Foreign exchange intervention and inflation targeting: The role of credibility

Foreign exchange intervention and inflation targeting: The role of credibility

Journal of Economic Dynamics & Control 106 (2019) 103716 Contents lists available at ScienceDirect Journal of Economic Dynamics & Control journal ho...

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Journal of Economic Dynamics & Control 106 (2019) 103716

Contents lists available at ScienceDirect

Journal of Economic Dynamics & Control journal homepage: www.elsevier.com/locate/jedc

Foreign exchange intervention and inflation targeting: The role of credibilityR Gustavo Adler a, Ruy Lama a,∗, Juan Pablo Medina b a b

International Monetary Fund, United States Business School, Universidad Adolfo Ibáñez, Chile

a r t i c l e

i n f o

Article history: Received 1 August 2018 Revised 2 July 2019 Accepted 2 July 2019 Available online 22 July 2019 JEL classification: E58 F31 F41 Keywords: Foreign exchange intervention Small open economy Learning Imperfect credibility Inflation targeting

a b s t r a c t We develop a small open economy model where the central bank operates under a flexible inflation targeting regime, i.e., monetary policy is aimed at stabilizing output and inflation. In this theoretical framework, we analyze to what extent foreign exchange intervention (FXI) can contribute to the central bank goals for different degrees of credibility. We find two key results. First, in a baseline scenario where the central bank is perfectly credible, FXI can improve macroeconomic outcomes by successfully stabilizing both output and inflation in response to foreign disturbances. Second, when central bank lacks credibility, FXI policies entail a trade-off by reducing output volatility at the expense of inducing higher inflation volatility. In this scenario, FXI policies prevent the central bank from achieving the goal of price stability. These results suggest that FXI is more likely to support an inflation targeting regime when the credibility of the central bank is high. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Over the last decade, many countries adopted inflation targeting (IT) as their monetary policy framework with great success in reducing inflation rates to single digits and in improving monetary policy credibility. While IT regimes are characterized for placing price stability as the overarching goal of monetary policy, in practice, central banks pursue additional short-run objectives such as output or exchange rate stabilization.1 Moreover, central banks in several emerging economies became actively involved in foreign exchange intervention (FXI) within an inflation targeting regime.2 In this context, a key question faced by policymakers is to what extent an active use of foreign exchange reserves is desirable or even compatible with the goal of price stability.

R This work has benefited from comments and discussions with Maury Obstfeld, Jonathan Ostry, Steve Phillips, Yossi Yakhin and other IMF colleagues. Remaining errors are ours. The views expressed in this paper are those of the authors and do not necessarily represent the views of the IMF, its Executive Board, or IMF management. ∗ Corresponding author. E-mail addresses: [email protected] (G. Adler), [email protected] (R. Lama), [email protected] (J.P. Medina). 1 In practice, central banks follow flexible IT regimes (Svensson, 2009), where monetary authorities might focus on stabilizing not only inflation but additional targets such as output, exchange rate, or financial stability. 2 See Ghosh et al. (2016).

https://doi.org/10.1016/j.jedc.2019.07.002 0165-1889/© 2019 Elsevier B.V. All rights reserved.

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G. Adler, R. Lama and J.P. Medina / Journal of Economic Dynamics & Control 106 (2019) 103716

Volatility of Inflation Expectations (1-year ahead, Percent)

18

1.8

1.6

CPI Inflation (YoY, Percent)

High Credibility Regimes

16 14

1.4 Low Credibility Regimes

1.2

12

1

10

0.8

8

0.6

6

0.4

4

0.2

2 0 Jan-02

0 Jan-02 Dec-03 Nov-05 Oct-07 Sep-09 Aug-11 Jul-13

Dec-03 Nov-05 Oct-07 Sep-09 Aug-11

Deviation from Inflation Target (Percent)

Detrended Foreign Exchange Reserves (Linear trend, Percent of GDP)

6

3

Jul-13

5 2 4 1 3 2

0

1 -1 0 -2 -1

-2 Jan-02 Dec-03 Nov-05 Oct-07 Sep-09 Aug-11 Jul-13

-3 Jan-02 Sep-03 May-05 Jan-07 Sep-08 May-10 Jan-12 Sep-13

Fig. 1. Empirical role of credibility in inflation targeting regimes.1 Source: Consensus Forecast, IFS, IMF Staff calculations. 1/ Inflation Targeting countries with a volatility of inflation expectations larger than the median are defined as “Volatile Expectations Regimes” and those with a volatility smaller than the median are defined as “Stable Expectations Regimes”. The countries in the volatile expectations sample are: Brazil, Czech Republic, Hungary, Indonesia, Phillipines, Romania, Thailand, Turkey and the UK. The countries in the stable expectations sample are: Canada, Chile, Colombia, Korea, Mexico, Norway, Peru, Poland, and Sweden. All variables displayed in the chart are a weighted average of each sample, where the weights are constructed using PPP GDP. The sample of inflation targeting countries is taken from Hammond (2012).

The starting point in our analysis is the role played by credibility in determining the effectiveness of macroeconomic policies. In countries where the central bank displays limited credibility, monetary policy tends to be less effective in achieving price stability.3 Furthermore, in a context of imperfect monetary policy credibility, other policies, such as FXI, might also display limited effectiveness in terms macroeconomic stabilization as inflation expectations are not fully anchored. Fig. 1

3 See Erceg and Levin (2003) for an analysis of the disinflation process of 1980s in the US, and De Michelis and Iacoviello (2016) for an assessment of reflationary policies implemented in recent years in Japan.

