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Monetary Policy and Uncertainty in an Empirical Small Open Economy Model

Alejandro Justiniano Federal Reserve Bank of Chicagoy Bruce Preston Columbia University and NBERz

March 14, 2008

Abstract This paper explores optimal policy design in an estimated model of three small open economies: Australia, Canada and New Zealand. Within a class of generalized Taylor rules, we show that to stabilize a weighted objective of output, consumer price in? ation and nominal interest variation optimal policy does not respond to the nominal exchange. This is despite the presence of local currency pricing and due, in large part, to observed exchange rate disconnect in these economies. Optimal policies that account for the uncertainty of model estimates, as captured by the parameters’ posterior distrbution, similarly exhibit a lack of exchange rate response. In contrast to Brainard (1967), the presence of parameter uncertainty can lead to more or less aggressive policy responses, depending on the model at hand.

This is a signi…cantly revised version of a paper circulated under the title “Small Open Economy DSGE Models: Speci…cation, Estimation and Model Fit” The authors thank Steve Durlauf, Marc Giannoni, Thomas . Lubik, Adrian Pagan, Frank Schorfheide, two anonymous referees, seminar participants at the Australian National University, Columbia University, the Federal Reserve Bank of Atlanta, and participants at the joint CAMA and Reserve Bank of New Zealand conference on “Macroeconometrics and Model Uncertainty” and especially our discussant Domenico Giannone. The usual caveat applies. The views expressed in this paper are those of the authors’and should not be interpreted as re? ecting the views of the Federal Reserve Bank of Chicago or any person associated with the Federal Reserve System. y Federal Reserve Bank of Chicago, Economic Research, 230 South LaSalle St., Chicago, IL 60604. E-mail: ajustiniano@frbchi.org. z Department of Economics, Columbia University, 420 West 118th St. New York, NY 10027. E-mail: bp2121@columbia.edu.

1

1

Introduction

Recent theoretical analyses have emphasized the importance of pricing to market assumptions for optimal exchange rate policy, monetary policy and macroeconomic dynamics. Whether a country has producer currency pricing or local currency pricing can give rise to rather di¤erent policy recommendations, even when the sole of objective of policy is to stabilize the aggregate in? ation rate. For instance, Devereux and Engel (2003) show in a two country model with local currency pricing that optimal monetary policy stipulates stabilization of the nominal exchange rate. Similarly, Monacelli (2005) shows that local currency pricing induces a tradeo¤ in stabilizing aggregate price in? ation and the output gap that is not present when the law of one price holds. Despite these theoretical contributions there has been relatively little work on policy evaluation in empirical small open economy models. This paper seeks to …ll this gap by exploring optimal policy design within an estimated structural model using data for Australia, Canada and New Zealand. Of particular interest is whether policies in a class of generalized Taylor rule optimally respond to exchange rate variations as predicted by theory. Moreover, we assess the consequence of various sources of model uncertainty for the design of optimal monetary policy. To our knowledge, this is the …rst such study in a fully estimated small open economy model.1 The analysis is pursued using generalizations of the small open economy framework proposed by Gali and Monacelli (2005) and Monacelli (2005), in which a small and large country each specialize in the production of a continuum of goods subject to imperfect competition and price rigidities.2 Following the latter, imports are subject to local currency pricing (through what could be considered a retail sector providing distribution services) giving rise to deviations from the law of one price. We depart from their framework, by considering incomplete asset markets, the addition of other rigidities — such as indexation and habit formation —

Levin, Onatski, Williams, and Williams (2005) pursue a similar analysis for the closed economy case. The model is technically a semi-small open economy model, as domestic goods producers have some market power. The model shall nonetheless be referred to as a small open economy. Note also that our analysis appeals to an earlier interpretation of the Gali and Monacelli (2005) of a small-large country pair, rather than as an analysis of a continuum of small open economies.

2 1

1

as well as a large set of disturbances which have been found crucial in taking closed economy models to the data as documented by, inter alia, Christiano, Eichenbaum, and Evans (2005) and Smets and Wouters (2003). Using the empirical model, the optimal policy rule within a generalized class of Taylortype rule is determined to minimize a weighted objective function in the variance of aggregate consumer price in? ation, output and interest rates, subject to the constraints imposed by the estimated model. The Taylor rule posits that nominal interest rates are adjusted in response to output, output growth, in? ation, nominal exchange rate growth, and past interest rates. Optimization occurs subject to two di¤erent assumptions about the central bank’ knowledge s of the economy. First, policy is determined assuming estimated model parameters are known with certainty to the policymaker. Second, we consider optimal policy that results from taking into account all uncertainty regarding model parameters by using the posterior distribution of our estimates. This is rendered feasible by adopting a Bayesian approach to inference. The central insights from our analysis are as follows. First, we …nd that optimal policies do not respond to the nominal exchange rate. This is true regardless of whether parameter uncertainty is taken into account or not. Furthermore, this result is robust to a wide range of weight combinations for the components of the loss function; to the set of observables used to estimate the model; and the precise shocks included in estimation. This …nding contrasts with Smets and Wouters (2002) which provides evidence that optimal policies stipulate a response to exchange rate variations. The …nding that it is not optimal to respond to exchange rate variation can be sourced to speci…c properties of the empirical model. There exists a “disconnect” between nominal exchange rate movements and the evolution of domestic series. Indeed, cost-push and risk premium shocks account for between 69 and 84 percent of variation in the exchange rate, while accounting for a substantially lower share of the variation in output, interest rates and aggregate in? ation across these three small open economies. Active stabilization of the nominal exchange rate exacerbates variability in output, in? ation and nominal interest rates by connecting the evolution of these series more tightly to cost-push and risk premium shocks. And, even if this disconnect were not too strong, active stabilization of nominal exchange rates

2

in the class of policies considered would still engender greater volatility in domestic variables. Second, the implications of parameter uncertainty for monetary policy design are ambiguous. Depending on the country, the weight given to output stabilization, and the speci…c policy coe? cient under consideration, policy can be more or less aggressive. The classic attenuation result of Brainard (1967) need not obtain, though our …ndings are consistent with multivariate generalizations of that analysis by Chow (1975). Similar results have also been documented for the closed economy case in the robust control literature — see Giannoni (2002). We conclude that the implications of parameter uncertainty for policy design need to be assessed on a case-by-case basis. In exploring the robustness of our conclusions, we give particular attention to the modeling of the foreign block, which in the baseline model is …tted to observed U.S. series. In an alternative model, the foreign block is treated as latent. By confronting the model with fewer observable series, greater ? exibility exists to …nd mechanisms that could warrant an exchange rate channel for output and in? ation stabilization. This is not the case and our characterization of optimal policy remains qualitatively unchanged. An emergent issue in our robustness analysis concerns the impact of parameter identi…cation for the design of optimal policy. For Australia, modeling the foreign block as latent gives rise to two modes with almost identical posterior densities. One is shown to favor a fairly high degree of nominal rigidities, while the other presents more persistent and volatile technology shocks. Although both modes con…rm our conclusion that it is not optimal to respond to exchange rate variations, the policy coe? cients on in? ation and output growth are di¤erent, and each policy engenders rather di¤erent losses. We source these discrepancies to changes in the implied contribution of shocks and the transmission mechanisms of disturbances. A number of recent papers have raised concerns about identi…cation in DSGE models — see Lubik and Schorfheide (2005) and Justiniano and Preston (2006) for discussions in the context of open economy models. More generally, Beyer and Farmer (2005), Fukac, Pagan, and Pavlov (2006), Canova and Sala (2005), Cochrane (2007) and Iskrev (2007) explore sources of identi…cation problems and their implications for inference and speculate on its consequences for policy evaluation. Our discussion provides a novel example of the problems that identi…-

3

cation pose for policy design and underscores that care is warranted in the estimation of this class of models. This paper most closely related to ours is Smets and Wouters (2002) and the references therein on policy evaluation in empirical small open economy models. Lubik and Schorfheide (2003) also consider whether there is evidence that Australia, Canada, New Zealand and the United Kingdom have had monetary policies that depend on nominal exchange rate variations. However, they do not address the question of optimal policy or the consequences of model uncertainty. Our analysis also builds on the ever growing literature on estimating small open economy models using Bayesian methods — see Ambler, Dib, and Rebei (2004), Bergin (2003, 2004), Del Negro (2003), Dib (2003), Ghironi (2000), Justiniano and Preston (2004, 2006), Lubik and Schorfheide (2003, 2005), Lubik and Teo (2005) and Rabanal and Tuesta (2005). The paper proceeds as follows. Section 2 lays out the theoretical model. Section 3 discusses the data. Section 4 outlines the estimation methodology and adopted priors. Section 5 presents the baseline estimation results and properties of the model implied second order moments. Section 6 presents the optimal policy exercises and assesses the implications of parameter uncertainty for policy design. Section 7 analyzes the robustness of our conclusions to the speci…cation of the foreign block. Section 8 concludes.

2

A Simple Small Open Economy Model

The following section sketches the derivation of key structural equations implied by the model proposed by Monacelli (2005) and its closely related precursor Gali and Monacelli (2005) when allowing for incomplete asset markets, habit formation and indexation of prices to past in? ation. These papers extend the microfoundations of the kind described by Clarida, Gali, and Gertler (1999) and Woodford (2003) for analyzing monetary policy in a closed economy setting to an open economy context. For additional detail the reader is encouraged to consult Monacelli (2005).

4

2.1

Households

1 X t=0

Households are assumed to maximize E0

t

~g;t "

"

(Ct 1

1

Ht )1

Nt1+' 1+'

#

where Nt is the labor input; Ht

hCt

is an external habit taken as exogenous by the

household; ; ' > 0 are the inverse elasticities of intertemporal substitution and labor supply respectively; and ~g;t is a preference shock. Ct is a composite consumption index " Ct = (1 ) CH;t +

1 1 1 1 1

CF;t

where CH;t and CF;t are Dixit-Stiglitz aggregates of the available domestic and foreign produced goods given by 21 3""1 Z " 1 CH;t = 4 CH;t (i) " di5

0

and

CF;t

where

is the share of foreign goods in the domestic consumption bundle;

21 3""1 Z " 1 = 4 CF;t (i) " di5

0

> 0 the elasticity

of substitution between domestic and foreign goods; and " > 1 is the elasticity of substitution between types of di¤erentiated domestic or foreign goods. Assuming the only available assets are one period domestic and foreign bonds, optimization occurs subject to the ? budget constraint ow Pt Ct + Dt + et Bt = Dt

1

(1 + ~t 1 ) + et Bt {

1

1 + ~t {

1

t

(At ) + Wt Nt +

H;t

+

F;t

+ Tt

for all t > 0, where Dt denotes the household’ holding of one period domestic bonds, and Bt s holdings of one period foreign bonds with corresponding interest rates ~t and ~t . The nominal { { exchange rate is et . Pt , PH;t , PF;t and P correspond to the domestic CPI, domestic goods ~ prices, the domestic currency price of imported goods and the foreign price respectively and are formally de…ned below. Wages Wt are earned on labor supplied and

H;t

and

F;t

denote

pro…ts from holding shares in domestic and imported goods …rms. Tt denotes lump-sum taxes and transfers. Following Benigno (2001), Kollmann (2002) and Schmitt-Grohe and Uribe (2003), the function

t

( ) is interpretable as a debt elastic interest rate premium given by h i ~t At + t = exp 5

where At et 1 Bt 1 ~ Y Pt 1

is the real quantity of outstanding foreign debt expressed in terms of domestic currency as a fraction of steady state output and ~ t a risk premium shock. The adopted functional form ensures stationarity of the foreign debt level in a log-linear approximation to the model. Implicitly underwriting this expression for the budget constraint is the assumption that all households in the domestic economy receive an equal fraction of both domestic and retail …rm pro…ts. Hence, nominal income in each period is Wt Nt + rium equals PH;t YH;t + (PF;t

H;t

+

F;t

which in equilib-

et Pt ) CF;t for all households. Absent this assumption, which ~

imposes complete markets within the domestic economy, the analysis would require modeling the distribution of wealth across agents. That same assumption also ensures that households face identical decision problems and therefore choose identical state-contingent plans for consumption. The household’ optimization problem requires allocation of expenditures across all types s of domestic and foreign goods, both intratemporally and intertemporally. This yields the following set of optimality conditions. The demand for each category of consumption good is CH;t (i) = (PH;t (i) =PH;t ) CH;t and CF;t (i) = (PF;t (i) =PF;t ) CF;t

for all i with associated aggregate price indexes for the domestic and foreign consumption bundles given by PH;t and PF;t : The optimal allocation of expenditure across domestic and foreign goods implies the demand functions CH;t = (1 where Pt = (1 ) (PH;t =Pt )

1 1

Ct

and CF;t =

(PF;t =Pt )

Ct

(1)

1 1 ) PH;t + PF;t

is the consumer price index. Allocation of expendi-

tures on the aggregate consumption bundle and optimal labor supply satisfy

t t

= ~g;t (Ct "

Ht )

1=

(2) (3)

= ~g;t Pt Nt' =Wt "

6

and portfolio allocation is determined by the optimality conditions ~ t et Pt

t Pt

= Et (1 + ~t ) { = Et [(1 + ~t ) {

~ t+1 t+1 et+1 Pt+1

t+1 Pt+1 ]

(4) (5)

for Lagrange multiplier equation.

t.

