-rate bound and the role of the exchange rate
for monetary policy in japan.” Journal of Monetary Economics, 50, 10711101.
Curdia, V. and M. Woodford (2009), “Credit frictions and optimal monetary policy.” mimeo, Columbia
University.
Eggertsson, G. (2006), “The deflation bias and committing to being irresponsible.” Journal of Money,
Credit, and Banking, 38, 283–321.
Eggertsson, G. and M. Woodford (2003), “The zero bound on interest rates and optimal monetary
policy (with discussion).” Brookings Papers on Economic Activity, 34, 139–235.
Gertler, M. and P. Karadi (2009), “A model of unconventional monetary policy.” mimeo, New York
University.
Medina, J and C. Soto (2007), “The chilean business cycle through the lens of a stochastic generalequilibrium model.” Central Bank of Chile Working Papers, No. 457.
Reynard, S. and A. Schabert (2009), “Modeling monetary policy.” mimeo, Swiss National Bank.
Sierra, J. (2008), “Imperfect financial integration, uncovered interest parity and central bank foreignexchange reserves.” mimeo, University of Southern California.
Sims, C. (2009), “Monetary and fiscal policy coordination.” Presented at the conference Quantitative
Approaches to Monetary Policy in Open Economies (FRB Atlanta).
Svensson, L. (2001), “The zero bound in an open economy: A foolproof way of escaping from aliquidity trap.” Monetary and Economic Studies, Institute for Monetary and Economic Studies,Bank of Japan, 19, 277–312.
Woodford, M. (2003), Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton
University Press.

15

A

Technical Appendix

A.1

The Lower Bound, Anticipating Policy Changes and Credibility in Linear
Rational Expectation Models

Here we describe the algorithm to compute the policy exercises in the paper.33 In a contemporaneous
work, Bodenstein et al. (2009) present a similar algorithm, considering how to compute impulses responses when, given an initial shock, the interest rate reaches the lower bound endogenously throughthe Taylor rule. Here, we additionally consider the possibility that the central bank chooses (exogenously, as a policy) to fix the interest rate, deviating from the Taylor rule, to a given value (possiblyzero) for a certain period of time. Moreover, we allow agents to assign an exogenous probability to
the central bank breaking its promise an returning to the usual rule before it was announced. We also
show how to compute the equilibrium trajectories when the unconventional policy is reversed, either
surprisingly or anticipatedly.
The first exercise we want to consider is when the central bank announces, at time t0 , that theinterest rate is going to stay at the lower bound until period t1 − 1, and starting at t1 the rate will be
determined by the Taylor Rule. The Taylor rule in “normal times” is constrained by the lower bound,so that the policy rate (measured as deviations from steady state) is
it = max{−i, itarget
},
t

(9)

where i is the steady state value of the policy rate and the targeted policy rate is a linear function ofany variable in the model (for instance, itarget= φi it−1 + φπ πt + φy yt ).34 In particular, this impliestthat, even though the central bank can guarantee to keep the interest rate at the lower bound up toperiod t1 − 1, the state of the economy could be such that the target rate is below −i at t1 . Therefore,
because the algorithm works with the idea of backward induction, we first describe how to computethe equilibrium for a given initial value of the state variables and the exogenous process when thepolicy rate is determined by the rule (9) (i.e. when the zero bound binds endogenously), and then we
explore the solution for the period in which the government sets the policy rate at the lower bound(i.e. when the zero bound is imposed exogenously, as a policy choice).
At any point in time in which the it = itarget
endogenously, the conditions characterizing a lineart
rational expectations equilibrium can be written as
AEt {zt+1 } = Bzt + Czt−1 + Det ,(10)

where zt is a vector collecting endogenous variables (including it and itarget), et are exogenous drivingtforces following the process et = ρet−1 + ut , ut is a vector of i.i.d random disturbances with zero mean,
and A, B, C, D are conformable matrices describing the linearized equilibrium conditions. In particular,
if n1 denotes the position in the vector zt of the variable it and n2 is the analogous for the variable itarget,t
there is an equation j the system such that (using Matlab notation) A(j, :) = C(j, :) = D(j, :) = 0
while B(j, n1 ) = −1, B(j, n2 ) = 1 and B(j, l) = 0 for l 6= {n1 , n2 }. On the other hand, when it = −i,33
Although we focus on applying the technique for the case analyzed in the paper, notice that this method is quite
general and can accommodate many examples in these lines.
34
It is however assumed that the constrain doesn’t bind at the steady state.

16

the equilibrium is characterized by the system
AEt {zt+1 } = B̃zt + Czt−1 + Det + c,

(11)

where B̃ differs from B only in that B̃(j, n2 ) = 0 and c is a column vector full of zeros except inposition j where c(j) = −i.
The algorithm works under the assumption (which we later discuss) that if, starting at any given
period t1 , there is a period t ≥ t1 such that it = itarget, then it+j = itarget
for j > 0 (i.e. if at somett+jpoint the constrain on the interest rate ceases to bind, it will not bind again after that). Let T denotethe minimum t ≥ t1 such the constrain is no longer binding (i.e. it = itarget
for t ≥ T and iT −1 = −i)tand assume for the moment that T is known. Therefore, the solution characterizing the equilibrium
for t ≥ T –obtained using the usual techniques– can written aszt = FT zt−1 + GT et .