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illustrates these points by showing one measure of imperfect credibility— the volatility of inflation expectations— in a sample of IT regimes in both advanced and emerging economies.4 While during the 20 0 0s, the vast majority of IT regimes were capable of anchoring inflation expectations, reflected in a lower volatility of inflation expectations, there are large differences between countries with high and low credibility (blue and red line, respectively). In the panel chart we observe that, on average, low credibility countries tend to have a higher inflation rate and, more importantly, exhibit larger and more persistent deviations from the inflation target. Moreover, low credibility countries tend to intervene more actively and frequently in the foreign exchange market. This particular feature of the data, the active use of FX reserves under imperfect credibility, motivates our analysis for understanding the macroeconomic implications of FXI under different degrees of credibility. With the goal of understanding the interaction between FXI and imperfect credibility, we develop a small open economy model with sticky prices á la (Calvo, 1983) and real wage rigidities as in Shimer (2012). As discussed by Blanchard and Gali (2007), the combination of price and wage rigidities breaks the “divine coincidence” existing in a model with only sticky prices. In a canonical New Keynesian model, inflation stabilization implies an output gap stabilization. On the contrary, under price and wage rigidities, the central bank needs to trade-off the objectives of output and inflation stabilization when setting the policy rate.5 Furthermore, under this trade-off, having two policy instruments instead of one will improve the trade-off by simultaneously stabilizing output and inflation. In our model we assume that the instruments available to the central bank are the short-term interest rate and FX reserves. The short-term interest rate will follow a Taylor-type rule, whereas the FX reserve will be chosen to ensure a minimization of the loss function, i.e., the sum of output and inflation volatility. The model assumes imperfect asset substitutability as in Chang et al. (2015), allowing sterilized FX intervention to have real effects in the economy through a portfolio balance channel.6 The model also allows for imperfect monetary policy credibility as in Erceg and Levin (2003), where households gradually learn about the true weights of the arguments in the Taylor-type rule.7 In response to shocks and policy actions, households gradually learn according to a signal extraction problem the systematic policy behavior of the central bank. We analyze the macroeconomic effects of FX intervention in response to a reduction in foreign interest rates. This scenario captures the external environment for many emerging economies in the aftermath of the global financial crisis, where global interest rates declined as advanced economies implemented quantitative easing policies. This reduction in the foreign interest rate results in a currency appreciation and a reduction of output and inflation in the absence of FX intervention policies. Then, we analyze the role of FX intervention in stabilizing output and inflation both under perfect and imperfect monetary policy credibility. Under perfect credibility, FX intervention can successfully stabilize output and inflation. Under imperfect credibility, since households are unable to distinguish the type of rule implemented by the central bank, the same magnitude of FX intervention can stabilize output via an exchange rate depreciation but generates a substantial increase in inflation relative to the perfect credibility case. Our results indicate that FX intervention can be useful for stabilizing output both under perfect and imperfect credibility. However, under a scenario of imperfect credibility, FXI amplifies inflation volatility. The policy implication of this result is that in countries where central banks are starting to build credibility, FX intervention has to be moderated and properly calibrated in order to ensure price stability. On the contrary, highly credible central banks can deploy FXI policies to stabilize output in a way that is compatible with a stable inflation rate. This paper is related to two strands of the literature: FX intervention and imperfect credibility. Related to the first one, we follow Ghosh et al. (2016), Benes et al. (2015), Canzoneri and Cumby (2014), and Liu and Spiegel (2015), by developing a small open economy model where FX intervention plays an important role as a macroeconomic stabilization tool. Related to the second one, we follow Evans and Honkapohja (2001), Erceg and Levin (2003), De Michelis and Iacoviello (2016), and Lemoine and Lindé (2016), introducing imperfect credibility as a learning process in the formation of expectations. Our main contribution to the existing literature is to quantitatively explore the role of FXI under learning and imperfect credibility. This is a relevant issue for many developing and emerging economies that resort to FXI policies in an environment of limited credibility. The remainder of the paper is organized as follows. Section 2 lays out the small open economy model and describes the calibration strategy. Section 3 discusses the simulations results of FXI under perfect and imperfect credibility. Section 4 concludes. 2. A two-sector small open economy model We develop a small open economy model featuring nominal and real rigidities along the lines of Christiano et al. (2005), Smets and Wouters (2007), Adolfson et al. (2008), and Altig et al. (2011). There are two sectors in the economy: a tradable 4

We rely on the sample of IT countries proposed by Hammond (2012). Implementing a Taylor-type rule that responds to output and inflation deviations is a plausible approach to decide on this policy trade-off. However, a calibrated interest rate rule does not necessarily coincides with the one that minimizes a conventional loss function. 6 The focus of our paper is on the role of sterilized FX interventions. However, in the paper we use interchangeably the terms sterilized FX intervention and FX intervention. 7 In the rest of the paper we interchangeably use the terms imperfect credibility and imperfect monetary policy credibility. 5

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(T) and non-tradable (N). We assume sticky prices in the non-tradable sector whereas the law of one price holds for goods produced in the tradable sector. We introduce real wage rigidities as in Blanchard and Gali (2007) and Shimer (2012). Finally, domestic and foreign bonds are assumed to be imperfect substitutes as in Chang et al. (2015), allowing FX intervention to have real effects in the economy through a portfolio balance channel. 2.1. Households Preferences of the representative household are defined over consumption and labor:



Ut = Et

∞ 



β u(Ct+i , Lt+i ) , i

(1)

i=0

where β ∈ (0, 1) is the subjective discount factor, Lt denotes labor effort and Ct consumption. Households have access to two types of assets: one-period non-contingent foreign and domestic bonds, Bt∗ and Bt . The budget constraint is given by:



 





∗ ∗ PtF Ct + Bt − Et Bt∗ = Bt−1 (1 + it−1 ) − Et Bt−1 1 + it−1  Bt−1 + Wt lt + t + Tt ,

(2)

where PtF is the price of final goods, it is the domestic interest rate, Et is the nominal exchange rate, Wt is the nominal wage, t are the profits generated by firms, and Tt are lump-sum transfers from the central bank. The return on foreign bonds is ∗ given by the foreign interest rate it∗ and a risk premium (Bt−1 ) which is a function of the aggregate stock of foreign debt ∗ ∗ ) and the aggregate stock of foreign debt (B∗ ) as given Bt−1 .8 The representative household takes all prices (PtF , Et , it−1 , it−1 t−1 ∗

to choose Ct , Bt , and Bt∗ to maximize (1) subject to (2).9 By introducing the foreign risk premium, (Bt−1 ), domestic and foreign bonds become imperfect substitutes, allowing sterilized FX intervention to have real effects in the economy through a portfolio balance channel.10 2.2. Real wage rigidities Real wages are sticky as in Blanchard and Gali (2007) and Shimer (2012), and are assumed to evolve according to the following adjustment process:

Wt = PtF



Wt−1 F Pt−1

χw

(wt∗ )1−χw ,

(3) u

where the parameter χw ∈ [0, 1] determines the degree of inertia in real wages and wt∗ = − uL,t is the equilibrium real wage C,t

determined by the household’s marginal rate of substitution between consumption and leisure. Eq. (3) states that the real wage gradually adjusts to the equilibrium real wage. When χw = 0, the real wage corresponds to the equilibrium under flexible wages. 2.3. Firms There are four types of firms in the economy: final good producers, intermediate good producers, retailers, and capital producers. Next, we describe the optimization problem for each of these firms. 2.3.1. Final good producers Producers of final goods (YtF ) combine tradable intermediate inputs (YtDT ) and non-tradable intermediate inputs (YtDN ) according to a constant elasticity of substitution production function:

YtF =



ηY ηY −1 ηY −1 η −1 Y αY1/ηY (YtDT ) ηY + (1 − αY )1/ηY (YtDN ) ηY ,

(4)

where α Y and ηY are the share of tradable inputs and the elasticity of substitution between tradable and non-tradable inputs, respectively. Producers choose the optimal combination of tradable and non-tradable intermediate inputs in order to maximize profits, taking prices as given.

8 Following Schmitt-Grohé and Uribe (2003) and Chang et al. (2015), the representative household does not internalize the effects of external borrowing on the risk premium. ∗ 9 Notice that in equilibrium, the aggregate stock of foreign debt (Bt ) will be equal to the value chosen by the representative household (Bt∗ ). 10 In Section 2.6 we discuss in more detail the transmission mechanism of FX intervention.