The latter condition when combined with (2) gives the usual Euler

2.2

Domestic Producers

There are a continuum of monopolistically competitive domestic …rms producing di¤erentiated goods. Calvo-style price-setting is assumed, allowing for indexation to past domestic goods price in? ation. Hence, in any period t, a fraction 1 fraction 0 <

H H

of …rms set prices optimally, while a

< 1 of goods prices are adjusted according to the indexation rule log PH;t (i) = log PH;t

1

(i) +

H

H;t 1

(6)

where 0 and

H;t

H

1 measures the degree of indexation to the previous period’ in? s ation rate

0

= log(PH;t =PH;t 1 ). Since all …rms having the opportunity to reset their price in

period t face the same decision problem they set a common price PH;t . The Dixit-Stiglitz aggregate price index therefore evolves according to the relation 2 !1 " 31=(1 H 0 (1 ") PH;t 1 5 PH;t = 4(1 + H PH;t 1 H ) PH;t PH;t 2 Firms setting prices in period t face a demand curve yH;T (i) = PH;t (i) PH;T PH;T PH;t

1 1

H

")

:

(7)

!

"

CH;T + CH;T

(8)

for all t and take aggregate prices and consumption bundles as parametric. Good i is produced using a single labor input Nt (i) according to the relation yH;t (i) = ~a;t Nt (i) where ~a;t is an " " exogenous technology shock. The …rm’ price-setting problem in period t is to maximize the expected present discounted s value of pro…ts Et

1 X T =t T t H Qt;T yH;T

(i) PH;t (i) 7

"

PH;T PH;t

1 1

H

PH;T M CT

#

where M CT = WT =(PH;T ~a;T ) is the real marginal cost function for each …rm, assuming " homogenous factor markets, subject to the demand curve, (8). The factor

T t H

in the …rm’ s

objective function is the probability that the …rm will not be able to adjust its price in the next (T t) periods. The …rm’ optimization problem implies the …rst order condition s " # 1 X PH;T 1 H H T t Et PH;T M CT = 0: H Qt;T yH;T (i) PH;t (i) PH;t 1 1 H T =t

(9)

2.3

Retail Firms

Retail …rms import foreign di¤erentiated goods for which the law of one price holds at the docks. In determining the domestic currency price of the imported good, …rms are assumed to be monopolistically competitive. This small degree of pricing power leads to a violation of the law of one price in the short run. Retail …rms face a Calvo-style price-setting problem allowing for indexation to past in? ation. Hence, in any period t, a fraction 1 0<

F F

of …rms set prices optimally, while a fraction

< 1 of goods prices are adjusted according to an indexation rule analogous to (6).

The Dixit-Stiglitz aggregate price index consequently evolves according to the relation 2 !1 " 31=(1 ") F 0 (1 ") PF;t 1 5 (10) PF;t = 4(1 + F PF;t 1 F ) PF;t PF;t 2 and …rms setting prices in period t face a demand curve CF;T (i) = PF;t (i) PF;T PF;T PF;t

1 1

F

!

"

CF;T

(11)

for all t and take aggregate prices and consumption bundles as parametric. The …rm’ prices setting problem in period t is to maximize the expected present discounted value of pro…ts " # 1 X PF;T 1 F T t Et eT PF;T (i) ~ H Qt;T CF;T (i) PF;t (i) PF;t 1 T =t subject to the demand curve, (11). The factor

T t F

in the …rm’ objective function is the s t) periods. The

probability that the …rm will not be able to adjust its prices in the next (T

…rm’ optimization problem implies the …rst order condition s " # 1 X PF;T 1 F H T t Et eT PH;T (i) = 0: ~ F Qt;T PF;t (i) PF;t 1 1 H T =t 8

2.4

International Risk Sharing

From the asset pricing conditions that determine domestic and foreign bond holdings, the uncovered interest rate parity condition Et

t+1 Pt+1 [(1

+ ~t ) {

(1 + ~t ) (~t+1 =~t ) { e e

t+1 ]

=0

(12)

follows, placing a restriction on the relative movements of the domestic and foreign interest rate, and changes in the nominal exchange rate. The real exchange rate is de…ned as qt ~ price fails to hold, we have ~ F;t et Pt =Pt : Since Pt = PF;t , when the law of one ~

et Pt =PF;t 6= 1, which de…nes what Monacelli (2005) calls ~

the law of one price gap. The models of Gali and Monacelli (2005) and Monacelli (2005) are respectively characterized by whether or not ~ F;t = 1.

2.5

General Equilibrium

Goods market clearing requires YH;t = CH;t + CH;t (13)

in the domestic economy. The model is closed assuming foreign demand for the domestically produced good is speci…ed as CH;t = where PH;t P Yt

> 0. This demand function is standard in small open economy models (see Kollmann

(2002) and McCallum and Nelson (2000)) and nests the speci…cation in Monacelli (2005) by allowing to be di¤erent from , the domestic elasticity of substitution across goods in the

domestic economy, in order to give additional ? exibility in the transmission mechanism of foreign disturbances to the domestic economy. However, our results are una¤ected by the parametrization of this demand function.3 Domestic debt is assumed to be in zero net supply so that Dt = 0 for all t.4

Constraining to equal results in identical insights from the estimation, and therefore we report results based on this more general speci…cation. 4 A similar condition holds for the foreign economy once it is noted that domestic holdings of foreign debt, Bt , is negligible relative to the size of the foreign economy.

3

9

The analysis considers a symmetric equilibrium in which all domestic producers setting prices in period t set a common price PH;t . Similarly, all domestic retailers choose a common price PF;t . Finally households are assumed to have identical initial wealth, so that each faces the same period budget constraint and therefore makes identical consumption and portfolio decisions. Finally, monetary policy is assumed to be conducted according to a Taylor-type rule discussed in the subsequent section. Fiscal policy is speci…ed as a zero debt policy, with taxes equal to the subsidy required to eliminate the steady state distortion induced by imperfect competition in the domestic and imported goods markets.

2.6

Log-linear approximation to the model

The empirical analysis employs a log-linear approximation of the model’ optimality conditions s around a non-stochastic steady state. We here discuss the key structural equations that emerge from this analysis. All variables are properly interpreted as log deviations from their respective steady state values. Relations pertaining to the domestic economy are discussed, followed by those for the foreign economy. A log linear approximation to the domestic household’ Euler equation (5) provides s ct hct

1

= Et (ct+1

hct )

1

(1

h)(it

Et

t+1 )

+

1

(1

h) ("g;t

Et "g;t+1 ) :

(14)

In the absence of habit formation, when h = 0, the usual Euler equation obtains. To derive a relationship in terms of domestic output, a log-linear approximation to the goods market clearing condition implies: (1 where

F;t

) ct = yt

(2

) st

F;t

yt

(15)

(et + pt )

pF;t

denotes the law of one price gap, the di¤erence between the world currency price and the domestic currency price of imports, and st = pF;t di¤erencing the terms of trade de…nition implies st =

F;t H;t :

pH;t gives the terms of trade. Time

(16)

10

Equilibrium domestic consumption depends on domestic output and three sources of foreign disturbance: the terms of trade, deviations from the law of one price and foreign output. The terms of trade and the real exchange rate are related according to qt = et + pt pt =

F;t

+ (1

) st

(17)

so that the real exchange rate varies with deviations from the law of one price and also di¤erences in consumption bundles across the domestic and foreign economies. A log-linear approximation to domestic …rms’optimality conditions for price setting and the price index, (7), imply the relation

H;t H;t 1

=

1 H

(1

H ) (1

H

) mct + Et (

H;t+1

H;t )

(18)

where mct = 'yt (1 + ') "a;t + st + (1 h)

1

(ct

hct 1 )

H;t

is the real marginal cost function of each …rm. Thus domestic price in? ation,

= pH;t

pH;t 1 , is determined by current marginal costs, expectations about in? ation in the next period and the most recent observed in? ation rate. The latter appears as a result of price indexation. In the case of zero indexation to past in? ation, = 0, the usual forward looking Phillips curve

arises. In contrast to a closed economy setting, domestic goods price in? ation depends on three sources of foreign disturbance. There is a direct and indirect e¤ect of the terms of trade on …rms’ marginal costs, with the latter operating through the terms of trade implications for equilibrium consumption. There are also the e¤ects of foreign output and deviations from the law of one price (recall relation (15)). The optimality conditions for the retailers’pricing problem yields

F;t F;t 1

=

1 F

(1

F ) (1

F

)

F;t

+ Et (

F;t

F;t+1

F;t )

+ "cp;t :

(19)

Here, in? ation in the domestic currency price of imports, current marginal cost conditions given by

F;t

= pF;t

pF;t 1 , is determined by

and expectations about next period’ in? s ation

rate. A cost-push shock has also been added, capturing ine? cient variations in mark-ups. Again, that prices are indexed to past in? ation induces a history dependence on the most 11

recent observed in? ation rate. The domestic CPI and home goods prices are related according to

t

=

H;t

+

st :

(20)

The CPI and domestic goods price in? ation di¤er insofar as imported goods prices deviate from domestic goods prices, with the di¤erence weighted by the importance of those goods in the CPI — recall equation (16). The uncovered interest-rate parity condition gives (it Et

t+1 )

it

Et

t+1

= Et qt+1

at

t

(21)

while the ? budget constraint implies ow c t + at =

1

at

1

st +

F;t

+ yt

(22)

where at = log(et Bt =(Pt Y )) is the log real net foreign asset position as a fraction of steady state output.5 The model is closed by specifying monetary policy which is conducted according to the Taylor-type rule it =

i it 1

+

t

+

y yt

+

y

yt +

e

et + "M;t :

(23)

The nominal interest rate is determined by past interest rates and also responds to the current all goods CPI in? ation rate, output, output growth and the change in the nominal exchange rate. The …nal term, "M;t , is a monetary policy shock or implementation error in the conduct of policy.6 The domestic block of the economy is therefore given by equations (14)-(23) in the unknowns ct ; yt ; it ; qt ; st ;

t; H;t ; F;t ; F;t ;

at : Combined with the processes for the exst =

ogenous disturbances f"a;t ; "M;t ; "g;t ; "s;t :"cp;t g and f t ; yt ; it g ; and the de…nitions

5 6

In steady state, the foreign economy is assumed to have a zero debt-to-gdp ratio. Policy is assumed to respond to the linear detrended level of output and the change in this measure as opposed to the model theoretic measure of the output gap. This is motivated by recent research suggesting that model theoretic output gap measures do not accord with more traditional measures of economic slack used by actual policymakers — see Neiss and Nelson (2005) and Andreas, Nelson, and Lopez-Salido (2005). This has relevance given our interest in assessing the historical stance of policy.