(12)

At period T − 1 the policy rate is at the lower bound and the equilibrium is characterized by(11). Moreover, because agents know at T , iT = iTtarget , they will use (12) to form expectations, i.e.
ET −1 {zT } = FT zT −1 + GT ρeT −1 . Therefore, we can write the equilibrium conditions at T − 1 asA(FT zT −1 + GT ρeT −1 ) = B̃zT −1 + CzT −2 + DeT −1 + c,

(13)

which can be rearranged to obtainzT −1 = (AFT − B̃)−1 CzT −2 + (AFT − B̃)−1 (D − AGT ρ)eT −1 + (AFT − B̃)−1 c,≡ FT −1 zT −2 + GT −1 eT −1 + HT −1 c,(14)provided AFT − B̃ being invertible. Working in this same way backwards until t1 we can compute theequilibrium for t ∈ [t1 , T − 1], which will have the form
zt = Ft zt−1 + Gt et + Ht c.

(15)

where
Jt ≡ (AFt+1 − B̃)−1 ,Ft ≡ Jt C,
Gt ≡ Jt (D − AGt+1 ρ),

(16)
−1

Ht ≡ Jt (−AHt+1 + I) for t ∈ [t1 , T − 2], HT −1 ≡ (AFT − B̃).

Therefore, using these matrices, for initial values of zt1 −1 and et1 −1 we can obtain the perfect foresightpath (impulse response) for a given shock ut1 , under the assumption that T is known. To find T , wecan just run a progressive search starting on t1 and find the first period for which iT = itarget
andT
iT −1 = −i.
Before moving to the case in which the central banks imposes the lower bound, two comments are
in order. First, notice that once T is found the assumption that it = itarget
for t ≥ T can be verifiedtnumerically by simulating the path for the variables for many periods beyond T (by construction,17

the assumption will hold in steady state). Second, as noticed also by Bodenstein et al. (2009), theextension for the case in which, given values zt1 −1 and ut1 , the lower bound is expected to be reached
in a period after t1 is straight forward.
To consider the case in which the central bank announces, at time t0 , that the interest rate isgoing to stay at the lower bound until period t1 − 1, and starting at t1 the rate will be determinedby the Taylor Rule, notice that the matrices in (16) characterize any backward induction problem inwhich the policy rate is expected to be at the lower last for a until a certain period of time. Therefore,
given an announced period t1 and an initial shock ut0 , we can proceed as follows. First, assumingthat T = t1 (i.e. that after the period the central has announced the state of the economy is suchthe constrain on the policy rate is not binding), compute the matrices characterizing the equilibrium
according to (16) and numerically verify whether it = itarget
for t ≥ t1 or not. If that is not the case,tincrease the candidate value for T until the condition is satisfied.35
We can also consider the role of exogenous credibility. In particular, we assume that in any every
period t ∈ [t0 , t1 − 2] when the interest rate is fixed, people assign a probability sequence pt to theevent that the central bank breaks its promise and sets the nominal interest rate according to the
Taylor rule staring on t + 1. Moreover, if this is the case, they believe that the Taylor rule will stayforever. Additionally, regardless of people not trusting the central bank, we assume that (ex-post) it
fulfills its promises.
This possibility is accommodated by simply modifying the equations in (16) to obtain,
Jt ≡ [A(p)tFt1 + (1 − pt )Ft+1 ) − B̃]−1 ,Ft ≡ Jt C,
Gt ≡ Jt [D − A(pt Gt1 + (1 − pt )Gt+1 )ρ],Ht ≡ Jt [−A(1 − pt )Ht+1 + I] for t ∈ [t0 , t1 − 2], Ht1 −1 ≡ (AFt1 − B̃)−1 .
Finally, to consider an unconventional policy in t0 , we just assign values at t0 for the shocks
affecting the different instruments in the central bank’s balance sheet. Additionally, the alternativeof undoing these policies is also considered by placing appropriate values for the sequence ut : if thechange is not pre-announced we just change the value in the corresponding period, and if the change
is pre-announced we can expand the exogenous vector et to consider anticipated shocks.

A.2

The Model

Here we present the equilibrium conditions of the model, as well as the calibration of the parameters.
It is a standard new Keynesian model for a small open economy with incomplete asset markets.
Households have separable preferences in consumption and leisure and can transfer resources overtime using an internationally traded bond, as well as the debt created by the central bank. They
consume a combination of domestic and foreign goods. There is an infinite number of monopolistic
intermediate good firms that use labor to produce using a constant returns-to-scale technology. They
are subject to a Calvo-type problem in setting prices, with full indexation to past inflation. The
35
Notice that if we want to consider to fix the interest rate not at zero but to any arbitrary value, this can be done bysetting an appropriate for the constant c.

18

equilibrium conditions associated with household optimization are,36
λt = c−σt ,λt = βi1,t Et(E.1)

wt = φ0 ltφ1 cσt ,!
BC /D BC ϕk
Dk,t
λt+1
k
BC /D BC
D1,t
1

πt+1

λt = βi∗t cpt EtSt+1 λt+1
St πt+1



Bt∗BC
∗BC
Bt−1

!ϕs

(E.2)
,

(E.3)

,

(E.4)


 η
η−1
 η−1
 η−1
1
1
H
F
η
η
η
η
.
ct = (1 − a) ct+ (a) ct
cFt = a pFt

−η

ct ,

H −η

cH
t = (1 − a) pt

(E.5)
(E.6)

ct .