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From the profit-maximizing condition, the price of the final good is given by:

PtF =

 1−η  1−ηY 1−1ηY Y αY PtT + (1 − αY ) PtN ,

(5)

where PtT and PtN are the price of tradable and non-tradable inputs, respectively. 2.3.2. Intermediate good producers Firms produce homogeneous intermediate tradable (YtT ) and non-tradable goods (YtN ) in a competitive market. The production function in each sector J = T , N, is given by:

 ηJ  N 1−ηJ

Yt J = AtJ KtJ J

Lt

J

,

(6)

J

where At , Kt , and Lt , denote aggregate productivity, capital, and labor inputs in each sector, respectively. The intermediate good producers choose the optimal combination of labor and capital to maximize their profits, taking as given the wage rate and the rental rate of capital. 2.3.3. Retailers in the non-tradable sector Firms in the retail sector sell non-tradable goods (YtDN ) in two separate stages. First, an assembler combines differentiated intermediate non-tradable goods indexed by j ∈ [0, 1] to produce YtDN . The technology is a constant elasticity of substitution function given by:

 YtDN

=

1 0

YtDN

( j)

N −1 N

  N−1 N

,

dj

(7)

where  N is the elasticity of substitution between a variety of goods. The resulting demand for the jth intermediate nontradable good is:



( j) =

YtDN

PtN ( j ) PtN

 −N

YtDN .

(8)

From the zero-profit condition we obtain the aggregate price of non-tradable goods:

 PtN

=

1

0

PtN

( j)

 1−1

N

1 −N

dj

.

(9)

Second, retailers purchase homogenous non-tradable intermediate goods and differentiate it into a continuum of goods. Each retailer sets their prices on a staggered basis as in Calvo (1983). Every period a fraction (1 − θN ) of retailers set their prices optimally while the remaining fraction are unable to change prices. The optimal price PtN∗ chosen by each retailer maximizes the expected present value of profits:



∞ 

Et



( θN )

i

N∗ t ,t +i Pt



WN Pt+ i



N Yt+ i



( j) ,

(10)

i=0

F ) and PW N is the wholesale price of the where t ,t +i is the stochastic discount factor defined as t ,t +i = β i (Ct /Ct+i )(PtF /Pt+ t i intermediate non-tradable good. The aggregate price of non-tradable goods evolves according to:

PtN =

 1−ε  1−ε p 1−1ε p p N θ Pt−1 + (1 − θ ) PtN∗ .

(11)

2.3.4. Capital producers Capital is sector-specific and there are firms designated to produce and rent capital to the intermediate good producers in the tradable and non-tradable sectors. The aggregate investment goods of each type of capital is a composite of tradable and non-tradable goods as in the case of the final good. The representative firm producing for a sector J = {T , N } solves the following problem:



J

Vt = max Et J Kt+ ,I J i t+i

∞ 



(

J J t ,t +i RK,t+i Kt+i



J F Pt+ i It+i

) ,

i=0

subject to the law of motion of physical capital:



J Kt+1 = (1 − δ )KtJ + S

ItJ J It−1



ItJ ,

(12)

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where Vt the present discounted value of profits, δ is the depreciation rate of capital in sector J, and S(.) is the adjustment cost function for investment.11 J

2.4. Monetary and foreign exchange reserves policies In the small open economy model, the central bank follows policy rules for the short-term interest rate (it ) and the stock of foreign exchange reserves (Ft∗ ). We quantitatively evaluate the role of FXI under the scenarios of monetary policy with perfect and imperfect credibility. Under perfect credibility, the interest rate rule followed by the central bank is fully observed by households and the macroeconomics outcomes will reflect a rational expectations equilibrium. Under imperfect credibility, the monetary rule perceived by the households is different from the one announced by the central bank, and the outcomes of the policy will be ultimately determined by a process of macroeconomic learning. We can interpret the scenario of imperfect credibility as a case of “fear of floating” prevalent in many emerging where the central bank announces that will allow the exchange rate to fluctuate, but in practice adjust their policies to reduce exchange rate volatility.12 2.4.1. Types of monetary policy rules We consider two types of monetary rules. Under the type 1 rule, the policy rate responds to detrended output and the deviations of inflation from its target. Under the type 2 rule, the policy rate not only responds to output and inflation deviations, but also to the foreign interest rate.13 Formally, the type 1 Taylor-type rule is defined as:



1 + it 1+i





=

1 + it−1 1+i

ψi  (1−ψ )ψy  (1−ψ )ψ i π i π Y t

t

π

Y

exp(εmp,t ),

(13)

where it , Yt , π t are the nominal interest rate, output, and the inflation rate, respectively. The parameter ψ i determines the degree of interest rate smoothing, and ψ y and ψ π denote the weights for output and inflation stabilization. The last term, εmp,t , is an i.i.d. shock, with mean 0 and standard deviation σ mp , that reflects transitory deviations of the interest rate from the policy rule. Alternatively, the type 2 monetary policy rule is defined as:



1 + it 1+i



 =

1 + it−1 1+i

ψi  ψy  ψ  ψ ∗ (1−ψi ) Y π π 1 + i∗ i t

Y

t

t

π

1 + i∗

exp(εmp,t ),

(14)

where the specification and coefficients are the same as in the monetary policy of type 1 in equation (13), except for the reaction to the foreign interest rate (ψi∗ > 0). One implication of the type 2 policy rule is that as the domestic interest rate follows the foreign interest rates, exchange rate fluctuations will be attenuated. Intuitively, as the domestic rate tracks the foreign rate, the expected depreciation implied from the uncovered interest rate parity condition will be smaller, resulting in lower exchange rate volatility. As discussed by Taylor (2014), this rule is consistent with the empirical characterization of the monetary policy in emerging economies where central banks attempt to stabilize the exchange.14 2.4.2. Perfect and imperfect monetary policy credibility Under perfect credibility monetary policy follows the type 1 policy rule (13), which is directly observed by households, and the outcomes are consistent with a rational equilibrium equilibrium. In contrast, when there is imperfect credibility, we assume that the central bank also implements the type 1 policy rule (13), but households assign a positive probability that in fact the central bank is implementing the type 2 rule (14). In this scenario, households observe the dynamics of the macroeconomic variables, and they gradually learn about the true arguments of the policy rule through a Bayesian updating process. This learning process is modelled as in Erceg and Levin (2003), where agents learn about the rule implemented by the central bank based on the log-deviations of the observed interest rate relative to systematic part of the type 1 policy rule (devmp, t ):



devmp,t = ln

1 + it 1+i





− ψi ln

1 + it−1 1+i



− ( 1 − ψi )

 π   Y  ψy ln t + ψπ ln t π Y

(15)

where devmp, t provides an imperfect signal of the deviations from the systematic policy behavior of the central bank of type 1. Observed deviations can be explained either by a monetary policy shock (ε mp,t ) or by the fact that the central bank is 11 Investment adjustment costs, as in Christiano et al. (2005), satisfy the following conditions: S (1 ) = 1, S (1 ) = 0, S (1 ) = −μS < 0. This assumption generates inertia in investment that is consistent with a time-to-build specification. 12 See Calvo and Reinhart (2002). 13 As it will be explained later, including the foreign interest rate in the Taylor-type rule empirically captures the behavior of central banks that systematically attempt to reduce exchange rate volatility by adjusting the monetary policy rate. 14 Empirically, there is a strong association between domestic and foreign interest rates in small open economies. Taylor (2014) argues that this correlation reflects the concern of central banks for exchange rate stabilization. For empirical estimates on the impact of foreign interest rates on policy rates see Clarida et al. (1998), Rey (2015), Caputo and Herrera (2017) and Gray (2012).