12

st

st

1

and

qt = qt

qt 1 , these relations constitute a linear rational expectations model

which can be solved using standard methods. Together these relations also comprise the equations used to construct the likelihood for estimation. The disturbances f"a;t ; "g;t ; "s;t g are assumed to be independent AR(1) processes and f"M;t g an i.i.d. process. The determination of the foreign block f t ; yt ; it g is discussed in the subsequent section. In estimation, we only make use of observable series for fyt ; it ;

t;

qt ; s t ;

t;

yt ; it g and therefore exploit only a

subset of cross-equation restrictions implied by the model.

2.7

The Foreign Economy

In Monacelli (2005) the foreign economy is speci…ed as the closed economy variant of the model described above. However, because the foreign economy is exogenous to the domestic economy, we have some ? exibility in specifying the determination of foreign variables. Rather than take a literal interpretation of the Monacelli model, we instead assume that the paths of f t ; yt ; it g are determined by a vector autoregressive processes of order two.

3

Data

For all three countries, estimation uses quarterly data on output, in? ation, interest rates, the real exchange rate and the terms of trade. GDP is per capita in log deviations from a linear trend. The in? ation series corresponds to the annualized quarterly log-di¤erence in the consumer price index (all goods), which includes both home and imported goods. For Australia, an adjustment is made to this series to take into account the e¤ects of the introduction of the goods and services tax in 2000-2001. For Canada, we use an in? ation

measure excluding food and energy, given numerous references to this core series in the conduct of monetary policy by the Bank of Canada. Similar considerations to those in Australia dictate adjusting the large outlier in the …rst quarter of 1991 with the use — for that year only — of a measure that also excludes the e¤ects of indirect taxes. Finally, we use the cash rate in Australia, and, for Canada and New Zealand, averages of 3-month bank rates (all expressed in annualized percentages) for interest rates. All Australian data were downloaded from the Statistical Tables published by the Reserve 13

Bank of Australia. For Canada and New Zealand all data were obtained from Data Stream International. We constructed a model consistent real exchange rate using U.S. price data discussed below, each country’ CPI — as described above — and the bilateral nominal s exchange rate. The real exchange rate is expressed in log-di¤erences for the estimation. The terms of trade are measured as the price of imports to exports using the corresponding price de? ators from the national accounts in each country. As with the real exchange rate, we use the log-di¤erence of this series when taking the model to the data. For speci…cations in which the foreign block is observable we assume it to be reasonably proxied by U.S. data. The U.S. series are the annualized quarterly log percentage change in the CPI, the log deviations of per capita GDP from a linear trend and the Fed Funds rate (annualized percentage), all taken from the Database at the Federal Reserve Bank of St. Louis. Our samples run from 1984:I until 2007:I for Australia, and 1988:III-2007:I for New Zealand, following the move in each country towards a ? exible exchange rate regime. For Canada, the sample covers the period 1982:I-2007:I, to coincide with the abandonment of monetary targeting with the Bank of Canada.7 In summary, for each country the model is taken to the data using 8 observable series and the same number of disturbances. We demean the series before the estimation.

4

Estimation

Our objective is not only to obtain point estimates for the parameters of the DSGE model speci…ed in the previous section, but also to provide accurate measures of uncertainty surrounding these estimates. Therefore, using Bayesian methods, we aim to characterize the posterior distribution of the model parameters 2 . Given a prior, ( ), the posterior density

is proportional to the product of the likelihood and the prior. As described by Schorfheide (2000), posterior draws for this density can be generated using a random walk metropolis algorithm and the state-space representation implied by the solution of the linear rational expectations model and the Kalman …lter. Measures of location and scatter are obtained

We use four observations before the start of the sample dates above listed to deal with the initialization of the Kalman …lter. These four initial data points are excluded from the computation of the likelihood and consequently from our estimates. Note that this does not represent the use of a training sample prior.

7

14

from the draws by computing, for instance, the median and standard deviations as well as posterior probability bands. Furthermore, given the draws, it is possible to characterize the posterior distribution of any functional of interest by computing the corresponding functional for each of the draws. This property will later be exploited to analyze the implication of model uncertainty on optimal policy. An optimization algorithm is used to obtain an initial estimate of the mode. We start the maximization algorithm from a number of random starting values — before launching the MCMC chains — and check that the optimization routine always converges to the same value.8 This is a useful diagnostic for the presence of identi…cation problems, conditional on a given set of priors. Indeed, our experience is that this is crucial to identifying local modes which may achieve almost identical values of the posterior with sometimes rather di¤erent con…gurations of coe? cients. Of course, this procedure remains silent on the role of priors in achieving local identi…cation, which may be discerned by looking at univariate or two dimensional plots of the likelihood or the Hessian. The existence of multiple modes, related identi…cation issues, and their implications for policy design are the focus of a later section. Having ensured a unique mode for the baseline model, the Hessian from the optimization routine is used as a proposal density, properly scaled to yield a target acceptance rate of 25%. For the MCMC results, …ve chains of 100,000 draws each were initialized by randomly selecting starting values (using an over dispersed normal density centered at the mode with a scaled-up Hessian as variance covariance matrix). For each chain, following a burn-in phase of 40,000 draws, convergence is monitored using CUMSUM plots and, for the overall chains, the potential scale reduction factors and con…dence interval variants of Brooks and Gelman (1998). The priors are described in the …rst three columns of Table 1. The same priors are used for all countries except for the openness parameter, ; which we calibrate to the average share of

For the baseline model discussed over 50 optimization runs were launched using random draws from the prior or an equally spaced grid covering the parameter space. All runs converged to the same mode. Note that obtaining di¤erent modes with substantially di¤erent values of the posterior/likelihood need not re? ect identi…cation issues but rather the properties of the optimization routine in place. In this respect, we di¤er from Canova and Sala (2005) in that we view the convergence to multiple modes with similar …t as problematic, not the convergence to multiple modes per se.

8

15

exports and imports to GDP in each country using national account data. Over our sample period this results in values for of 0.185, 0.28 and 0.29 for Australia, Canada and New

Zealand. Attempts to estimate this parameter often led to implausibly low values. We adopt fairly loose Gamma priors, with large tails, for the inverse Frisch elasticity of labor supply as well as the elasticity of substitution between domestic and foreign goods, considering the diverse estimates emerging from macro and micro studies. Similarly, our prior for the intertemporal elasticity of substitution easily accommodates values of 1 or 0.5 as used in the international business cycle literature, as well as substantially larger estimates that may result from the absence of capital and the consumption of durables in our model — see Rotemberg and Woodford (1999). Priors for the Calvo price parameters assume the presence of nominal rigidities, centered at a compromise between traditionally large values obtained in macro studies and recent evidence of greater ? exibility in prices using disaggregated data for the U.S. — see Bils and Klenow (2004). For imported goods, it may be reasonable to assume a lower degree of stickiness. Nonetheless, estimated open economy models tend to produce fairly large deviations from the law of one price. Therefore, just as in the case of domestic prices, we opt for a compromise in choosing our prior. We follow Lubik and Schorfheide (2003) in specifying the prior for the parameters of the Taylor rule, except for output growth which is not considered in their analysis. Habit and indexation have been found to be crucial for …tting closed economy models which suggests considering possibly large values for the parameters governing these intrinsic mechanisms of persistence. However, a prori it is possible that the dynamics in the foreign block may provide an alternative source of persistence in the model. To allow for this possibility, we specify very ? priors on habit as well as the indexation coe? cients of both domestic at and imported goods. The exogenous stochastic disturbances (risk premium, technology, preference and import cost-push shocks) are assumed to be fairly persistent, re? ected in a beta prior with a mean of 0.8 for the autoregressive coe? cients. For the VAR(2) in the foreign block, we choose priors suggested by pre-sample individual autoregressions.9

For the …rst order autoregressive coe? cients, we specify a N (0:59; 0:22 ) for in? ation and N (0:9; 0:12 ) for 2 output and interest rates. Second order own lags have a N (0; 0:25 ) prior, while the o¤-diagonal elements of

9

16

Finally, the priors for the standard deviations of the shocks are the same for foreign and domestic shocks. To allow for a wide set of values a priori we specify Inverse-Gamma 1 densities, with in…nite variance by …xing the degrees of freedom at 2. The scale parameters are chosen to obtain a mean of 0.5. We do not normalize the impact of any shocks as is sometimes done in closed economy models.

5

Results

The following section details a number of properties of the estimated models. The baseline estimates are presented for each country and the model’ ability to …t particular second order s characteristics of the data discussed.

5.1

Estimates

Table 1 reports the estimation results for the baseline model in which the foreign block is observed. The intertemporal elasticity of substitution is a little below unity, taking values around 0.75 for Australia and New Zealand, and a larger value of 1.1 in Canada. The inverse elasticity of labor supply, a parameter notoriously poorly identi…ed in DSGE models, takes values slightly above unity, although has fairly wide posterior probability bands. Optimal price setting in the production of home goods displays some variation across countries. At the median of our parameter estimates, …rms reoptimize prices approximately every 5, 3 and 3 quarters in Australia, Canada and New Zealand, respectively. The latter numbers accord well with survey evidence for the U.S. in Blinder, Canetti, Lebow, and Rudd (1998) and values reported in Woodford (2003). Prices in the imported goods sector for these countries are adjusted more frequently than home goods prices, being reoptimized on average every 2.2, 1.7 and 1.4 quarters. The elasticity of substitution between domestic and foreign goods is somewhat low, with median estimates between 0.6 and 0.76, despite a prior that allows for far larger values. These values have relevance for papers such as Obstfeld and Rogo¤ (2000) which proposes a

the …rst and second lag matrices are speci…ed a priori as N (0; 0:32 ) and N (0; 152 ) respectively. Results using a prior centered at the pre-sample OLS estimates of a VAR(2) did not alter our results although it induced some convergence problems in the mcmc chains in the case of Canada.