(E.7)

i1,t
Mt P y −ϕm
)
.
=(
M
P
Pt yt Mit

(E.8)

ytH = zt lt − κ.

(E.9)

mct =

wt
,
pH
t zt

(E.10)

−ǫThose related to firm’s choices are37

ft1 = mct p̃H
t


H 1−ǫ

ft2 = p̃t

−ǫytH + θβEt





ytH

ǫ−1
ǫ


λ

t+1

 λt

+ θβEt


λ

πt
πt+1

t+1

 λt



πt
πt+1

p̃H
t
p̃H
t+1

1−ǫ

!−ǫ
p̃H
t
H
p̃t+1

ft1 = ft2 .

The conditions associated with rest of the world variables are F ∗
cpt = ψd e−pt bt+1 +b̄ − 1 ,St πt∗
pFt=.
St−1 πt
pFt−1

36

pH
t
pH
t+1
!1−ǫ

!−1−ǫ
pH
t
H
pt+1

1
ft+1

!−ǫ





,

2
ft+1





(E.11),(E.12)(E.13)(E.14)(E.15)

Variables without time subscripts denote steady state values. Prices have been rescaled by the price index. Recall
that in the case of (E.3), (E.4) and (E.8), which display the deviations from the standard model, these are imposed
ad-hoc over the usual equilibrium conditions and are not derived from microfoundations.
37
The assumption of full indexation to past inflation yields that, up to first order, the distortion that in principle
price stickiness introduces (i.e. a gap between production and absorption) is not relevant even if steady state inflation isdifferent from zero.

19

pH
t

cH∗
=
t

pft

!−η∗

yt∗ .

(E.16)

Market clearing conditions are
1=θpH
t−1 πt−1
pH
t πt

!1−ǫ

+ (1 − θ) p̃H
t

1−ǫ

.

(E.17)

H∗
ytH = cHt + ct ,(E.18)H∗
F F
yt = c t + p H
t ct − p t ct .

(E.19)H∗
F F
tbt = pHt ct − p t ct .

(E.20)

pFt b∗t+1 = pFtb∗t ∗i cpt−1 + tbt .
πt∗ t−1(E.21)

Finally, monetary policy is as described in the text.
We calibrate the parameters following the structural estimation for Chile performed by Medina
and Soto (2007), and the steady state ratios of the variables in CB balance sheet are taken from theaverage of the series used for estimation described in the text. The time period is meant to be aquarter. Table 1 presents the parameters and steady state variables that were calibrated. Finally, we
assume that the frictions disappear in steady state. This implies that we need to introduce a demand
for real balances in steady state, which we choose to be M/P = y(i)−µm .

A.3

σ
η
θ
ρi
i∗

1
1.12
0.74
0.74
1.0025l

0.2

Table 1: Calibration
φ1
0.84
a
ǫ
11
η∗
ψd
0.01
µm
απ
1.67
αy
i
1.016
π
D1CB +DkCB
M

1.8

DkCB
D1CB

0.34
0.79
0.2
0.39
1.00740.8

Estimation

Here we describe the details of the estimation of equations (4) to (6). Before presenting the results,
it is relevant to mention several data limitations. First, even though there is data available on the
total stock of bonds of the Central Bank of Chile (CBCh) on circulation, the data is not rich enoughto obtain a long series for the stock of bonds at different maturities. In addition, although nowadays
the CBCh has available a rich maturity structure for its bonds, this has not been always the case, andonly the two-years bond in pesos (BCP2) has a long enough series of interest rates.
P is measured by the monetary policy rate set by the CBCh, i
In term of equation (4), iM
1,tt
38
is the 90-days deposit rate, M is the monetary base, output is the monthly index of economic38

All rates are quarterly to be consistent with the calibration of the model.

20

activity (IMACEC) and the price level is the core consumer price index (IPCX1).39 The equation was
estimated by GMM, using as instruments lags one to five of the regressor, the J. P. Morgan Emerging
Markets Bond Index Global (EMBI) index, the Federal Funds Rate, the rate on a 5- years bond inpesos (BCP5) and the 90-days lending rate, and allowing also for an AR(1) error term.40 The result
obtained was,41
P
i1,t − iM= −0.0025 − 0.0014(mt − pt − yt ) + ut ,t
(0.0017)(0.0007)ut = 0.8056ut−1 + et ,(0.0416)2

R (Adj) = 0.587, J − stat = 0.175, Obs = 60.
For equation (5), the foreign interest rate is the Federal Funds Rate and the country premium ismeasured by the EMBI, the stock of government foreign assets Bt∗CB was measured the net foreign asset
element in CBCh balance sheet, and the three-month-ahead expectation of the nominal exchange rate
comes from the Expectations Survey collected by the CBCh. The equation was estimated by GMM
using the same instruments as in the previous equations and allowing also for an AR(1) error term.
The results were,st − Et {st+1 } − i∗t − cpt + i1,t = −0.0037 + 0.2047(∆b∗BC) + ut ,t(0.001)(0.036)ut = 0.2615ut−1 + et ,(0.0307)

R2 (Adj) = 0.014, J − stat = 0.115, Obs = 76.
Finally, for equation (6) the rate of nominal 2-year bonds was used as a measure for ik,t and the
expectations about the future short rate also comes from the Expectations Survey. Because, as weˆBCcommented before, there is no data on the stock of bonds at different maturities we measure dˆBCk,t − d1,t
as the difference between the total stock of bonds in pesos and the stock of 90-days commercial andconsumer deposits. The equation was estimated by GMM using the same instruments as in theprevious equations and allowing also for an AR(1) error term. The estimation yields,
ik,t −

k−1
X
j=0


BC
Et {i1,t+j } = −0.001 + 0.017 dBC
k,t − d1,t ut ,(0.001)(0.001)

ut = 0.2615ut−1 + et ,(0.0307)2

R (Adj) = 0.671, J − stat = 0.179, Obs = 76.