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following the type 2 policy rule. Households assign probabilities to each type of central bank (pr1,t and pr2,t , ) in order to forecast the expected path of the short-term interest rate and use Bayesian inference to update their beliefs about these probabilities and the policy shock over time.15 According to this inference process, households can optimally predict the future path of the monetary policy rate based on the available public information.16 For the implementation of the Bayesian inference, the vector of state variables is defined as:

 ξt = pr1,t

εmp,t

pr2,t



(16)

Upon observing devmp, t , private agents infer the vector ξ t using the Kalman filter:

 ξt |t = pr1,t |t

pr2,t |t

εmp,t |t



(17)

where prj,t|t corresponds to the probability of the central banks being of type j in period t, based on the information available up to t; and ε mp,t|t is the inferred monetary policy shock based on the same information set.17 2.4.3. Foreign exchange intervention In our simulation, we consider two cases for the foreign exchange intervention (FXI). In one case we simply assume no ∗ FXI, i.e., (Ft∗ = F ), and the FX reserves are held constant. In the other case, the foreign exchange intervention policy follows a rule, which is a function of the foreign interest rate:18

Ft∗ F





=

1 + it∗

−θi∗

1 + i∗

,

(18)

where Ft∗ is the stock of foreign exchange reserves, it∗ the international interest rate, and θi∗ governs the intensity of FX ∗ interventions. F and i∗ are the steady state values of the foreign exchange reserves and the foreign interest rate, respectively. This rule captures the rationale of a FX reserves policy that “leans against the wind”, meaning that an accumulation of FX reserves by the central bank takes place during episodes of capital inflows, i.e., when international interest rates are low and the real exchange rate appreciates.19 The central bank’s budget constraint is given by: ∗ ∗ Et Ft∗ − Bt = Et Ft−1 (1 + it−1 ) − Bt−1 (1 + it−1 ) − Tt ,

(19) Et Ft∗

Sterilized FX interventions are conducted by issuing Bt units of domestic bonds and purchasing units of risk-free ∗ (1 + i∗ ) on the stock of reserves foreign assets (Et Ft∗ = Bt ). Each period the central bank earns interest income Et Ft−1 t−1 from the previous period, and pays Bt−1 (1 + it−1 ) to domestic bond holders. Profits or losses are rebated to households through lump-sum transfers Tt . 2.5. Market clearing conditions In each period, markets for labor, capital, domestic and international bonds, intermediate and final goods clear. The market clearing condition for labor is given by:

Lt = LtN + LtT ,

(20)

The market clearing condition for non-tradable goods is:

YtDN tN = YtN ,

(21)

tN

where captures a deadweight loss term of price dispersion of retailers in the non-tradable sector. The aggregate domestic demand for final goods satisfies:

YtF = Ct + ItT + ItN .

(22)

15 This process is similar to the one proposed by Erceg and Levin (2003) to model imperfect credibility in the context of quantifying the costs disinflationary processes. These authors propose a signal-extraction problem as in Evans and Honkapohja (2001), where households optimally learn based on the state of the economy and observed policy actions. 16 The initial prior probability of being type 1 is set to pr1,0|0 = 0.5. Of course, this choice is important for the numerical simulation. We use an intermediate value of the two polar cases. If pr1,0|0 = 1 the simulation coincides with the case of perfect credibility since the prior and posterior probabilities assigned by households to the monetary policy type 1 will be always equal to 1. In contrast, when pr1,0|0 = 0 households will not learn that the true monetary policy is type 1 since they do not assign any probability to that possibility. Hence, our choice gives a moderate and transitory role for the problem of imperfect credibility of monetary policy. 17 The Kalman filter provides the optimal inference process about the unobservable states in linear models. See Appendix B for a detailed description of the simulations under imperfect credibility with Bayesian learning. 18 Alternative specifications of the FX reserves rule give quantitatively similar results to the baseline policy rule stated in Eq. (18). This is due to the fact that the simulations are conducted under the assumption that only foreign interest rate disturbances are operating. 19 See Ghosh et al. (2016).

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Fig. 2. Transmission mechanism of foreign exchange intervention.

The law of one price holds for tradable goods:

PtT = Et Pt∗ ,

(23)

Pt∗

where is the price of the tradable goods in foreign currency. Combining the households and government budget constraints, we obtain the balance of payment equation that describes the dynamics of the external position:









∗ ∗ ∗ ∗ Et (Ft∗ − Bt∗ ) = (1 + it−1 ) Et Ft−1 −  Bt−1 Et Bt−1 + PtT YtT − PtT YtDT .

(24)

Notice that the return on FX reserves does not include a risk premium, since we assume that the reserves are invested in risk-free foreign bonds. 2.6. Transmission mechanism of foreign exchange intervention As in Chang et al. (2015), the key margin that governs the transmission mechanism of FXI is the degree of asset substi ∗ tution between domestic and foreign bonds, which in our model is determined by the endogenous risk premium  Bt−1 . As indicated in Eq. (19), an accumulation of FX reserves is financed by increasing the supply of domestic bonds which are purchased by households. In the case of perfect asset substitution ( = 1), households will respond to this excess of supply of bonds by borrowing from the rest of the world, fully offsetting the impact of FX reserves accumulation.20 Since the net foreign asset position of the economy (FX reserves minus foreign debt) is unchanged, both the exchange rate and the current account will remain constant. This is consistent with result of “Wallace neutrality” which states that open market operations are neutral in the absence of financial frictions.21   ∗ When households face an endogenous risk premium  Bt−1 , the increase in households foreign debt will induce a higher risk premium that prevents a full rebalancing of the portfolio. The higher risk premium will increase the effective cost of borrowing in foreign bonds such that households will borrow from abroad less than in the case of perfect asset substitution. As a result, the increase in foreign debt does not fully offset the increase in FX reserves, implying an increase in the net foreign asset position of the economy. This, in turn, generates a current account improvement and an exchange rate depreciation. As it will be discussed in the calibration section, the larger the degree of asset substitution the stronger the transmission mechanism of FXI. Fig. 2 summarizes the transmission mechanism of FXI under perfect and imperfect asset substitution. 2.7. Calibration The model is calibrated at a quarterly frequency for a prototypical small open economy. Consistent with an annual real interest rate of 4%, we set β = 0.99. Household preferences are represented by the functional form:

u(Ct , Lt ) = log(Ct ) − ϕ

(Lt )1+ν , 1+ν

(25)

20 In equilibrium, households will borrow from the rest of the world to offset the excess supply of bonds since their desired savings rate remains unchanged as long as their net wealth is constant. 21 See Wallace (1981) and Curdia and Woodford (2011).