17

model in which a fairly large elasticity of substitution between domestic and foreign goods — together with transaction costs — help explain a number of prominent puzzles in international macroeconomics. In estimated open economy models inference on this parameter has tended to produce either very small elasticities, particularly with complete markets, or seemingly implausibly large values — see Rabanal and Tuesta (2005) and Adolfson, Laseen, Linde, and Villani (2005) respectively. Habit formation appears to play a less prominent role than in other studies, having a maximum value of 0.33 in Australia. Even more surprisingly, price indexation presents a limited source of endogenous persistence in both domestic and imported goods sectors, with coe? cients values of at most 0.11. These …ndings contrast with many closed economy analyses — see, for example, Christiano, Eichenbaum, and Evans (2005) and Smets and Wouters (2003) — and the closely related open economy analysis Justiniano and Preston (2006).10 The di¤erences relative to closed economy models are driven by the fact that the open economy dimension of the model explains some domestic ? uctuations. The di¤erences relative to open economy models is most likely due to the chosen set of shocks: in particular, due to the presence of a cost-push shock in the imported good sector as opposed to the pricing of domestically produced goods. Because of this assumption, the persistence of home goods in? ation is in large part explained by real factors according to the assumed theory of marginal cost; that is, the autocorrelation of technology and preference shocks imparts inertia in domestic in? ation rather than relying on a cost-push shock for the high frequency variation for the change in home goods prices and a high degree of indexation for its persistence. Regardless of these modeling assumptions, the results on optimal policy are una¤ected.11 The policy parameters bear some resemblances across countries. Di¤erences emerge in the responses to in? ation, the nominal exchange rate and output growth. The response to

An earlier version of this paper discussed this property of the estimates in great detail, using posterior odds ratios to examine relative …t across a range of models. Due to the number of results now reported, this discussion is excluded, but such model comparison exercises would reveal that models that excluded price indexation provide a superior chacterization of the data for these three economies. The inclusion or exclusion of these model features matters not for our policy conclusions. 11 Had a cost-push shock been included in home goods pricing, as in Justiniano and Preston (2006), it would have explained a signi…cant part of in? ation variation and real factors would be less important. Moreover, an earlier version of this paper excluded the cost-push shock in imports, yielding higher estimates of indexation but with the same insights on policy design

10

18

in? ation is largest in New Zealand and smallest in Australia. The reverse is true for the coe? cient on output growth. The estimated responses to the level of output are small, consistent with substantial evidence from closed economy models. Finally, the response to the nominal exchange rate is largest in Canada, with a coe? cient of 0.29. This is consistent with the …ndings of Lubik and Schorfheide (2003). As expected, the estimates of the foreign block (excluded due to space considerations) are remarkably similar across countries. Even though the foreign block is exogenous, in the sense that economic developments in each small country under consideration cannot feedback into the foreign series, it is not true that the foreign block is exogenous econometrically speaking. The cross-equation restrictions that result from uncovered interest parity tie the estimates of the foreign data generating process to domestic parameters. Finally, the cost-push, preference and the risk premium disturbances are highly persistent, having autoregressive coe? cients between 0.87 and 0.96 across all three countries. The estimated standard deviations are for the most part plausible, with the biggest di¤erences across countries emerging for the cost-push shock which is the most volatile disturbance for all economies. The standard deviations are 1.6, 2.01 and 7.27 for Australia, Canada and New Zealand. It is worth bearing in mind that the terms of trade and exchange rates (nominal and real) in these countries are quite volatile, particularly for New Zealand. Moreover, as we do not to normalize these shocks — e.g. modify them such that they enter the corresponding equation with a unit coe? cient — their scale is a¤ected by other estimates and hence di? cult to interpret at face value.

5.2

Second order properties

Table 2 presents a set of second order moments for the data and the corresponding statistics implied by the estimated model. We report medians as well as (5; 95) percent probability bands for the moments of DSGE which account for both parameter and small sample uncertainty.12 Providing information on second order properties provides a measure of absolute …t rather than posterior odds ratios, for example, that characterize relative …t.

For each paramater draw obtained with the MCMC chains we simulate 500 samples of length equal to the data after discarding the …rst 50 observations.

12

19

Taking Australia …rst, the small open economy model matches the second order properties of the data quite well. The median implied standard deviations for in? ation, the real exchange rate and output are very close to their empirical counterparts. While this is not true for interest rates and the terms of trade, the empirical standard deviations are nonetheless contained in the 90 percent posterior bands generated by the model, albeit close to their edges. As for persistence, the model does very well once again for in? ation and output, with the 90 percent interval for the remaining observable series encompassing the autocorrelations in the data except for the interest rate which is marginally outside its band. Although we use …rst di¤erences in the real exchange rate and the terms of trade in the estimation, we wish to check that the model can account for the persistence in their levels, since this has been a challenge for open economy models. In both cases the autocorrelation in the data fall comfortably within the estimated 90 percent bands for the same parameter in the model. For Canada, the model provides a similarly reasonable characterization of the data. The model matches the volatility of in? ation, output, real exchange rate and interest rate. The only exception is the terms of trade which is somewhat over predicted in the model. The serial correlation properties are for the most part well matched, except for the autocorrelation in the real exchange rate, which is outside the posterior bands particularly for …rst di¤erences. Finally, for New Zealand similar remarks to Australia apply for the standard deviations. For the autocorrelations, the model matches the corresponding sample moments, with the exception of real exchange rate and terms of trade growth. This is not surprising, given the random walk hypothesis of the exchange rate, and the associated di? culty that structural models have …tting the persistence and volatility of these series — see, for example, Chari, Kehoe, and McGrattan (2002) and Justiniano and Preston (2006) for calibration and estimation based studies. Overall, the model performs reasonably well for all three countries, perhaps with the exception of the serial correlation properties of the log di¤erence in the real exchange rate, a feature shared by most structural and reduced form open economy models. Nonetheless in all countries the level of the real exchange rate is correctly characterized as a very persistent process.

20

6

Monetary Policy Design and Uncertainty

Recent theoretical analyses have emphasized the importance of pricing to market assumptions for optimal exchange rate and monetary policy. Whether a country has producer currency pricing or local currency pricing can give rise to di¤erent policy recommendations, even when the sole objective of policy is to stabilize the aggregate in? ation rate. For instance, Devereux and Engel (2003) show in a two country model with local currency pricing that optimal monetary policy stipulates stabilization of the nominal exchange rate. Similarly, Monacelli (2005), in a model nested by the one estimated in this paper, shows that deviations from the law of one price lead to a trade-o¤ in the stabilization of in? ation and output in the absence of ine? cient variations in markups. His analysis overturns the closed economy result that stabilizing the in? ation rate serves to simultaneously stabilize economic activity and introduces an explicit motive to respond to the exchange rate even when consumer prices are the sole objective of policy. Despite these theoretical contributions there has been relatively little work on policy evaluation in empirical open economy models. In the small open economy literature, Smets and Wouters (2002) consider the implications of imperfect pass through for optimal monetary policy, demonstrating that welfare maximizing policies introduce a motive to stabilize the exchange rate (see also the references therein). Lubik and Schorfheide (2003), rather than explore the question of optimal policy, instead seek to identify if in the small open economies considered here (as well as the United Kingdom) there is evidence that monetary authorities have responded to nominal exchange rate ? uctuations. They …nd that only in the case of Canada does there exist strong evidence supporting such responses. The following sections build on these analyses by considering optimal policy in our estimated model. Two exercises are pursued. First we look at the design of optimal monetary policies within the class of Taylor-type rule adopted in the empirical model. Policy rule coe? cients are chosen to minimize a quadratic loss function assuming that the remaining estimated model parameters take their median values. This elucidates whether optimal policy requires nominal interest rates to be adjusted in response to nominal exchange rate ? uctuations or not. 21

Second, we determine the optimal policy rule that takes into account all parameter uncertainty implied by the estimated model. That is, we compute the policy rule that minimizes the expected loss, where expectations are also taken with respect to the posterior distribution of the remaining model parameters. As explained later, this analysis is facilitated by a Bayesian approach, which also allows taking into account the covariance between inferred parameters when quantifying the dispersion around our estimates. This permits addressing an old question of whether parameter uncertainty leads to more cautious policy prescriptions, as suggested by the seminal analysis of Brainard (1967).

6.1

The Optimal Policy Problem

The policymaker seeks to minimize the objective function W0 = E0 where 0 <

1 X t=0 t

Lt

(24)

< 1 coincides with the household’ discount factor and s Lt =

t 2

+

y yt

2

+

i it

2

(25)

is the period loss at any date t

0. The policymaker is therefore assumed to stabilize variation

in aggregate consumer price in? ation, output and nominal interest rates, where the weights

x; i

> 0 determine the relative priority given to each of these objectives. To simplify goes to unity. This transforms

further, we consider the limiting case of this objective when

the analysis of the loss function (25) into the analysis of the objective W0 ( ) = var ( t ) + a weighted sum of variances, and on model parameters. The assumption of arbitrary weights ( y ;

i ), y var (yt )

+

i var (it )

makes explicit the dependency of the variance calculation

and the assertion that consumer price in? a-

tion, output and nominal interest rate variation ought to be stabilized is questionable. To address these concerns the robustness of our conclusions is gauged by analyzing the above loss function as the weights ( y ;

i)

are varied over a …ne grid on the unit square. Our con-

clusions are largely una¤ected by the precise choice of weights in the objective function. For 22

presentation purposes, we focus on how varying the relative weight on output stabilization a¤ects outcomes, since this dimension has played a prominent role in the analysis of optimal policy — see Svensson (1999, 2000). Attention is restricted to optimal policies within a class of Taylor-type rules of the form it =

i it 1

+

t

+

y yt

+

y

yt +

e

et :

(26)

As in estimation, policy is assumed to adjust nominal interest rates in response to contemporaneous values of in? ation, output, output growth, the nominal exchange rate growth and lagged observations of the nominal interest rate. Note that the response coe? cients in equation (26) are not multiplied by (1 of rules having

i i)

as in (23) since we wish to consider the possibility

very close to one. Care should be taken in comparing the optimal policy

coe? cients described in subsequent sections to the corresponding estimated policy parameters of section 5. To …x notation, partition the estimated parameters for a given model as where

i; s

= f p;

p

sg

collects structural parameters other than those determining policy, denoted

y; y; e

= =

;

.

Conformably partition the associated parameter space as

f

p;

13 s g.

Let

s

denote the estimated median value of the structural parameters.

6.2

Optimal Policy under Parameter Certainty

In our …rst policy experiment the optimal policy coe? cients are chosen assuming the structural parameters are known and equal to

s;

the median of the MCMC draws. Thus the e¤ects of

parameter uncertainty are ignored and optimal policy is determined as

p

= arg min W0

p2 p

p

j

s

where the minimization is subject to the constraints that policy is given by (26) and aggregate dynamics are as determined in section 2. The …nal restriction placed on the policy design is that the coe? cient on the lagged nominal interest rate must satisfy 0 super-inertial interest rate rules is left for future research.

Note that the MCMC posterior simulator produces joint and marginal posterior densities which validate this approach.

13

i

1. The study of

23

Table 3 provides results for this optimal policy problem for three di¤erent objective functions, which di¤er according to the weight assigned to output stabilization.14 Consider the results for Australia when output is assigned a weight of

y

= 0 in the objective function.

Optimal policies are highly inertial, characterized by a unit coe? cient on the lagged interest rate, prescribing the stance of policy in terms of the evolution of the …rst di¤erence of nominal interest rates, rather than the level. The optimal response to in? ation may seem smaller relative to typical estimates of this parameter and the estimated policy reaction function in section 5. Recall, however, that these are not multiplied by one minus the coe? cient on lagged interest rates, and that optimal policies exhibit a greater degree of inertia. In contrast, the response to output, output growth and the nominal exchange rate is zero to the second decimal place. The second and third columns give results for an objective function that places greater weight on output stabilization. The response to output and output growth tends to rise with greater concern for output variability. For a unit weight on output in the objective function, both coe? cients are roughly ten times that observed in the …rst column. Concomitantly, the variances of in? ation and output under optimal policy increase and decline respectively as greater weight is placed on output stabilization. Hence a Taylor frontier is mapped out, delineating the inherent in? ation-output stabilization trade-o¤ present in this model. Regardless of the relative weights appearing in the objective function, it is never optimal to respond strongly to nominal exchange rate variations. This last result is particularly surprising: despite the open economy dimension of the model and the existence of deviations from the law of one price, optimal policy does not prescribe a direct response to exchange rate ? uctuations to ensure that its objectives of stable output and in? ation are met. This is at odds with the theoretical literature which underscores models characterized by local currency pricing should give cause to respond to exchange rate ? uctuations. Furthermore, it also suggests the …nding of Lubik and Schorfheide (2003), of

As throughout the paper, the variances correspond to median volatilities from 100 samples of length equal to the data after discarding the …rst 50 observations. Qualitatively, results are unchanged when using the asymptotic variances instead. The approach pursued here has the advantage of incorporating small sample uncertainty into the analysis.