39

These two series have been seasonally adjusted. The source for all the series is the Central Bank of Chile. Variables
are measured in logs.
40
Adding lags of the dependent variable as regressors did not improve the goodness of fit.
41
Newey-West robust standard errors in parenthesis

21

Figures
Figure 1: Purchase of Foreign Assets Financed with a Monetary Expansion% Deviation from SS∆B ∗B C = ∆M ⇒ y∆B ∗B C = ∆M ⇒ π

4

7

3.5

6

10

5

8

4

6

3

4

2

2

0.5

1

0

0

0

−2

3

∆B ∗B C = ∆M ⇒ ∆S

12

2.5
2
1.5
1

−4

∆B ∗B C = ∆M ⇒ rer

6% Deviation from SS

B

x 10
2

5

0

4

−2

3

−4

2

−6

1

−8

∆B ∗B C = ∆M ⇒ i 1∆B ∗B C = ∆M ⇒ r

0
−0.2
−0.4
−0.6
−0.8

0

0

5

10
Quarters

15

−10

−1
−1.2

0

5

10
Quarters

15

−1.4

0

5

10

15

Quarters

Note: y is real GDP, π is inflation, ∆S is nominal depreciation, rer is the real exchange
rate, rt ≡ i1,t − Et πt+1 is the ex-ante real interest rate. All variables are in log-deviations
from steady state. The the shock is permanent increase in Bt∗CB equivalent to a 10% of
P remains in its steady state value forever. The
nominal GDP in steady state, while iMtsolid line is the benchmark case, while the dashed-dotted line is the case with ϕd = ϕs = 0.

22

Figure 2: Purchase of Debt Financed with a Monetary Expansion∆M ⇒ y4
3.5% Deviation from SS∆M ⇒ π

6

10

5

3

8

4

2.5

∆M ⇒ ∆S

12

6
2

3
4

1.5

2

2

1
0.5
0

0

0

−2

−4

∆M ⇒ rer

5

1∆M ⇒ i 1

x 10
2∆M ⇒ r

0

0

−0.2

−2

−0.4

−4

−0.6

−6

−0.8

−8

−1

−10

−1.2

% Deviation from SS

4

3

2

1

0

0

5

10
Quarters

15

−12

0

5

10
Quarters

15

−1.4

0

5

10

15

Quarters

Note: The solid line corresponds to a purchase of short debt, D1,t , and the dashed-dottedline is purchase of long debt, Dk,t . The the shock is permanent increase in Mt equivalent
P remains in its steady state value
to a 10% of nominal GDP in steady state, while iMtforever.

23

Figure 3: Purchase of Long Debt Financed with New Short Debt

% Deviation from SS∆D1 = −∆Dk ⇒ y
0.05

0.25

0.04

0.04

0.03

0.03

0.02

0.02

0.01

0.01

0

0

0

−0.05

−0.01

−0.01

0.2

∆D1 = −∆Dk ⇒ ∆S

0.05

0.15
0.1
0.05

−4

∆D1 = −∆Dk ⇒ rer

0.015

x 10
6

−3

∆D1 = −∆Dk ⇒ i 1

x 10
8∆D1 = −∆Dk ⇒ r

7

5

0.01
% Deviation from SS∆D1 = −∆Dk ⇒ π

0.3

6
4

5

0.005
3

4

0

3

2

2
−0.005

−0.01

1

0

5

10
Quarters

15

0

1
0

5

10
Quarters

15

0

0

5

10

15

Quarters

Note: The the shock is permanent increase in D1,t equivalent to a 10% of nominal GDP
P remains in its steady state value forever.
in steady state, while iMt24

Figure 4: Purchase of Foreign Assets Financed with new Debt
∆B ∗B C ⇒ y
0.15∆B ∗B C ⇒ π

0.5∆B ∗B C ⇒ ∆S

1

0.1
% Deviation from SS

0.5
0.05
0

0
0
−0.05

−0.5

−0.1
−1
−0.15
−0.2

−0.5

−4

∆B ∗B C ⇒ rer

0.6

−1.5

x 10
8

∆B ∗B C ⇒ i 1∆B ∗B C ⇒ r

0.6

0.5

0.5

% Deviation from SS

6
0.4

0.4
4

0.3
0.2

0.3
0.2

2

0.1

0.1
0

0
−0.1

0

0

5

10
Quarters

15

−2

0

5

10
Quarters

15

−0.1

0

5

10

15

Quarters

Note: The solid line corresponds to new short debt, D1,t , and the dashed-dotted line is
new long debt, Dk,t . The the shock is permanent increase in Bt∗CB equivalent to a 10% of
P remains in its steady state value forever.
nominal GDP in steady state, while iMt25

Figure 5: Purchase of Foreign Assets Financed with a Monetary Expansion. Different T ’s, pt = 0.