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9

Table 1 Baseline calibration. Parameter

Value

Description

β

0.99 0.60 0.50 0.50 6 0.75 0.875 0.30 0.40 0.02 2.5 0.7 1.5 0.5 1 9.7 0.2 0.95

Discount factor Labor supply elasticity Share of tradable inputs – final good sector Elasticity of substitution – final good sector Elasticity of substitution – non-tradable sector Calvo parameter –non-tradable sector Wage rigidity parameter Capital share –non-tradable sector Capital share – tradable goods sector Depreciation rate Investment adjustment cost Interest rate smoothing coefficient – Taylor Rule Inflation coefficient –Taylor Rule Output coefficient – Taylor Rule Foreign interest rate coefficient – Taylor Rule Foreign interest rate coefficient – FXI Rule Foreign risk premium elasticity Persistence of foreign interest rate

1/ν

αY ηY εN θN χw ηN ηT δ μS ψi ψπ ψy ψi∗ θi∗ ϱ

ρi∗

where the inverse of the Frisch elasticity of the labor supply is set to ν = 5/3. The share of tradable goods in the final goods basket is 50% (αY = 0.5) whereas the elasticity of substitution between tradable and non-tradable goods is ηY = 0.5. The elasticity of substitution among differentiated intermediate non-tradable goods is such the average markup in that sector is 20% (N = 6). Consistent with the standard parametrization of New-Keynesian models, we set the frequency of price adjustment to four quarters (θN = 0.75). Real wage rigidities are set to χw = 0.875, which is consistent with a half-life duration of wage adjustment of 5 quarters. The elasticity of the investment adjustment cost is μS = 2.5 consistent with the parametrization in Christiano et al. (2005). A critical parameter in the model is the risk premium elasticity of the foreign debt ( = ( /)B∗ ). This elasticity controls the strength of the sterilized FX intervention in the model economy by introducing portfolio rigidities in the stock of household’s foreign bond holdings. We calibrate ϱ based on the empirical evidence of Bayoumi et al. (2015), who find that an increase of 1% of GDP in the stock of foreign reserves improves the current account balance by 0.4% of GDP. Consistent with this evidence we set  = 0.2. The foreign interest rate coefficient ψi∗ in the monetary policy rule (14) is set to 1 based on the empirical evidence from Caputo and Herrera (2017), who estimate a Taylor-type rules in a panel of 23 advanced and emerging economies.22 The source of fluctuations in the model are shocks to the foreign interest rate, which follow an AR(1) process with persistence ρi∗ = 0.95. The interest rate rule is calibrated to standard parameter values (ψi = 0.7, ψπ = 1.5 and ψy = 0.5).23 Finally, the elasticity of FX reserves to the foreign interest rate θi∗ is chosen to minimize a loss function based on output and inflation volatility (L = var (yt ) + var (πt )).24 , 25 Table 1 summarizes the parameter values for the baseline calibration.

3. Foreign exchange intervention and the role of credibility In this section we analyze the macroeconomic effects of FXI under perfect and imperfect credibility. For both cases, we consider a scenario of a transitory but persistent decline of one percent in foreign interest rates which captures an episode of capital inflows. The role of FX reserves in this context is to stabilize output and inflation in response to an environment of low foreign interest rates. This section is organized as follows. First, we analyze separately the model dynamics under perfect and imperfect credibility. We then explore the gains of macroeconomic stabilization of FXI, measured by the second moments of output and inflation, under perfect and imperfect credibility. Finally, we present a robustness analysis.

22 Notice that in our simulations under imperfect credibility we assume that the central bank is of type 1, but households perceive with a positive probability that the central bank could be of type 1 or 2. The uncertainty regarding the type of central bank gives rise to the problem of imperfect credibility. As a result, the transmission mechanism of implementing the rule of type 1 under perfect credibility will be different from the case of imperfect credibility. 23 Notice that the calibrated output coefficient in our Taylor-type rule (ψy = 0.5) is higher than the one considered in the original Taylor rule specification (ψy = 0.5/4 = 0.125). However, these two parameter values yield quantitatively similar results when the policy rule features interest rate smoothing. 24 We have also computed the optimal value of θi∗ based on maximizing the household’s welfare, obtaining quantitatively similar results. 25 We conducted additional simulations, available upon request, where we explore alternative rules for the case of imperfect credibility. The main results related to the macroeconomic impact of FXI under imperfect credibility remain as long as the weight on the foreign interest rate argument in the type 2 policy rule (the alternative rule anticipated by households with a positive probability) is higher than in the type 1 policy rule (the rule announced by the central bank).

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3.1. Perfect credibility Under perfect credibility the central bank follows the policy rules (13) and (18) for setting the short-term interest rate and foreign exchange reserves policy. Fig. 3 shows the model dynamics in response to a transitory reduction of foreign interest rates under perfect credibility and for the cases of FXI and constant foreign exchange reserves (No FXI). Under constant FX reserves, the only policy response to the foreign interest rate shock is the adjustment of the short-term interest rate according to the Taylor-type rule. In the absence of FXI (black line), a lower foreign interest rates appreciates the real exchange rate, generating a reallocation of resources from the tradable to the non-tradable sector, a decline in the current account balance, and a transitory reduction in the inflation rate. Furthermore, there is a decline in GDP as the reduction in tradable output is greater than

No FXI RES/GDP

Percent

5

0

0

-0.5 5

Percent

10

15

20

Inflation rate

1

5

0.5

0.1

0

0

-0.5

-0.1

10

15

20

15

20

GDP

0.2

-1

-0.2 5

10

15

20

RER

1

Percent

Real interest rate

0.5

-5

5

0.5

0.5

0

0

-0.5

-0.5

10

CA/GDP

1

-1

-1 5

10

15

20

Tradable GDP

1

Percent

FXI

5

10

15

20

Non-tradable GDP

0.5

0.5 0

0

-0.5 -1

-0.5 5

10

15

Quarters

20

5

10

15

20

Quarters

Fig. 3. Effects of a reduction in the foreign interest rate under perfect credibility. No FXI vs FXI.

G. Adler, R. Lama and J.P. Medina / Journal of Economic Dynamics & Control 106 (2019) 103716

Type 1 RES/GDP

Percent

0.2

-1 5

Percent

10

15

20

Inflation rate

2

5

10

15

20

15

20

GDP

1

0

0

-1

-2 5

10

15

5

20

RER

1 Percent

Real interest rate

0

-0.2

0

10

CA/GDP

1

0

-1

-1 5

10

15

20

Tradable GDP

1 Percent

Type 2 1

0

11

5

15

20

Non-tradable GDP

2

0

10

0

-1

-2 5

10 15 Quarters

20

5

10 15 Quarters

20

Fig. 4. Comparing the effects of a reduction in the foreign interest rate for monetary policy type 1 and 2 under perfect credibility.

the increase in non-tradable production.26 In response to lower output and inflation, the central bank lowers the nominal interest rate according to the policy rule (13), resulting in reduction of the real interest rate that partially offsets the contractionary impact of the external shock. Notice that in our baseline calibration the monetary stimulus induced by the policy rule is not strong enough to fully stabilize the economy. A policy of FXI in this scenario (blue line) engineers an exchange rate depreciation through a portfolio balance channel that limits the reallocation of factors across sectors, resulting in a stabilization of output, the current account, and inflation. Note that most macroeconomic variables are stabilized with an FXI operation of about 2 percent of GDP. 3.2. Imperfect credibility Under imperfect credibility, the central bank also follows the policy rules (13) and (18), however in this case since the monetary policy rule is not credible, households learn about the central bank’s interest rate rule as in Erceg and Levin (2003).27

26 Notice that the effect of interest rate shocks on GDP depends ultimately on the dynamics of the tradable and non-tradable sectors. While a lower foreign interest rate appreciates the exchange rate and lowers tradable output, it also boosts consumption and investment, resulting in an expansion of domestic demand and non-tradable output. The final impact of GDP will depend on the relative strength of these two opposing forces. For a discussion on the impact of capital inflows on GDP see Blanchard and Gali (2007). 27 Recall that a detailed explanation of the learning process can be find in Section 2.4.2. and Appendix B.