14

24

little evidence that the Reserve Bank of Australia has responded to exchange rate ? uctuations, is part of an optimal policy framework, at least in this restricted family of Taylor-type rules. The broad theme of these results are applicable to Canada and New Zealand. Policies are highly inertial, leading to a di¤erence rule for the nominal interest rate. Both countries respond more aggressively to in? ation than does Australia, and both show little response to the level of real economic activity regardless of the objective function. Higher preference for output stabilization is associated with stronger responses to output growth. Again there is little evidence supporting the desirability of policies responding to the nominal exchange rate. These conclusions are valid regardless of the weight placed on output and interest rate stabilization. Figure 1 plots the optimal exchange rate coe? cient as the weights ( y ; Australia, Canada and New Zealand.15 Further insights into the characteristics of these optimal rules emerge from comparing the standard deviations under optimal policy and the volatility observed in the data. For a zero weight on output stabilization, the standard deviations of in? ation implied by these policy rules are 0.13, 0.03 and 0.06 for Australia, Canada and New Zealand — see Table 3. Comparison to Table 2 makes clear that optimal policy implies very limited variation in in? ation relative to historical data, roughly approximating in? ation targeting. As output stabilization becomes relatively important, the case for strict in? ation targeting weakens for obvious reasons — though the implied volatilities for in? ation are close those observed in the data for the case of Australia and Canada. This is consistent with the notion of ? exible in? ation targeting — see Svensson (1997, 1999).

i)

are

varied on the unit square. This coe? cients attains maximum values of 0.01, 0.05 and 0.02 for

6.3

Sourcing the Result

The striking result from these optimal policy exercises is the lack of response of nominal interest rates to exchange rate ? uctuations. One interpretation of this …nding is that the tradeNote that the coe? cient magnitudes themselves are not su? cient to infer the relevance of the exchange rate — one also must consider the magnitude of exchange rate variations. Furthermore, a one standard deviation change in nominal exchange rate growth would imply – partial equilibrium– the most a 10 basis in at point increase in nominal interest rates for the case of Canada. The response coe? cients for Australia and New Zealand are considerably smaller.

15

25

o¤ generated by deviations from the law of one price is not particularly important for imported goods price in? ation dynamics and therefore CPI in? ation dynamics. However, this is not generically true for the presented theoretical model. Consequently, it is worth considering further why the empirical model identi…es a parameter con…guration that engenders optimal policies without an active role for exchange rate stabilization. The …nding that it is not optimal to respond to the exchange rate can be sourced to two features of the empirical model. First, there exists a “disconnect” of the real and nominal exchange rates from the remaining domestic series — see Obstfeld and Rogo¤ (2000), among others, for a detailed discussion. Indeed, variance decompositions reveal that cost-push shocks in the imported goods sector and risk premium shocks together account for 84, 69 and 83 percent of the variation in nominal exchange rates in Australia, Canada and New Zealand, with almost identical shares for the real exchange rate. At the same time, both shocks play a substantially more muted role for in? ation, output and domestic interest rates.16 A consequence of this disconnect is that by responding to the exchange rate, monetary policy ties the evolution of the domestic economy to cost-push and risk premium shocks, and may give rise to increased variability. Second, even if risk premium and cost-push shocks are not negligible for output, in? ation and interest rate variations, policy responses to stabilize the exchange rate exacerbate variability in these series. Figures 2 and 3 shed light on these mechanisms, presenting impulse responses of various series for cost-push and risk premium shocks in the case of Australia. Similar insights hold for the other two countries. Three impulse responses are shown for each variable, each being associated with three di¤erent policy coe? cients on the exchange rate. The baseline with

e

= 0 corresponds to the optimal policy coe? cients when

y

= 0:5 and re-

maining model parameters as shown in Table 1.17 The second and third impulse responses are generated assuming, counterfactually, that …xed.

16 For Australia the combined variance share for risk premium and cost-push shocks in output, in? ation and interest rates are 16, 17 and 22 percent respectively; for Canada and the same series order: 3, 15 and 16 percent; while for New Zealand these shocks combined explain 1, 21 and 19 percent respectively. 17 Similar insights result from using the estimated — as opposed to optimal — policy coe? cients.

e

= 0:2 and

= 0:4; holding all other parameters

26

Consider the case of the cost-push shock when

e

= 0 (solid lines). An innovation to

this disturbance causes an appreciation (i.e. decline) in the exchange rate (nominal and real) and a negative deviation in the law of one price gap. Because the latter is the marginal cost of imported goods some of the direct e¤ect of the cost-push shocks on imported goods price in? ation is o¤set. Regardless, imported goods prices rise substantially leading domestic demand to shift towards domestically produced goods, although price responses in the home goods sector are rather muted. Nominal and real interest rates fall slightly to counteract the rise in all goods in? ation engendered by this shock. Increasingly strong responses to the exchange rate (dashed lines) tend to counteract the degree of exchange rate appreciation, which reduces the decline in the marginal costs of imported goods and leads to larger price pressures in this sector. Greater declines in real rates also exacerbate domestic price in? ation: a given sized cost-push shock is therefore more in? ationary. Moreover, the stronger response to the exchange rate triggers larger variations in nominal interest rates and output. As a result, responding to exchange rates induces increased variability and larger losses in equation (25). In the case of risk premium shocks, the depreciation (i.e. increase) in the exchange rate calls for an interest rate tightening. This has two counteracting e¤ects on in? ation. On the one hand, responding to exchange rate movements serves to stabilize imported goods price in? ation. On the other hand, higher nominal and real interest rates tend to cause a contraction in domestic activity: output and domestic in? ation fall. This might suggest that there is some scope to stabilize in? ation through an exchange rate channel. However, two points should be made. First, responding more aggressively to exchange rate variations leads to larger movements in nominal rates and a larger contraction in domestic activity — these e¤ects outweigh the positive stabilizing in? uence on import goods price in? ation leading to larger losses. Second, the optimal policy rules determined in the previous section are not conditional on a given shock. They are unconditional optimal policies. While our discussion of the e¤ects of cost-push shocks and risk premium shocks can provide intuition for why more aggressive exchange rate policy is undesirable, it by no means rules out the possibility that, conditional

27

on a single shock, there may be welfare improvements from managing exchange rate variations. However, taking into account all sources of variation and the associated property of exchange rate disconnect, our results suggest that stabilizing exchange rate ? uctuations is undesirable. These …ndings di¤er from Smets and Wouters (2002) which presents evidence in an empirical small open economy model with local currency pricing that optimal policy does respond to exchange rate ? uctuations.18 While the precise details of the underlying models di¤er, they do have the same basic elements. There are two sources of discrepancy in the two studies worth mentioning. First, Smets and Wouters use the theoretical based output gap for their analysis, while we work with detrended output — see footnote 6. Second, our analyses di¤er in the estimation methodology. Smets and Wouters estimate a small subset of model parameters by matching impulse response functions. Our conjecture is that confronting the model with data on a greater number of dimensions, as done in the likelihood-based estimation procedure of this paper, engenders considerably di¤erent second order moments which in turn delivers di¤erent optimal policy prescriptions. Given these di¤erences in policy implications, future research should attempt to sort out the e¤ects of these alternative assumptions and estimation procedures on the characterization of optimal policy.

6.4

Optimal Policy under Parameter Uncertainty

We now determine the optimal policy that takes into account the e¤ects of uncertainty regarding

s

on the choice of optimal policy coe? cients. The policy problem is: Z ^ = arg min W0 p j s p( s j Yt )d s p

p2 p s

where minimization is subject to the same constraints as before and where p( s j Yt ) is the estimated posterior distribution of the structural parameters. In determining the optimal policy coe? cients the policymaker integrates out the uncertainty surrounding structural parameters by making use of the posterior distribution for these parameters obtained from model estimation.19 In contrast to section 6.2 this problem accounts for the covariance across all estimated

This accords with the analysis of Batini and Pearlman (2007) which considers the role of balance sheet e¤ects in a calibrated model. 19 It is important to note that this second approach to policy design, which entails discarding the draws of the policy parameters and retaining those of the non-policy block to represent p( s j Yt ), is consistent with our

18

28

model parameters, including the standard deviations of the shocks, which are part of

s:

Columns 4 - 6 of Table 3 report results of this exercise based on 5,000 draws for three objective functions.20 For Australia, and a zero weight on output in the objective function, there is little evidence of attenuated policy responses once parameter uncertainty is taken into account — compare column 1. Indeed, optimal policies are virtually identical. As the preference for output stabilization increases, the response to both output growth and in? ation rise, while other coe? cients are roughly unchanged. Hence, optimal policy under parameter uncertainty demands more aggressive policy in response to in? ation variations when output stabilization is relatively important. Concomitantly, the in? ation and output variances attached to these policies are lower and higher than when model parameters are known with certainty to the policymaker. Brainard’ seminal insight appears not to hold true s in this estimated small open economy model. Note that uncertainty does e¤ect outcomes, judging from the substantially larger losses in output in the last three columns, which is rationalized by the larger responses to in? ation already mentioned. For Canada, results are broadly similar. In the case of a low weight on output stabilization,

y

= 0, there is little change in the optimal policy coe? cients relative to the certain parameter

case. As output stabilization becomes a greater priority, policy becomes more aggressive when compared to the certain parameter case not only for output but, as in Australia, for in? ation as well. As before, uncertainty does not engender attenuated policy responses. And, in contrast to the Australian case, the variability of output need not increase once uncertainty is taken into account. New Zealand reveals yet a di¤erent pattern of results. For objective functions giving less weight to output stabilization, policy response coe? cients tend to be attenuated. This is true for both

y

= 0 and

y

= 0:5. When a unit weight is given to output stabilization,

estimation of the DSGE models. This is because Bayesian MCMC methods yield draws that correspond to the marginal densities of the model parameters. What we would have not been able to do, given our approach to inference, is to make any statements that required the conditional densities, say p( s j Yt ; p ), since we do not have samples from these ordinates in the estimation. 20 As in section 6.2 we also account for small sample uncertainty. For each parameter draw we generate 100 arti…cial samples of length equal to the data, after discarding the initial 50 observations. Optimal policy hence minimizes the average loss over 250,000 samples. Computational capacity prevents using all parameter draws generated by the MCMC. However, the dispersion in a sample of 5000 is almost identical to that in the pooling of all draws since the former are closer to an ideal independent sample.

29

the optimal policy coe? cients under uncertainty are roughly equal to those obtained ignoring the dispersion in the non-policy parameters. Despite this near equality on policy coe? cients, taking into account uncertainty produces larger output losses. Taken together the results indicate that parameter uncertainty fails to have clear implications for the design and outcomes of simple optimal monetary rules. Depending on the country at hand, more or less aggressive policy responses might obtain. As Chow (1975) notes, in a multivariate setting the conclusions of Brainard (1967) for attenuation in policy need not hold, depending on the covariance properties of the uncertain model parameters. Similarly, the robust control literature on optimal policy design, demonstrates that model uncertainty can lead to more aggressive policy settings — see Giannoni (2002). In addition, the associated losses may be larger or smaller once we account for parameter uncertainty, with di¤erences stemming mostly from the variability of output. It follows that resolution of the implications of uncertainty for policy design is largely an empirical matter. What is clear from the present analysis is that regardless of whether policymakers face parameter uncertainty or not, the optimal coe? cients on the exchange rate are always small. This is because of the exchange rate disconnect property and the additional variability in output, in? ation and interest rates engendered by stabilizing the exchange rate in this estimated model described earlier.

7

Robustness and Identi…cation

This section turns to some robustness exercises and discussion of identi…cation in our empirical model.