% Deviation from SS∆B ∗B C = ∆M ⇒ y∆B ∗B C = ∆M ⇒ πt

1

2

0.8

1.5

0.6

1

0.4

0.5

∆B ∗B C = ∆M ⇒ ∆S

4
3
2
1
0

0.2

0

0

0

5

10

15

−0.5

−1

0

5

15

−2

0

5

Quarters

Quarters∆B ∗B C = ∆M ⇒ rer

2

10

15

∆B ∗B C = ∆M ⇒ rt

∆B ∗B C = ∆M ⇒ iM P

0.025

10
Quarters

0.4% Deviation from SS

0.3
1.5

0.02

1

0.015

0.5

0.01

0

0.005

0.2
0.1
0
−0.1
−0.2
−0.5

0

5

10
Quarters

15

0

0

5

10
Quarters

15

−0.3

0

5

10

15

Quarters

Note: The solid, dashed-dotted, dashed and dotted lines correspond, respectively to T =2, 4, 6, 8. The the shock is permanent increase in Bt∗CB equivalent to a 10% of nominal
P remains in its steady state value until T .
GDP in steady state, while iMt26

Figure 6: Purchase of Foreign Assets Financed with a Monetary Expansion. T = 6, pt = pt , differentp’s.
∆B ∗B C = ∆M ⇒ y∆B ∗B C = ∆M ⇒ πt

0.5

1.5

0.4

1

∆B ∗B C = ∆M ⇒ ∆S

2.5

% Deviation from SS

2
1.5
0.3

0.5

0.2

0

0.1

−0.5

1
0.5
0
−0.5
−1

0

0

5

10

15

−1

0

5

Quarters∆B ∗B C = ∆M ⇒ rer

1.4

10

15

5

10

15

∆B ∗B C = ∆M ⇒ rt

∆B ∗B C = ∆M ⇒ iM P

0.025

0.3
0.25

0.02
% Deviation from SS

0

Quarters

1.2

0.2

1

0.15
0.015

0.8
0.6

0.1
0.05

0.01

0

0.4

−0.05

0.005
0.2
0

−1.5

Quarters

−0.1
0

5

10
Quarters

15

0

0

5

10
Quarters

15

−0.15

0

5

10

15

Quarters

Note: The solid, dashed-dotted, dashed and dotted lines correspond, respectively to p =0, 0.3, 0.6, 0.9. The the shock is permanent increase in Bt∗CB equivalent to a 10% of nominal
P remains in its steady state value until T .
GDP in steady state, while iMt27

Figure 7: Purchase of Long Debt Financed with New Short Debt. Different T ’s, pt = 0.
−∆Dk = ∆D1 ⇒ y
0.4−∆Dk = ∆D1 ⇒ π

0.2% Deviation from SS

0.35

0.3

0.3

0.15

0.25

0.25

0.2

0.2
0.1

0.15

0.05

0.05

0.15
0.1

0.1
0.05

0

0
−0.05

0

5

10

15

0

0

5

10

15

−0.05

0

−3

−∆Dk = ∆D1 ⇒ rer

0.12

2

MP
x 10 −∆Dk = ∆D1 ⇒ i

5

10

15

−∆Dk = ∆D1 ⇒ r

0.005

0.1
% Deviation from SS−∆Dk = ∆D1 ⇒ ∆S

0.35

0
1.5

0.08

−0.005

0.06

1

−0.01

0.04

−0.015
0.5

0.02
0

−0.02

0

5

10
Quarters

15

0

0

5

10
Quarters

15−0.025

0

5

10

15

Quarters

Note: The solid, dashed-dotted, dashed and dotted lines correspond, respectively to T =2, 4, 6, 8. The the shock is permanent increase in Bt∗CB equivalent to a 10% of nominal
P remains in its steady state value until T .
GDP in steady state, while iMt28

Figure 8: Purchase of Long Debt Financed with New Short Debt. T = 6, pt = pt , different p’s.
−∆Dk = ∆D1 ⇒ y
0.4−∆Dk = ∆D1 ⇒ π

0.16−∆Dk = ∆D1 ⇒ ∆S

0.3

0.14

0.25

% Deviation from SS

0.3
0.12

0.2

0.1

0.2

0.15
0.08
0.1

0.1

0.06
0.05

0.04

0

0

0.02
−0.1

0

5

10

15

1.6

0.08
% Deviation from SS

0

5

10

15

−0.05

−3

−∆Dk = ∆D1 ⇒ rer

0.09

0

x 10
2

10

15

−∆Dk = ∆D1 ⇒ r

0
−2

1.2

0.06

5

−3

MP
x 10 −∆Dk = ∆D1 ⇒ i

1.4

0.07

0

−4

1

0.05

−6
0.8

0.04

−8
0.6

0.03

−10

0.4

0.02
0.01

0.2

0

0

0

5

10
Quarters

15

−12
−14
0

5

10
Quarters

15

−16

0

5

10

15

Quarters

Note: The solid, dashed-dotted, dashed and dotted lines correspond, respectively to p =0, 0.3, 0.6, 0.9. The the shock is permanent increase in Bt∗CB equivalent to a 10% of nominal
P remains in its steady state value until T .
GDP in steady state, while iMt29