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G. Adler, R. Lama and J.P. Medina / Journal of Economic Dynamics & Control 106 (2019) 103716

To build intuition on the transmission mechanism of foreign disturbances under imperfect credibility, we first compare the responses of the type 1 and 2 monetary policy rules assuming perfect credibility. In practical terms, the case of imperfect credibility analyzed in this section can be interpreted as the combination of the outcomes under the type 1 and 2 policy rules. Fig. 4 presents the responses without FXI comparing the type 1 and 2 policy rules under perfect credibility. Notice that the type 2 rule, relative to the type 1, prevents an exchange rate appreciation with a higher level of inflation. This is a reflection of a lower real interest rate path and a more depreciated real exchange rate relative to the outcomes under the type 1 rule. The latter helps to cushion the fall in tradable output, but with a larger expansion of non-tradable output and a higher inflation rate.

Perfect Credibility RES/GDP

0.1

Imperfect Credibility

0.5

0

0

-0.05

-0.5

Percent

0.05

-0.1

-1 5

Percent

10

15

20

Inflation rate

1 0.5

0.1

0

0

-0.5

-0.1

10

15

20

15

20

GDP

-0.2 5

10

15

20

RER

1

Percent

5 0.2

-1

5

0.5

0.5

0

0

-0.5

-0.5

10

CA/GDP

1

-1

-1 5

10

15

20

Tradable GDP

1

Percent

Real interest rate

1

5

0.5

0

0

-0.5

-0.5

-1

15

20

Non-tradable GDP

1

0.5

10

-1 5

10

15

Quarters

20

5

10

15

20

Quarters

Fig. 5. Role of credibility in shaping the effects of a reduction in the foreign interest rate without FXI.

G. Adler, R. Lama and J.P. Medina / Journal of Economic Dynamics & Control 106 (2019) 103716

No FXI RES/GDP

Percent

5

-1 5

Percent

10

15

20

Inflation rate

1

5

10

15

20

15

20

GDP

0.5

0

0

-0.5

-1 5

10

15

5

20

RER

0.5

Percent

Real interest rate

0

-5

10

CA/GDP

1

0

0

-0.5

-1 5

10

15

20

Tradable GDP

1

Percent

FXI 1

0

13

5

15

20

Non-tradable GDP

1

0

10

0

-1

-1 5

10

15

Quarters

20

5

10

15

20

Quarters

Fig. 6. Role of credibility in shaping the effects of a reduction in the foreign interest rate with FXI.

Fig. 5 compares the model dynamics under perfect (black line) and imperfect (red line) credibility in the absence of FXI, i.e., the central bank only relies on the interest rate rule for macroeconomic stabilization purposes. This allows us to better understand how imperfect credibility affects the model dynamics in response to a foreign interest rate shock. The key channel through which imperfect credibility operates in the economy is through higher expected inflation. Even though the true weights of the monetary policy rule are captured by Eq. (13), the lack of credibility implies that with a positive probability households anticipate a central bank of type 2 is implementing the policy rule (14). The type 2 monetary policy rule implies a lower interest rate in the future, as the policy rate responds systematically to the foreign interest rates. Consequently, in anticipation of a more accommodative monetary policy stance, inflation expectations increase. This implies that in equilibrium the real interest rate will be lower. Relative to the case of perfect credibility, the lower real interest results in higher inflation and GDP on impact, and a more moderate real exchange rate appreciation. Fig. 6 shows the impulse response function under FXI and imperfect credibility. The case of constant FX reserves under imperfect credibility (red line) is the same simulation shown in Fig. 5. FXI under imperfect credibility (green line) generates a large real exchange rate depreciation that boosts tradable production and an improvement of the current account balance.28 While FXI under imperfect credibility can largely stabilize GDP, this is achieved at the cost of an increase in the inflation rate of 1% point. The large response in the inflation rate is explained by the lack of credibility of monetary policy. As the central bank intervenes in the foreign exchange market, it engineers an exchange rate depreciation that stimulates output and inflation. Since households assign a positive probability to the fact that the central bank is of type 2 (i.e., the possibility that the central bank is concerned about minimizing deviations of the policy rate with respect to the foreign interest 28

For comparability, we consider the same magnitude of FXI used in the case of perfect credibility case (2 percent of GDP).

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G. Adler, R. Lama and J.P. Medina / Journal of Economic Dynamics & Control 106 (2019) 103716

A. No FXI

1

B. FXI

1 Probb type 1 Probb type 2

Probb type 1 Probb type 2

0.8

Probabilities

Probabilities

0.8

0.6

0.4

0.2

0.6

0.4

0.2

0

0 0

5

10

15

20

25

0

0.12

5

10

15

25

0.12 MP shocks

MP shocks

0.1

0.1

0.08

0.08

Percent

Percent

20

0.06

0.06

0.04

0.04

0.02

0.02

0

0 0

5

10

15

20

25

Quarters

0

5

10

15

20

25

Quarters

Fig. 7. Inferred probabilities of types of monetary policy rules and inferred discretionary policy shocks.

rate), the implication of imperfect credibility is a more accommodative monetary policy stance after FXI is implemented, resulting in lower real interest rates and higher inflation. To complete the analysis of imperfect credibility, Fig. 7 shows the inferred probabilities of type 1 and 2 under imperfect credibility as well as the inferred monetary policy shocks. We notice that agents gradually learn over time the true type of monetary policy rule. At the same time, the inferred monetary policy shock shows that agents eventually realize that the monetary policy response is part of the systematic component of the rule and not the discretionary one. 3.3. Stabilization gains derived from foreign exchange intervention In this subsection we evaluate the stabilization gains from FXI over the business cycle by analyzing the second moments of output and inflation. We simulate shocks to the foreign interest rate and compute the unconditional standard deviations of output and inflation under two monetary regimes (perfect and imperfect credibility) and two policy scenarios (No FXI and FXI). Figure 8 (Panel I) summarizes the results in terms of the output and inflation volatility space, measured by the standard deviation of both variables. Panel A depicts the regime of perfect credibility, where points A and B describe the scenario of no FXI and FXI, respectively. Consistent with impulse response functions presented in Fig. 3, we observe that FXI can simultaneously reduce output and inflation volatility under perfect credibility. The simulation highlights the result that under perfect credibility, FXI represents a highly effective macroeconomic stabilization policy tool capable of reducing overall volatility. On the contrary, panel B illustrates the limitations of FXI under imperfect credibility. Points C and D correspond to the cases of no FXI and FXI, respectively. Under the regime of imperfect credibility, FXI is capable of reducing output volatility, but at the expense of higher inflation volatility. Consistent with Fig. 6, FXI under imperfect credibility amplifies inflation volatility over the business cycle. Overall, these results are consistent with the empirical evidence shown in Fig. 1, which illustrates that active FX intervention in an environment of imperfect credibility is associated with higher inflation volatility.