7.1

Unobserved Foreign Block

Rather than modeling the foreign block as being driven by a VAR in observed U.S. in? ation, output and nominal interest rates, we instead treat this component of the model as unobserved following the analysis of Lubik and Schorfheide (2003). Two observations motivate this alternative speci…cation. First, while for Canada the use of U.S. data as proxy for the foreign block may be plausible, it seems less appropriate in the case of Australia and New 30

Zealand where construction of trade-weighted indices of the relevant foreign variables — including, for instance, Japan — would be more desirable. Furthermore, this renders the model more agnostic about the precise nature of the foreign disturbances and allows evaluating the sensitivity of results to the choice of observables used in estimation. Second, and related to this last point, the estimated model of section 2 is prone to some of the di? culties detailed in Justiniano and Preston (2006). In particular, variance decompositions reveal a limited role for foreign sourced disturbances in the evolution of domestic variables. The following investigates whether it is this feature of the model which engenders a negligible role for stabilizing exchange rate ? uctuations in the design of optimal policy rules. We assume that foreign output, in? ation and interest rate shocks follow second order autoregressive processes. The priors used in estimation coincide with those employed in section 4, with appropriate adjustments arising from the di¤erent treatment of the foreign block. Table 4 presents the resulting estimates. For all three countries, while the intertemporal elasticity of substitution is very similar to the baseline model (observable foreign block), the inverse Frisch elasticity is slightly higher here. The Calvo parameters are quite stable as well, except for the degree of stickiness in home goods prices for Canada which is substantially larger with an unobserved foreign block. Canada also exhibits a greater degree of habit persistence and more aggressive responses to in? ation relative to the baseline model. Regarding the properties of shocks, risk premium disturbances are less persistent for all three countries, while disturbances to foreign interest rates are somewhat more volatile. These parameter shifts largely take place to exploit the ? exibility permitted by having an unobserved foreign block. Because the model is no longer constrained to …t the U.S. time series it is free to exploit the variation inherent in these shocks to …t the domestic observable series. In particular, the restriction imposed by interest parity would seem to be substantially loosened here. Not surprisingly, foreign disturbances are now found to explain a greater fraction of the variation in domestic observables than in the model with an observable foreign block. Given these estimates, we revisit the optimal policy exercises conducted earlier: the results

31

are reported in Table 5. Casual inspection reveals the optimal policy coe? cients on the nominal exchange rate to be less than 0.02 when model parameters are known with certainty to policymakers and less than 0.05 when model parameters are uncertain. As noted earlier, for these response coe? cients, a one standard deviation movement in the exchange rate implies a very small change in nominal interest rates. The intuition for this …nding is similar to the baseline case: exchange rate disconnect divorces movements in the exchange rate from movements in other domestic series. With the foreign block unobserved this disconnect is less striking, particularly for output, than when the foreign block is observed. Nonetheless, having monetary policy respond to exchange rate movements forces in? ation and interest rates to inherit the variability of risk premium and particularly cost-push shocks. This increase in their variance results in larger losses.21 As to the question of whether parameter uncertainty leads to cautious or aggressive policy, the results portray a mixed message once again. Depending on the country; the weight given to output stabilization; and the particular policy coe? cient under consideration, policy can be more or less aggressive. This is consistent with the theory referenced earlier. As for

outcomes, the resulting losses may di¤er, sometimes substantially, once parameter uncertainty is accounted for. This is mostly due to the variance of output and aligns well with the changes in optimal coe? cients. We conclude that the policy implications of parameter uncertainty are model and data speci…c and must be examined carefully on a case-by-case basis.

7.2

Matters of Identi…cation

A number of recent papers have addressed identi…cation problems and conditions for identi…cation in medium scale dynamic stochastic general equilibrium models. Lubik and Schorfheide (2005) and Justiniano and Preston (2006) discuss speci…c identi…cation issues in open economy models. More general discussions are provided by Beyer and Farmer (2005), Fukac, Pagan, and Pavlov (2006), Canova and Sala (2005), Cochrane (2007) and Iskrev (2007). These papers explore a range of identi…cation issues emerging from both the nature of estimation —

In an earier version of this paper we did not use the terms of trade and had also arbitrarily removed a few shocks, to force an even greater role for the unobserved foreign disturbances. Nonetheless, optimal policy was once again characterized by a lack of response to the exchange rate.

21

32

method of moments and likelihood based estimation — and economic structure. Adolfson and Linde (2007) performs a number of Monte Carlo exercises to examine local identi…cation in a medium scale small open economy model. Collectively, these papers underscore that identi…cation problems can plague estimation of models of the kind developed here. While considerable care was taken to ensure estimation resulted in a unique mode for our baseline model, when the foreign block is treated as unobserved a non-trivial identi…cation issue arises for Australia. Two modes are estimated that achieve almost identical posterior densities. Table 6 reports parameters that exhibit di¤erences across these two modes, together with the associated log posteriors. Most notable are the higher degree of nominal rigidity in home good prices,

H,

for the …rst mode and the greater persistence and volatility of

a

technology shocks for the second mode (

and sda )

The variance decompositions in Table 7 evidence that these two parameter con…gurations imply rather di¤erent contributions of shocks for output and in? ation. Preference shocks explain almost half of in? ation variability in the second mode, compared to 34 percent of its variance in the …rst mode. The reverse pattern is true for technology shocks (20 versus 38 percent). In contrast, technological disturbances explain the bulk of output variations in the second mode (92 percent variance share) while they retain an important but more modest role in the …rst mode (38 percent). It is di? cult to isolate how individual parameters a¤ect these results. Scrutiny of unreported impulse response functions suggests that for in? ation the changing contribution of shocks is mostly attributable to di¤erences in the estimated Calvo parameter for home goods. Indeed, the lower degree of price stickiness in the second mode rationalizes larger responses to preference shocks all else equal, and a more muted response to technology disturbances. This is a salient di¤erence of the impulse response functions across these two modes. As for output, the higher variance and autocorrelation of technology shocks accounts, at least in part, for the drastic increase in the contribution of these shocks for the second mode, despite the lower degree of nominal rigidities. Table 8 characterizes optimal policy assuming policymakers treat as certain the parameters from each individual mode, as opposed to the median of the draws reported in table 5.

33

While policy remains highly inertial — a di¤erence rule for the nominal interest rate is optimal — the prescribed optimal policy coe? cients are rather di¤erent for in? ation and output growth. The …rst mode has much weaker response coe? cients to in? ation for all weights on output stabilization. In contrast, optimal policy tends to respond more strongly to output growth. Intuitively, it would be reasonable to conjecture — given the di¤erences in estimated

H; a

and sda — that optimal policy would prescribe strong responses to in? ation in the …rst

mode and a weaker response in the second. That this is not the case stems from the changing contribution of shocks adduced above, which calls for more activist monetary policy in response to preference shocks — hence, variations in in? ation — in the second mode. Overall, as evidenced by the …nal row of Table 8, which reports the losses, the policy implications of these two parameter con…gurations are clearly di¤erent. Comparing the optimal policy results of Table 5 to those in Table 8 permits an additional insight on how identi…cation impacts policy design. The calculations in Table 5 were based on estimates from the MCMC Metropolis-Hastings algorithm using as starting values for the multiple chains draws around the …rst, highest, mode in Table 7. Focusing on the in? ation response coe? cients in each of these tables, an interesting pattern emerges: for the cases in which

y

> 0, the optimal coe? cients of Table 5 lie between the policy coe? cients associated

with each of the two modes reported in Table 8. The identi…cation problem a¤ects inference in the neighborhood of the …rst mode as the MCMC algorithm takes some draws from the posterior distribution of the second mode.22 Indeed, the posterior distribution of

a,

for in-

stance, is clearly bimodal. Even though it may appear that local identi…cation is achieved, a second local peak a¤ects inference and policy design. This example underscores that identi…cation problems can have implications for policy design. Moreover, it emphasizes the importance of using additional data to mitigate identi…cation issues. In our baseline, using observed series to …t the foreign block of this small open economy model, helps to better disentangle the e¤ects of various latent variables and exogeWe use a t distribution, rather than a normal, as a proposal for the MCMC and use very low degrees of freedom to allow for possibly large steps that may facilitate the transition across modes. We have also tried starting the MCMC sampler around the second, lower, mode. In both cases the draws are still mostly drawn from the higher mode although for some parameters Kernel estimates reveal the presence of a second peak.

22

34

nous shocks. By dropping these observables, there is insu? cient information in the domestic series, the terms of trade and the real exchange rate to pin down the e¤ects of the various disturbances. This leads to the possibility of multiple modes. As a …nal example, an earlier version of this paper estimated the unobserved foreign block model without using terms of trade data but reducing the number of domestic shocks. In this case identi…cation problems appeared to be ameliorated. Nonetheless, the absence of a response to the nominal exchange rate in optimal policy was seen once again, for the reasons discussed earlier.

8

Conclusions

This paper analyzes optimal policy design in an estimated small open economy for Australia, Canada and New Zealand. Motivated by the theoretical literature on local currency pricing, the central question is whether optimal policy responds to nominal exchange rate variations in a class of generalized Taylor rules. The role of parameter uncertainty in policy design is also evaluated. The central …ndings are twofold. First, within the class of rules that we consider, it is not optimal for policy to respond to nominal exchange rate variations. This is true regardless of country, whether policymakers face parameter uncertainty or not, the precise set of observables and shocks used to estimate the model, as well as the relative weight of the objectives in the loss function. This result is somewhat surprising given the presence of frictions in import goods markets that generate departures from the law of one price. Several recent papers have focused on this aspect of the speci…cation to provide a rationale for managing exchange rate ? uctuations in order to achieve in? ation and output stabilization. Second, parameter uncertainty may lead policymakers to respond more or less aggressively to variables that appear in their policy rule. Depending on the country; model; and speci…c policy weights under consideration, either outcome is possible. This suggests that generic empirical implications of parameter uncertainty for policy design are unlikely to be available, consistent with the theoretical predictions of Chow (1975). Finally, we provide an example of how parameter identi…cation may a¤ect policy design 35

and its associated outcomes. A more thorough and general analysis of this last issue is required given the growing role of DSGE models as inputs for the conduct of monetary policy in various central banks.

References

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Schorfheide, F. (2000): “Loss Function-Based Evaluation of DSGE Models,” Journal of Applied Econometrics, 15, 645– 670. Smets, F., and R. Wouters (2002): “Openness, imperfect exchange rate pass-through and monetary policy,”Journal of Monetary Economics, 49, 947– 981. (2003): “An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area,”Journal of the European Economic Association, 1(5), 1123– 1175. Svensson, L. E. (1997): “In? ation Forecast Targeting: Implementing and Monitoring In? ation Targets,”European Economic Review, 41, 1111– 1146. (1999): “In? ation Targeting as a Monetary Policy Rule,” Journal of Monetary Economics, 43, 607– 654. (2000): “Open-Economy In? ation Targeting,” Journal of International Economics, 50, 155– 183. Woodford, M. (2003): Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton University Press.