Figure 9: Purchase of Foreign Assets Financed with a Monetary Expansion, Pre-announced reversal,
different k’s∆B ∗B C = ∆M ⇒ y% Deviation from SS

0.05∆B ∗B C = ∆M ⇒ π

0.5∆B ∗B C = ∆M ⇒ ∆S

1

0

0.5

−0.05

0
0

−0.1

−0.5

−0.15

−1

−0.2

0

10

15

∆B ∗B C = ∆M ⇒ rer

0.2% Deviation from SS

5

−0.5

0

5

10

15

∆B ∗B C = ∆M ⇒ i M P

0.015

−1.5

0

5

10

15

∆B ∗B C = ∆M ⇒ r

0.5

0
0.01

−0.2

0
−0.4

0.005

−0.6
−0.8

0

% Deviation from SS

10

5

10
Quarters∆B ∗B C = ∆M ⇒ M

15

0

0

6

8

5

10
Quarters∆B ∗B C = ∆M ⇒ B ∗B C

15

−0.5

0

5

10

15

Quarters

4

6
2
4
0

2
0

0

5

10
Quarters

15

−2

0

5

10

15

Quarters

Note: The solid, dashed-dotted, dashed and dotted lines correspond, respectively to k =2, 3, 4, 5, with T = 6. The the shock is permanent increase in Bt∗CB equivalent to a 10%
P remains in itsof nominal GDP in steady state, that is fully reversed in T − k, while iMtsteady state value until T .

30

Figure 10: Purchase of Foreign Assets Financed with a Monetary Expansion, Pre-announced gradualreversal, different k’s∆B ∗B C = ∆M ⇒ y% Deviation from SS

0.2
0∆B ∗B C = ∆M ⇒ ∆S

1

0
0

−0.2
−0.5
−0.4

−1
−1

−0.6
−0.8

0

5
∆B

0.5
% Deviation from SS∆B ∗B C = ∆M ⇒ π

0.5

10

∗B C

15

= ∆M ⇒ rer

−1.5

0

5
∆B

0.03

10

∗B C

= ∆M ⇒ i

15

−2

0

MP

5
∆B

1

10

∗B C

15

= ∆M ⇒ r

0
0.02

0.5

0.01

0

−0.5
−1
−1.5

0
∆B

10
% Deviation from SS

5

10

15

Quarters= ∆M ⇒ M

0

∗B C

4

6

3

4

2

2

1
0

5

10
Quarters

15

0

5
∆B

5

8

0

0

0

∗B C

10

15

Quarters= ∆M ⇒ B ∗B C

5

10

−0.5

0

5

10

15

Quarters

15

Quarters

Note: The solid, dashed-dotted, dashed and dotted lines correspond, respectively to k =2, 3, 4, 5, with T = 6. The the shock is permanent increase in Bt∗CB equivalent to a 10% of
P remainsnominal GDP in steady state, that is reversed slowly startying at T − k, while iMtin its steady state value until T .

31

Figure 11: Purchase of Foreign Assets Financed with a Monetary Expansion, Pre-announced reversal,
pt = pt , different p’s∆B ∗B C = ∆M ⇒ y% Deviation from SS

0.1
0∆B ∗B C = ∆M ⇒ ∆S

2

0.5

1

0

0

−0.5

−1

−0.1
−0.2
−0.3
−0.4

0

5
∆B

0.5
% Deviation from SS∆B ∗B C = ∆M ⇒ π

1

10

∗B C

15

= ∆M ⇒ rer

−1

0

5
∆B

0.01

10

∗B C

= ∆M ⇒ i

15

−2

0

MP

5
∆B

0.5

10

∗B C

15

= ∆M ⇒ r

0.008
0

0.006
0
0.004

−0.5

0.002
−1

0
∆B

10
% Deviation from SS

5

10

15

Quarters= ∆M ⇒ M

0

0

10

15

Quarters∆B ∗B C = ∆M ⇒ B ∗B C

∗B C

6

8

5

−0.5

0

5

10

15

Quarters

4

6
2
4
0

2
0

0

5

10
Quarters

15

−2

0

5

10

15

Quarters

Note: The solid, dashed-dotted, dashed and dotted lines correspond, respectively to p =0, 0.3, 0.6, 0.9. The the shock is permanent increase in Bt∗CB equivalent to a 10% of nominalP
GDP in steady state, that is fully reversed in T − k, with T = 6 and k = 4, while iMtremains in its steady state value until T .

32

Figure 12: Purchase of Foreign Assets Financed with a Monetary Expansion, Unexpected reversal,
different k’s∆B ∗B C = ∆M ⇒ y% Deviation from SS

0.6
0.4

∆B ∗B C = ∆M ⇒ ∆S

3

1

2

0.5

1

0

0

−0.5

−1

0.2
0
−0.2

0

5

10

15

−1

0

5

10

15

−2

0

−3

∆B ∗B C = ∆M ⇒ rer

1.5% Deviation from SS∆B ∗B C = ∆M ⇒ π

1.50x 10 ∆B ∗B C = ∆M ⇒ i M P

1015∆B ∗B C = ∆M ⇒ r

0.4

1

5

0.2
−0.5

0.5

0
0

−1
−0.2

−0.5
−1

0

5

10

15

−1.5

0

5

∆B ∗B C = ∆M ⇒ M% Deviation from SS

10

10

15

Quarters

Quarters

0

5

10

15

Quarters∆B ∗B C = ∆M ⇒ B ∗B C

6

8

−0.4

4

6
2
4
0

2
0

0

5

10
Quarters

15

−2

0

5

10

15

Quarters

Note: The solid, dashed-dotted, dashed and dotted lines correspond, respectively to k =2, 3, 4, 5, with T = 6. The the shock is permanent increase in Bt∗CB equivalent to a 10% of
nominal GDP in steady state, that is fully reversed in T − k but is not anticipated, while
P remains in its steady state value until T .
iM
t