G. Adler, R. Lama and J.P. Medina / Journal of Economic Dynamics & Control 106 (2019) 103716

I. Foreign interest rate shocks

II. Foreign demand shocks

III. Domesc tradable producvity shocks

Fig. 8. Responses to a sequence of two foreign interest rate shocks and gains in credibility. No FXI versus FXI.

15

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G. Adler, R. Lama and J.P. Medina / Journal of Economic Dynamics & Control 106 (2019) 103716

No FXI

FXI

Percent

RES/GDP

Real interest rate

5

1

0

0

-5

-1 5

10

15

20

5

10

Percent

Inflation rate 1

0.5

0

0

-1 10

15

20

5

10

RER

15

20

CA/GDP

0.5

1

0

0

Percent

20

-0.5 5

-0.5

-1 5

10

15

20

5

Tradable GDP Percent

15

GDP

10

15

20

Non-tradable GDP

1

1

0

0

-1

-1 5

10 15 Quarters

20

5

10 15 Quarters

20

Fig. 9. Output and inflation volatility with different shocks.

3.4. Robustness analysis In this subsection, we analyze the robustness of our main results. First, we evaluate how the volatilities of inflation and output computed in Fig. 8 (Panel I) vary with changes in key parameter values of the model. Second, we analyze how the effects of FXI under imperfect credibility are modified when the small open economy faces a sequence of shocks and the credibility of the central bank improves over time. Third, we extend the model to include additional external and domestic shocks to explore how our results depend on the underlying shocks hitting the economy. Tables 2–5 in the Appendix A present the standard deviations of output and inflation for different parameter values related to: (i) price stickiness (θ N ); (ii) real wage rigidities (χ w ); (iii) persistency of foreign interest rate shock (ρi∗ ); and (iv) the elasticity of the risk premium (ϱ). For the range of parameter values considered, we find consistently that FXI reduces both output and inflation volatility under perfect credibility, whereas under imperfect credibility FXI reduces output volatility while increases inflation volatility. Hence, we find that our main conclusions remain robust to changes in key parameter values. We also explore the implications of learning over time in response to a sequence of foreign interest rate shocks. As shown in Fig. 7, the probability that agents assign to type 1 rule increases over time, and after 16 quarters the credibility is substantially higher since agents give a probability close to 1 to the type 1 rule. To explore the implications of this learning process in the trade-off of using FXI under imperfect credibility, we assume that a second negative foreign interest rate shock occurs after the 16th quarter. For this second shock, we use the posterior probability inferred for agents in quarter 16 to implement the Bayesian learning going forward. The result of this simulation is shown in Fig. 9, where we compare the case without FXI and with FXI. Since the credibility is substantially higher when the second shock hits the economy,

G. Adler, R. Lama and J.P. Medina / Journal of Economic Dynamics & Control 106 (2019) 103716

17

the additional effects of using FXI closely resembles the case of perfect credibility. Hence, FXI is more effective for reducing overall macroeconomic volatility when the second foreign interest rate shock materializes. Finally, we evaluate the robustness of our results to alternative shocks, extending the model to the cases of a foreign demand and domestic tradable productivity shocks.29 In order to implement the case of imperfect credibility, we proceed in a similar fashion as in the case of the foreign interest rate shock. In particular, we assume that there are two types of monetary policy rules. The type 1 rule is the same as equation (13), but the type 2 rule reacts to the specific shock hitting the economy:



1 + it 1+i



 =

1 + it−1 1+i

ψi  ψy  ψ  ψ (1−ψi ) Y π π z z t

Y

t

π

t

z

exp(εmp,t ),

(26)

where zt refers to the specific shock incorporated into the model. The magnitude of the coefficient ψ z is calibrated such that the shock zt has the same impact on the exchange rate as the foreign interest rate shock under the interest rate rule (14). Similarly, as in the baseline model, an active FXI rule will be function of the specific shock analyzed zt :

Ft∗ F



 =

exp(zt ) exp(z )

θz ,

(27)

Also, for each type of shock, we compute the optimal magnitude of the FXI under perfect credibility, parameter θ z , in order to minimize the same loss function based on the sum of the volatilities of output and inflation. Fig. 8 (Panels II and III) reproduce the same plots of macroeconomic volatility in Fig. 8 (Panel I) for the cases of foreign demand shocks and domestic tradable productivity shocks, respectively. Although the magnitudes of the volatilities are different under these two alternative shocks, our main results remain robust. Under perfect credibility, FXI can stabilize inflation and minimize the loss function, whereas under imperfect credibility FXI policies lead to higher inflation volatility.

4. Conclusions In this paper we studied the macroeconomic effects of FX intervention under perfect and imperfect credibility. We find two key results. First, under perfect credibility FXI unambiguously improves macroeconomic outcomes by stabilizing output and inflation in response to foreign interest rate shocks. Second, under imperfect credibility, FXI remains effective in stabilizing output but at the expense of inducing higher inflation volatility. These results suggest that the stabilization benefits from relying on FXI are greater for credible central banks. Going back to our initial motivation presented in Fig. 1, our model illustrates a mechanism in which active FXI policies can increase inflation volatility. While there might be merits in implementing FXI even in a environment of low monetary policy credibility (e.g., Dutch Disease considerations or balance sheet effects), those additional objectives could make more difficult for the central bank to achieve the goal of price stability. Central banks might be able to reap the full macroeconomic stabilization benefits derived from FXI policies by first improving their credibility.

Appendix A. Sensitivity Analysis

Table 2 Sensitivity to price rigidities (θ N ).

σy Regime θ N Perfect credibility No FXI FXI Imperfect redibility No FXI FXI

29

See Appendix C for more details of this model extension.

σπ

Low 0.66

Base 0.75

High 0.875

Low 0.66

Base 0.75

High 0.875

0.26 0.11

0.27 0.11

0.29 0.08

0.86 0.11

0.83 0.11

0.79 0.09

0.40 0.17

0.40 0.17

0.34 0.22

0.70 1.34

0.66 1.25

0.47 1.00

18

G. Adler, R. Lama and J.P. Medina / Journal of Economic Dynamics & Control 106 (2019) 103716 Table 3 Sensitivity to wage rigidities (χ w ).

σy Regime χ w Perfect credibility No FXI FXI Imperfect redibility No FXI FXI

σπ

Low 0.66

Base 0.875

High 0.95

Low 0.66

Base 0.875

High 0.95

0.24 0.11

0.27 0.11

0.36 0.13

0.74 0.12

0.83 0.11

0.93 0.13

0.23 0.18

0.40 0.17

0.54 0.22

0.67 1.22

0.66 1.25

0.67 1.30

Table 4 Sensitivity to persistence of capital flows (ρi∗ ).