40

Table 1: Prior Densities and Posterior Estimates for Baseline (Observed Foreign Block)

Posterior

/2

Prior

Prior Density 1/ Mean Std [5,95] Prob [5,95] Prob Median Std Median Std Median Std

AUSTRALIA

CANADA

NEW ZEALAND

Coefficients

[5,95] Prob

Inverse Intertemporal Elasticity of Substitution

σ G 0.88 1.26 0.68 0.41 0.80 0.30 0.06 0.10 0.74 2.01 0.08 0.04 [ 0.08 , 0.21 ] 0.74 0.23 [ 0.38 , 1.15 ] 0.29 0.67 0.06 [ 0.32 , 0.51 ] 0.08 [ 0.68 , 0.95 ] 0.06 [ 0.20 , 0.40 ] 0.05 [ 0.01 , 0.17 ] 0.09 [ 0.02 , 0.31 ] 0.04 [ 0.66 , 0.80 ] 0.15 [ 1.78 , 2.27 ] 0.03 [ 0.04 , 0.13 ] 0.07 [ 0.20 , 0.42 ] 0.16 [ 0.44 , 0.96 ] 0.05 [ 0.60 , 0.77 ] 0.68 0.29 0.67 0.08 0.11 0.11 0.82 2.33 0.06 0.07 0.45 0.57 [ 0.54 , 2.42 ] 1.12 0.18 [ 0.63 , 1.22 ] 1.42 G B B G B B B B G G G G 0.25 0.13 0.25 0.13 0.25 0.13 0.09 0.14 1.50 0.30 1.83 0.50 0.25 0.84 0.50 0.25 0.07 0.07 [ 0.01 , 0.22 ] 0.03 [ 0.78 , 0.88 ] 0.20 [ 1.52 , 2.18 ] 0.05 [ 0.03 , 0.20 ] 0.50 0.25 0.05 0.05 [ 0.01 , 0.16 ] 0.50 0.25 0.33 0.09 [ 0.17 , 0.47 ] 1.50 0.75 0.58 0.07 [ 0.52 , 0.74 ] 0.50 0.10 0.55 0.06 [ 0.45 , 0.65 ] 0.50 0.10 0.79 0.08 [ 0.60 , 0.87 ] 1.50 0.75 1.12 0.57 [ 0.45 , 2.27 ] 1.20 0.40 1.31 0.31 [ 0.89 , 1.89 ] φ θH θF η h δH δF ψi ψπ ψy ψΔe ψΔy

0.33 [ 0.94 , 2.01 ] 0.65 [ 0.41 , 2.44 ] 0.07 [ 0.57 , 0.79 ] 0.06 [ 0.19 , 0.38 ] 0.07 [ 0.58 , 0.81 ] 0.05 [ 0.02 , 0.18 ] 0.09 [ 0.02 , 0.32 ] 0.11 [ 0.02 , 0.35 ] 0.03 [ 0.77 , 0.86 ] 0.24 [ 1.99 , 2.78 ] 0.03 [ 0.03 , 0.12 ] 0.03 [ 0.03 , 0.13 ] 0.13 [ 0.25 , 0.68 ]

Inverse Frisch

Calvo domestic prices

Calvo import prices

Elasticity H-F goods

Habit

Indexation domestic

Indexation foreign

Taylor rule, smoothing

Taylor rule, inflation

Taylor rule, output

Taylor rule, exchange rate

Taylor rule, output growth

Table 1: Prior Densities and Posterior Estimates for Baseline (Observed Foreign Block)

Posterior

/2

Prior

Prior Density 1/ Mean Std Median Std [5,95] Prob Median Std [5,95] Prob Median Std [5,95] Prob

AUSTRALIA

CANADA

NEW ZEALAND

Coefficients

Technology

ρa B B B B I I I I I I I I 0.50 inf 1.58 0.50 inf 0.35 0.50 inf 0.16 0.50 inf 0.26 0.03 [ 0.22 , 0.32 ] 0.03 [ 0.12 , 0.22 ] 0.09 [ 0.22 , 0.52 ] 0.51 [ 0.99 , 2.59 ] 0.50 inf 0.37 0.11 [ 0.27 , 0.62 ] 0.50 inf 0.12 0.01 [ 0.10 , 0.13 ] 0.15 0.42 0.29 0.17 0.20 2.01 0.50 inf 0.48 0.04 [ 0.43 , 0.55 ] 0.52 0.50 inf 0.35 0.03 [ 0.31 , 0.40 ] 0.36 0.50 0.25 0.94 0.04 [ 0.87 , 0.97 ] 0.97 0.01 [ 0.94 , 0.99 ] 0.03 [ 0.32 , 0.41 ] 0.04 [ 0.46 , 0.59 ] 0.01 [ 0.14 , 0.17 ] 0.09 [ 0.27 , 0.56 ] 0.03 [ 0.25 , 0.36 ] 0.02 [ 0.13 , 0.20 ] 0.03 [ 0.15 , 0.26 ] 0.53 [ 1.30 , 3.02 ]

[

0.80 0.80 0.80 0.10 0.94 0.02 [ 0.89 , 0.97 ] 0.95 0.03 [ 0.90 , 0.98 ] 0.95 0.98 0.34 0.48 0.10 0.77 0.23 0.22 0.23 7.27 0.10 0.93 0.02 [ 0.88 , 0.96 ] 0.95 0.02 [ 0.92 , 0.98 ] 0.94

0.10

0.69

0.13 [ 0.50 , 0.92 ]

0.90

0.04 [ 0.83 , 0.95 ]

0.85

0.06 [ 0.73 , 0.93 ] 0.02 [ 0.90 , 0.97 ] 0.03 [ 0.90 , 0.98 ] 0.01 [ 0.97 , 0.99 ] 0.03 [ 0.30 , 0.39 ] 0.04 [ 0.42 , 0.55 ] 0.01 [ 0.08 , 0.12 ] 0.24 [ 0.49 , 1.24 ] 0.03 [ 0.19 , 0.28 ] 0.04 [ 0.17 , 0.31 ] 0.05 [ 0.17 , 0.32 ] 2.86 [ 4.40 , 12.85 ]

Preferences

ρg ρrp ρcp sdπ* sdy* sdi* sda sdmp sdg sdrp sdcp

Risk premium

Import cost-push shock

sd foreign inflation

sd foreign output

sd foreign interest rates

sd technology

sd taylor rule

sd preferences

sd risk premium

sd import cost-push

1/ Distributions, N Normal, B Beta, G Gamma, I Inverse-Gamma 1. Calibrated β=0.99 and χ=0.01. Also, the share of openness is calibrated to the average share of exports and imports to GDP in our sample, which equals 0.185 for Australia, 0.28 for Canada and 0.29 for New Zealand.

2/ Corresponds to median and posterior percentiles from 5 MCMC chains of 100,000 draws each, in which 40,000 draws were used as an initial burn-in phase, and only one in every ten draws retained from the remaining 60,000, in each chain. Convergence diagnostics were assessed using trace plots and the potential scale reduction factors for the variance and 95% posterior intervals.

Table 2: Data and model implied standard deviations and first order autocorrelations

Median and [5,95] posterior band implied by the estimated baseline model

/1

AUSTRALIA

Inflation Real exchange rate (fd) Interest Rate Output Terms of trade (fd) Data Standard Deviation 0.76 4.72 1.09 1.98 1.98 Data Autocorrelations 0.63 0.15 0.97 0.92 0.44 0.93 0.93 Model Standard Deviation 0.79 [ 0.63 4.78 [ 4.04 0.70 [ 0.46 1.82 [ 1.27 2.43 [ 1.96 Model Autocorrelation 0.59 [ 0.42 0.00 [ -0.17 0.91 [ 0.82 0.88 [ 0.78 0.56 [ 0.40 0.92 [ 0.81 0.96 [ 0.91 , , , , , 1.00 5.66 1.10 2.79 3.01 ] ] ] ] ]

/2

Inflation Real Exchange Rate (fd) Interest Rate Output Terms of trade (fd) Real exchange rate (level) Terms of trade (level)

, , , , , , ,

0.74 0.18 0.96 0.94 0.70 0.97 0.98

] ] ] ] ] ] ]

CANADA

Inflation Real exchange rate (fd) Interest Rate Output Terms of trade (fd) Data Standard Deviation 0.61 2.25 0.88 2.88 1.32 Data Autocorrelations 0.66 0.27 0.92 0.97 0.18 0.98 0.98 Model Standard Deviation 0.61 [ 0.48 2.42 [ 2.06 0.65 [ 0.42 2.34 [ 1.53 1.74 [ 1.45 Model Autocorrelation 0.60 [ 0.42 0.07 [ -0.10 0.91 [ 0.81 0.93 [ 0.87 0.49 [ 0.33 0.93 [ 0.85 0.95 [ 0.90 , , , , , 0.78 2.84 1.05 3.80 2.08 ] ] ] ] ]

Inflation Real Exchange Rate (fd) Interest Rate Output Terms of trade (fd) Real exchange rate (level) Terms of trade (level)

, 0.75 0.23 , 0.96 , 0.97 , 0.63 , 0.97 , 0.98

] ] ] ] ] ] ]

NEW ZEALAND

Inflation Real exchange rate (fd) Interest Rate Output Terms of trade (fd) Data Standard Deviation 0.56 4.05 0.73 2.34 2.09 Data Autocorrelations 0.40 0.41 0.92 0.88 -0.05 0.96 0.87 Model Standard Deviation 0.77 [ 0.61 4.38 [ 3.68 0.76 [ 0.48 2.80 [ 1.88 2.55 [ 2.11 Model Autocorrelation [ 0.36 0.55 0.02 [ -0.15 0.93 [ 0.85 0.89 [ 0.79 0.39 [ 0.21 0.93 [ 0.83 0.94 [ 0.88 , , , , , 0.99 5.22 1.23 4.48 3.08 ] ] ] ] ]

Inflation Real Exchange Rate (fd) Interest Rate Output Terms of trade (fd) Real exchange rate (level) Terms of trade (level)

, 0.72 0.19 , 0.97 , 0.95 , 0.54 , 0.97 , 0.97

] ] ] ] ] ] ]

/1 Model standard deviations and first order autocorrelations are computed by generating, for each parameter draw, 100 replications of length equal to the sample size for each country, after discarding the first 50 observations. For each replication and parameter pair we compute the standard deviation and autocorrelations. We report medians and [5,95] posterior bands of the implied statistics. /2 fd corresponds to the log first-difference

Table 3: Optimal Policy and Uncertainty for Baseline

MEDIAN OF DRAWS /1

Relative Weight on Output

OVER DRAWS /2

Relative Weight on Output

0

Coefficients Interest Rate Inflation Output Nominal Exchange Rate Output Growth Variance Inflation Interest rates Output Loss Coefficients Interest Rate Inflation Output Nominal Exchange Rate Output Growth Variance Inflation Interest rates Output Loss Coefficients Interest Rate Inflation Output Nominal Exchange Rate Output Growth Variance Inflation Interest rates Output Loss

0.5

1.00 0.91 0.06 0.00 2.02 0.59 0.35 1.44 1.66

1

Panel A: Australia 1.00 0.80 0.10 0.00 3.15 0.87 0.47 0.89 2.22 Panel B: Canada

0

1.00 1.02 0.00 0.00 0.25 0.11 0.29 5.18 0.40

0.5

1.00 1.14 0.06 0.00 2.06 0.53 0.34 2.39 2.06

1

1.00 1.07 0.08 0.01 3.07 0.78 0.45 1.80 3.04

1.00 1.01 0.00 0.00 0.28 0.13 0.31 4.38 0.44

1.00 2.18 0.00 0.00 0.25 0.03 0.20 7.38 0.23

1.00 2.27 0.01 0.02 2.57 0.39 0.20 5.59 3.39

1.00 1.57 0.04 0.01 3.10 0.85 0.30 4.83 5.97 Panel C: New Zealand

1.00 2.09 0.00 0.00 0.24 0.03 0.22 7.14 0.25

1.00 2.38 0.01 0.00 2.72 0.38 0.22 5.31 3.26

1.00 1.85 0.03 0.02 3.62 0.79 0.32 4.62 5.73

1.00 1.49 0.00 0.00 0.18 0.06 0.34 10.53 0.40

1.00 1.91 0.03 0.02 1.88 0.72 0.28 7.81 4.90

1.00 1.48 0.07 0.01 2.56 1.56 0.37 6.53 8.45

1.0 1.4 0.0 0.0 0.1 0.07 0.32 11.59 0.39

1.0 1.7 0.0 0.0 1.7 0.73 0.28 8.63 5.33

1.0 1.5 0.1 0.0 2.5 1.54 0.37 7.40 9.31

1/ Optimal coefficients are obtained by minimizing the weighted sum of variances for inflation, nominal interest rates and output, with equal weights on inflation and interest rates, but varying the relative weight on output. All parameters other than those in the Taylor-type rule are fixed at the median of the MCMC estimates. The variances are obtained by simulation with the same settings as reported in Table 2. 2/ In optimizing over the draws, we use a subset of 5000 draws, taken at equally spaced intervals, from the generated samples obtained with the MCMC simulator. For each candidate set of policy parameters we compute the loss over these draws and average the resulting loss. Once again, variances are obtained by simulation.