33

Figure 13: Purchase of Foreign Assets Financed with a Monetary Expansion, Unexpected gradualreversal, different k’s∆B ∗B C = ∆M ⇒ y% Deviation from SS

0.5

0

−0.5

0

% Deviation from SS

5

10

15

∆B ∗B C = ∆M ⇒ rer

2

1

2

0

0

−1

−2

−2

0

5

10

15

∆B ∗B C = ∆M ⇒ iM P

0.06

∆B ∗B C = ∆M ⇒ ∆S

4

−4

0

5

10

15

∆B ∗B C = ∆M ⇒ r

0.4

1

0.2
0.04

0

0
0.02

−1
−2

−0.2

0

5
∆B

10
% Deviation from SS∆B ∗B C = ∆M ⇒ π

2

10

15

Quarters= ∆M ⇒ M

0

∗B C

4

6

3

4

2

2

1
0

5

10
Quarters

15

0

5
∆B

5

8

0

0

0

∗B C

10

15

Quarters= ∆M ⇒ B ∗B C

5

10

−0.4

0

5

10

15

Quarters

15

Quarters

Note: The solid, dashed-dotted, dashed and dotted lines correspond, respectively to k =2, 3, 4, 5, with T = 6. The the shock is permanent increase in Bt∗CB equivalent to a 10%
of nominal GDP in steady state, that is reversed slowly startying at T − k but is not
P remains in its steady state value until T .
anticipated, while iMt34

Figure 14: Driving imp
to zero. Different T ’s, pt = 0.
t
y
20

15
% Deviation from SS

π

25
20

40

15

30

10

20

5

10

0

0

10

5

0

−5

0

5

10

15

−5

0

5

RER

20

10

15

−10

0

5

Quarters

Quarters

iM P

0

15

10

15

10

15

Quarters

r

1

−0.2
% Deviation from SS

∆S

50

0

−0.4

−1

−0.6
−2
10

−0.8
−3
−1

5

−4

−1.2

−5

−1.4
0

0

5

10
Quarters

15

−1.6

0

5

10
Quarters

15

−6

0

5
Quarters

Note: The solid, dashed-dotted, dashed and dotted lines correspond, to the response of
P to zero and keeping at this value until, respectively, T = 1, 2, 3, 4.
driving iMt35

Figure 15: Driving imp
to zero. T = 3, pt = pt , different p’s.
t
y
12

π

20

∆S

35
30

10
% Deviation from SS

15
25

8

20

10

6

15
4

5

10

2

5

0
0
−2

0
0

5

10

15

−5

0

5

15

0

5

RER

iM P

0

10

15

10

15

Quarters

r

1

−0.2
% Deviation from SS−5

Quarters

Quarters

15

10

0

−0.4
10

−1

−0.6
−0.8

−2

−1

5

−3

−1.2
−4

−1.4
0

0

5

10
Quarters

15

−1.6

0

5

10
Quarters

15

−5

0

5
Quarters

Note: The solid, dashed-dotted, dashed and dotted lines correspond, to the response of
P to zero and keeping at this value until T = 3, with p = pt and, respectively,
driving iMttp = 0, 0.3, 0.6, 0.9.

36Figure 16: Purchase of Foreign Assets Financed with a Monetary Expansion. T = 6, pt = 0. Robustness with respect to ϕm .
∆B ∗B C = ∆M ⇒ y
0.8∆B ∗B C = ∆M ⇒ π

2% Deviation from SS

0.7

3

1.5

0.6

2

0.5

1

0.4

1
0.5

0.3

0

0.2

0

−1

0.1
0

0

5

10

15

−0.5

0

5

Quarters

10

15

−2

0

5

Quarters∆B ∗B C = ∆M ⇒ rer

1.6

10

15

Quarters∆B ∗B C = ∆M ⇒ i M P

0.035

1.4
% Deviation from SS∆B ∗B C = ∆M ⇒ ∆S

4

∆B ∗B C = ∆M ⇒ r

0.5

0.03

0.4

0.025

0.3

0.8

0.02

0.2

0.6

0.015

0.1

0.01

0

0.005

−0.1

1.2
1

0.4
0.2
0
−0.2

0

5

10
Quarters

15

0

0

5

10
Quarters

15

−0.2

0

5

10

15

Quarters

Note: The solid line correspond to the benchmark case (ϕm = −0.0014), while the dasheddotted and dashed are the cases with ϕm = −0.0021 and ϕm = −0.0007, respectively. The
the shock is permanent increase in Bt∗CB equivalent to a 10% of nominal GDP in steadyP remains in its steady state value until T .
state, while iMt37

Figure 17: Purchase of Long Debt Financed with New Short Debt. T = 6, pt = 0. Robustness withrespect to ϕk .
−∆Dk = ∆D1 ⇒ y
0.6−∆Dk = ∆D1 ⇒ π