σy Regime ρi∗ Perfect credibility No FXI FXI Imperfect credibility No FXI FXI

σπ

Low 0.7

Base 0.95

High 0.98

Low 0.7

Base 0.95

High 0.98

0.21 0.15

0.27 0.11

0.33 0.22

0.68 0.27

0.83 0.11

0.89 0.18

0.23 0.05

0.40 0.17

0.49 0.24

0.41 0.42

0.66 1.25

0.75 1.49

Table 5 Sensitivity to portfolio balance channel (ϱ).

σy Regime ϱ Perfect credibility No FXI FXI Imperfect credibility No FXI FXI

σπ

Low 0.1

Base 0.2

High 0.3

Low 0.1

Base 0.2

High 0.3

0.36 0.19

0.27 0.11

0.23 0.16

1.10 0.56

0.83 0.11

0.70 0.34

0.50 0.33

0.40 0.17

0.35 0.13

0.50 0.77

0.66 1.25

0.76 1.55

Appendix B. Simulation Under Imperfect Credibility In this appendix we describe the simulation under imperfect credibility, based on the log-linear approximation of the equilibrium conditions of the model. In particular, conditional of being a type j central bank ( j = 1, 2), the solution of the model can be written as:

 Xt = P Xt−1 + Qi∗ , j it∗ + Qmp εmp,t

(28)

where  Xt is a vector containing all endogenous variables,  it∗ is the corresponding log-deviation of the foreign interest rate, and ε mp,t is the monetary policy shock. P, Qi∗ , j , and Qmp are a matrix and two column vectors, respectively, which are functions of the structural parameters of the model. Using this notation, the dynamics of the model under perfect credibility are characterized by (28) for the case j = 1. In this case, the private sector infers the probabilities of the two types of central bank and the size of the monetary policy shock in response to a foreign interest rate shock:





 Xt = P Xt−1 + pr1,t |t Qi∗ ,1 + pr2,t |t Qi∗ ,2  it∗ + Qmp εmp,t |t

(29)

where prj,t|t and ε mp,t|t are the Bayesian inference of the probability of being type j and the size of the monetary policy shock. Using the Kalman filter, this inference is updated following:



⎤ ⎡ pr1,t |t it∗ ρi ∗ ⎣ pr i∗ ⎦ = ⎣ 0 2,t |t t

εmp,t |t

0

0

ρi ∗ 0

⎤   ∗ pr1,t −1|t −1 it−1 ∗  it∗ − ( pr1,t −1,t −1 + pr2,t −1|t −1 )ρi∗ it−1 ∗ ⎦+K 0⎦⎣ pr2,t −1|t −1 , it−1 g ∗ devmp,t + (1 − ψi )ψi∗ pr2,t −1|t −1 ρi∗ it−1 0 εmp,t −1|t −1 0

⎤⎡

where Kg is the Kalman gain matrix. When the monetary authority follows rule (13) under imperfect credibility, devmp,t = 0, ∀t. However, the credibility problem prevents immediate full inference ( pr1,t |t = 1 and pr2,t |t = εmp,t |t = 0) by the private sector, and the latter only learns gradually over time the true type of the central bank.

G. Adler, R. Lama and J.P. Medina / Journal of Economic Dynamics & Control 106 (2019) 103716

19

To obtain the Kalman gain matrix we define the matrices Fξ , Qξ , and Hξ as follows:30

⎡ ρi ∗ ⎣ Fξ = 0  Hξ =

0

0



0⎦, Qξ = ⎣

ρi ∗

0



0

0

pr1,0|0 σi2∗ 0

1

0

0

( 1 − ψi )ψi∗

1

0

pr2,0|0 σi2∗

0

0



1

0 0

σ

⎤ ⎦

2 mp

where prj,0|0 is the prior probability of being monetary authority type j and σi2∗ is the variance of the innovations in the foreign interest rate. Thus, Kg is obtained iterating the following process: 1. Initialization: obtain 0 as the solution of 0 = Fξ 0 Fξ + Qξ

2. Iteration: Given t−1 compute Kg,t and t as:



Kg,t = Fξ t−1 Hξ Hξ t−1 Hξ

−1

    t = Fξ − Kg,t Hξ t−1 Fξ − Hξ Kg,t + Qξ 3. Iterate over step 2 until difference between t−1 and t is close to zero. Appendix C. Model extension with additional shocks In this appendix, we present the model extension that incorporates a standard foreign economy block and domestic tradable shocks. The foreign economy is modelled as Galí (2008). In log-linear form, the foreign block is characterized by the following equations: ∗ ∗ yt∗ = Et [yt+1 ] − σ ∗ (it∗ − Et [πt+1 ] ) + σ ∗ (g∗t − g∗t+1 )

(30)

 1   ∗ ∗ ∗ ∗ ∗ πt∗ = β ∗ Et [πt+1 ] + κ∗ + ν y − ( 1 + ν ) a t t σ∗ ∗ ∗ ∗ ∗ ∗ it = ψπ πt + ψy yt

(31) (32)

g∗t = ρg∗ g∗t−1 + εg∗,t

(33)

yt∗

πt∗

it∗

g∗t

at∗

where is foreign output, is foreign interest rate, is foreign inflation, is a foreign demand shock, and is a foreign productivity shock. Equation (30) is the IS curve, (31) is the New-Keynesian Phillips curve, and (32) is the monetary policy rule. β ∗ is the discount factor of foreign households, σ ∗ is the elasticity of intertemporal substitution, ν ∗ is the inverse of elasticity of the foreign labor supply, κ ∗ is the slope the Phillips curve in the foreign economy, ψπ∗ and ψy∗ are the reaction coefficient to inflation and output in the monetary policy rule. For the domestic tradable sector productivity shock, we assume that it follows an autoregressive process of order one: T log(AtT ) = (1 − ρaT ) log(AT ) + ρaT log(At−1 ) + εaT ,t

(34)

To implement the imperfect credibility in monetary policy in the case of these additional shocks, we proceed in the same way as with the base model. In particular, for shock zt = { g∗t , log(AtT )}, we consider that central bank type 2 has the following rule for setting the interest rate:



1 + it 1+i



 =

1 + it−1 1+i

ψi  ψy  ψ  ψz (1−ψi ) Yt πt π exp(zt ) exp(εmp,t ), π exp(z ) Y

(35)

Also, with FXI, its rule is modelled as

Ft∗ F





=

exp(zt ) exp(z )

θz

,

(36)

where zt is the alternative shock analyzed.31 30

For further details on the algorithm see Hamilton (1994). Notice that the interest rate and FX intervention rules depend on the relevant exogenous shock zt . Consistent with the work conducted by Erceg and Levin (2003), De Michelis and Iacoviello (2016) and Lemoine and Lindé (2016), the learning process derived from the Kalman filter is conducted only relative to the exogenous processes. This setup is not designed for modeling a learning process on endogenous variables, such as the foreign interest rate when we consider a structural foreign economy block (Eqs. (30)–(33)). For this reason, when we consider alternative shocks, we directly incorporate the exogenous process zt in the FXI and interest rate rules. We then calibrate the policy response to the exogenous shock (ψ z ) to ensure that the type 2 interest rate rule captures a scenario of fear of floating, i.e., the central bank attempts to smooth exchange rate fluctuations in response to that particular shock. 31

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