0.2% Deviation from SS

0.5

0.25
0.15

0.4

0.2

0.3

0.15
0.1

0.2

0.1

0.1

0.05

0.05

0
−0.1

0

0

5

10

15

0

0

5

Quarters

0.06

0.04

0.02

0

5

10
Quarters

15

−0.05

0

5

15

−3

MP
x 10 −∆Dk = ∆D1 ⇒ i

x 10
2

1.6

0

1.4

−2

1.2

−4

1

−6

0.8

−8

0.6

−10

0.4

−12

0.2

−14

0

0

5

10
Quarters

10

15

Quarters

−3

1.8

0.08

0

10
Quarters−∆Dk = ∆D1 ⇒ rer

0.1% Deviation from SS−∆Dk = ∆D1 ⇒ ∆S

0.3

15

−16

0

−∆Dk = ∆D1 ⇒ r

5

10

15

Quarters

Note: The solid line correspond to the benchmark case (ϕk = 0.017), while the dasheddotted and dashed are the cases with ϕm = 0.019 and ϕm = 0.015, respectively. The theshock is permanent increase in Bt∗CB equivalent to a 10% of nominal GDP in steady state,
P remains in its steady state value until T .
while iMt

38

Figure 18: Purchase of Foreign Assets Financed with a Monetary Expansion. T = 6, pt = 0. Robustness with respect to the Taylor rule.
∆B ∗B C = ∆M ⇒ y
0.5∆B ∗B C = ∆M ⇒ π

1.5∆B ∗B C = ∆M ⇒ ∆S

2.5
2

% Deviation from SS

0.4
1

1.5
1

0.3
0.5

0.5

0

−0.5

0.2

0

0.1

−1
0

0

5

10

15

−0.5

0

5

Quarters

15∆B ∗B C = ∆M ⇒ iM P

0.025

5

10

15

∆B ∗B C = ∆M ⇒ r

0.3
0.25

0.02
% Deviation from SS

0

Quarters

1

0.2

0.8

0.15
0.015

0.6
0.4

0.1
0.05

0.01

0

0.2
0.005

−0.05

0

−0.15

0
−0.2

−1.5

Quarters∆B ∗B C = ∆M ⇒ rer

1.2

10

−0.1
0

5

10
Quarters

15

0

5

10
Quarters

15

0

5

10

15

Quarters

Note: The solid line correspond to the benchmark case (ρi = 0.74, απ = 1.67 and απ =0.39), while the dashed-dotted, dashed and dotted are the cases with ρi = 0.37, απ = 3.39
and απ = 0.78, respectively. The the shock is permanent increase in Bt∗CB equivalent to a
P remains in its steady state value until T .
10% of nominal GDP in steady state, while iMt39

Figure 19: Purchase of Long Debt Financed with New Short Debt. T = 6, pt = 0. Robustness withrespect to the Taylor rule.
−∆Dk = ∆D1 ⇒ y
0.4−∆Dk = ∆D1 ⇒ π

0.2% Deviation from SS

0.35

0.25

0.3

0.15

0.2

0.25

0.15

0.2
0.1
0.15

0.1

0.1

0.05

0.05

0.05

0

0
−0.05

0

5

10

15

0

0

5

Quarters

10

15

−0.05

−3

2.5

−3

MP
x 10 −∆Dk = ∆D1 ⇒ i

x 10
5

2

0

0.06

1.5

−5

0.04

1

−10

0.02

0.5

−15

0

5

10
Quarters

5

15

0

0

5

10
Quarters

10

15

Quarters

0.08

0

0

Quarters−∆Dk = ∆D1 ⇒ rer

0.1% Deviation from SS−∆Dk = ∆D1 ⇒ ∆S

0.3

15

−20

0

−∆Dk = ∆D1 ⇒ r

5

10

15

Quarters

Note: The solid line correspond to the benchmark case (ρi = 0.74, απ = 1.67 and απ =0.39), while the dashed-dotted, dashed and dotted are the cases with ρi = 0.37, απ = 3.39
and απ = 0.78, respectively. The the shock is permanent increase in Bt∗CB equivalent to a
P remains in its steady state value until T .
10% of nominal GDP in steady state, while iMt40

Figure 20: Driving imp
to zero. T = 3, pt = 0. Robustness with respect to the Taylor rule.
t
y
0.12

π

0.2

0.3

0.1
% Deviation from SS

0.15

0.25

0.08

0.2

0.1

0.06

0.15
0.04

0.05

0.1

0.02

0.05

0
0
−0.02

0
0

5

10

15

−0.05

0

5

rer

0.16

−0.0020.12−0.0040.1−0.0060.08−0.008

0.06

−0.01

0.04

−0.0120.02−0.0140

5

10Quarters

15

0

5

15

−0.016

10

15

10

15

Quarters

iM P

0

0.14

0

10

−0.05

Quarters

Quarters% Deviation from SS

∆S

0.35

r

0.01
0
−0.01
−0.02
−0.03
−0.04

0

5

10
Quarters

15

−0.05

0

5
Quarters

Note: The solid line correspond to the benchmark case (ρi = 0.74, απ = 1.67 and απ =0.39), while the dashed-dotted, dashed and dotted are the cases with ρi = 0.37, απ = 3.39
P to zero and to keep it at thisand απ = 0.78, respectively. The experiment is to drive iMtvalue until T = 3, with pt = 0.

41