Table 4: Posterior Estimates for Unobserved Foreign Block

/1

Posterior

/2

AUSTRALIA

CANADA

NEW ZEALAND

Coefficients Median Std [5,95] Prob Median Std Median Std 2.01 ] [5,95] Prob 1.50 ] [5,95] Prob 1.79 ]

Inverse Intertemporal Elasticity of Substitution

σ 1.37 1.15

2.37 ] 2.74 ]

0.35 [ 0.88 0.60 [ 0.46 0.08 [ 0.61

0.87 ] 0.89 ]

0.91 1.46 0.82 0.38

0.48 ]

0.30 [ 0.53 0.65 [ 0.64 0.06 [ 0.69 0.06 [ 0.28 0.07 [ 0.61

0.82 ]

1.24 1.29 0.65 0.29 0.73 0.09 0.13 0.07 [ 0.53 0.05 [ 0.21 0.08 [ 0.62 0.06 [ 0.02 0.12 [ 0.03 0.69 [ 0.46

2.71 ]

0.30 [ 0.80

Inverse Frisch

φ θH 0.80 0.52

0.62 ]

Calvo domestic prices

0.77 ]

Calvo import prices

θF 0.07 [ 0.40 0.07 [ 0.55

0.77 ]

0.38 ]

Elasticity H-F goods

η 0.63 0.35

0.50 ]

0.69 0.54 0.05 0.11 0.77 1.79 0.10 0.33 0.10 [ 0.02 0.04 [ 0.70 0.18 [ 1.53 0.04 [ 0.04 0.07 [ 0.23 0.05 [ 0.01 0.09 [ 0.39

0.68 ]

0.87 ]

Habit

h 0.09 [ 0.19 0.05 [ 0.01

0.16 ]

0.21 ]

Indexation domestic

δH 0.05 0.07

0.25 ]

0.15 ]

0.39 ]

Indexation foreign

δF 0.08 [ 0.01 0.03 [ 0.77

0.88 ]

0.33 ]

0.10

0.82 ]

0.10 [ 0.02 0.81

2.10 ]

0.33 ]

Taylor rule, smoothing

ψi 0.84 1.82

2.19 ]

0.03 [ 0.76 2.26

0.16 ]

0.86 ]

Taylor rule, inflation

ψπ 0.22 [ 1.48 0.06 [ 0.03 0.04 [ 0.07 0.23 [ 0.36

0.22 ]

0.25 [ 1.90 0.05

0.46 ]

2.71 ]

Taylor rule, output

ψy 0.09 0.13 0.71 ψΔe ψΔy

0.03 [ 0.02 0.06 0.03 [ 0.03

0.11 ]

Taylor rule, exchange rate

0.20 ]

0.12 ]

Taylor rule, output growth

1.14 ]

0.63

0.16 [ 0.39

0.91 ]

0.42

0.13 [ 0.23

0.65 ]

Table 4: Posterior Estimates for Unobserved Foreign Block

/1

Posterior

/2

AUSTRALIA

CANADA

NEW ZEALAND

Coefficients Median Std [5,95] Prob Median Std Median Std 0.92 ] [5,95] Prob 0.91 ] [5,95] Prob 0.95 ]

Technology

ρa 0.66 0.92

0.95 ] 0.94 ]

0.14 [ 0.46 0.02 [ 0.87 0.10 [ 0.64

0.95 ] 0.94 ]

0.78 0.90 0.81 0.98

0.99 ]

0.10 [ 0.59 0.03 [ 0.85 0.12 [ 0.58 0.01 [ 0.95 0.27 [ 0.16

0.84 ]

0.87 0.92 0.81 0.98 0.27 0.58 0.21 0.11 [ 0.58 0.01 [ 0.97 0.19 [ 0.15 0.81 [ 0.18 0.10 [ 0.14 0.03 [ 0.87

0.96 ]

0.06 [ 0.76

Preferences

ρg ρrp 0.86 0.96

0.98 ]

Risk premium

0.94 ]

Import cost-push shock

ρcp 0.03 [ 0.90 0.11 [ 0.15

0.50 ]

0.99 ]

sd foreign inflation

sdπ* 0.26 0.34

1.47 ] 1.07 ]

0.30 0.41 0.17 0.27 0.25 0.20 0.18 2.36 0.08 [ 0.20 0.03 [ 0.21 0.03 [ 0.15 0.04 [ 0.12 0.71 [ 1.50 0.04 [ 0.12

0.25 ]

0.66 ]

sd foreign output

sdy* 0.38 [ 0.16 0.09 [ 0.17

0.47 ]

0.29 [ 0.17

2.60 ]

sd foreign interest rates

sdi* 0.28 0.37

0.62 ]

0.37 ]

sd technology

sda 0.11 [ 0.28 0.03 [ 0.22

0.32 ]

0.45 ]

0.90

0.31 ]

0.34 [ 0.53 0.23

0.26 ]

1.65 ]

sd taylor rule

sdmp 0.26 0.17

0.23 ]

0.03 [ 0.19 0.24

0.26 ]

0.28 ]

sd preferences

sdg 0.03 [ 0.13 0.11 [ 0.16 0.73 [ 1.11

0.51 ]

0.04 [ 0.17 0.22

3.79 ]

[

0.32 ]

sd risk premium

sdrp 0.30 1.86 sdcp

0.08 [ 0.14 7.10

0.37 ]

sd import cost-push

3.45 ]

2.13 [ 4.33 10.97 ]

]

1/ Priors and calibrated parameters are the same as in table 1, except for the foreign block which is now driven by independent AR(2) processes for the latent foreign variables.

2/ Corresponds to median and posterior percentiles from 5 MCMC chains of 100,000 draws each, in which 40,000 draws were used as an initial burn-in phase, and only one in every ten draws retained from the remaining 60,000, in each chain. Convergence diagnostics were assessed using trace plots and the potential scale reduction factors for the variance and 95% posterior intervals.

Table 5: Optimal Policy and Uncertainty when Foreign Block is Unobserved MEDIAN OF DRAWS

Relative Weight on Output

/1

OVER DRAWS /2

Relative Weight on Output

0.0

Coefficients Interest Rate Inflation Output Nominal Exchange Rate Output Growth Variance Inflation Interest rates Output Loss

0.5

1.0

0.0

0.5

1.0

1.00 0.92 0.00 0.00 0.24 0.13 0.30 4.37 0.43

1.00 0.81 0.09 0.01 1.88 0.58 0.38 1.22 1.57

Panel A: Australia 1.00 1.00 0.73 0.71 0.00 0.14 0.01 0.00 0.14 3.01 0.82 0.50 0.70 2.02 0.15 0.42 6.91 0.56

1.00 0.82 0.07 0.04 1.58 0.56 0.51 3.80 2.97

1.00 0.79 0.10 0.05 2.58 0.87 0.64 3.18 4.69

Coefficients Interest Rate Inflation Output Nominal Exchange Rate Output Growth Variance Inflation Interest rates Output Loss

1.00 1.05 0.00 0.00 0.13 0.06 0.21 8.95 0.27

1.00 0.59 0.05 0.01 1.94 0.78 0.42 1.96 2.17

Panel B: Canada 1.00 1.00 0.90 0.53 0.00 0.06 0.02 0.00 0.01 3.29 1.13 0.64 1.14 2.91 0.06 0.24 11.04 0.31

1.00 0.83 0.04 0.02 1.98 0.78 0.41 3.43 2.91

1.00 0.85 0.01 0.02 3.59 1.23 0.67 2.46 4.35

Coefficients Interest Rate Inflation Output Nominal Exchange Rate Output Growth Variance Inflation Interest rates Output Loss

1.00 1.52 0.00 0.00 0.12 0.04 0.27 13.01 0.31

1.00 1.86 0.02 0.02 1.75 0.63 0.24 10.44 6.08

Panel C: New Zealand 1.00 1.00 1.38 1.56 0.00 0.05 0.00 0.01 0.10 2.58 1.44 0.33 9.20 10.96 0.06 0.27 12.24 0.33

1.00 2.01 0.01 0.02 1.62 0.58 0.24 9.91 5.78

1.00 1.54 0.01 0.02 2.13 1.25 0.29 8.92 10.46

1/ Optimal coefficients are obtained by minimizing the weighted sum of variances for inflation, nominal interest rates and output, with equal weights on inflation and interest rates, but varying the relative weight on output. All parameters other than those in the Taylor-type rule are fixed at the median of the MCMC estimates in Table 4. The variances are obtained by simulation with the same settings as reported in Table 2. 2/ In optimizing over the draws, we use a subset of 5000 draws, taken at equally spaced intervals, from the generated samples obtained with the MCMC simulator. For each candidate set of policy parameters we compute the loss over these draws and average the resulting loss. Once again, variances are obtained by simulation.

Table 6: Selected Coefficients from the Two Modes for Australia when Foreign Block is Unobserved /1 Coefficients

Inverse Intertemporal ES Calvo domestic prices Elasticity H-F goods Habit Taylor rule, inflation Taylor rule, output growth Technology sd technology sd import cost-push σ θH η h ψπ ψΔy ρa sda sdcp

First Mode

1.43 0.82 0.59 0.38 1.71 0.75 0.65 0.31 1.88

Second Mode

1.30 0.62 0.71 0.23 1.91 0.52 0.93 0.51 2.22

Log Posterior

-870.07

-870.54

/1 Priors are as in table 4.

Table 7: Variance Decomposition for all-goods Inflation and Output in Australia for Two Modes when Foreign Block is Unobserved /1 Panel A. First Mode Foreign Shocks Monetary Policy Risk Premium Import Costpush

Series \ Shock Inflation Output

Neutral

Preference

0.11 0.08

0.38 0.30

0.10 0.08

0.34 0.25

0.02 0.00

0.04 0.29

Panel B. Second Mode Series \ Shock Foreign Shocks Neutral Monetary Policy Preference Risk Premium Import Costpush

Inflation Output

0.10 0.01

0.20 0.92

0.19 0.01

0.46 0.02

0.03 0.00

0.02 0.03

/1 Stationary variance decomposition at each of the modes reported in Table 6 for Australia

Table 8: Optimal Policy for Two Modes in Australia when Foreign Block is Unobserved

First Mode

Relative Weight on Output

Second Mode

Relative Weight on Output

0.0 Coefficients Interest Rate Inflation Output Nominal Exchange Rate Output Growth Variance Inflation Interest rates Output Loss 0.13 0.28 4.24 0.41 0.79 0.00 0.00 0.20

0.5

1.0

0.0

0.5

1.0

0.74 0.07 0.00 1.98

0.68 0.09 0.00 3.34

1.76 0.00 0.00 0.00

2.01 0.00 0.00 1.69

1.83 0.01 0.02 2.71

0.49 0.39 1.14 1.45

0.67 0.49 0.71 1.87

0.02 0.27 6.13 0.29

0.22 0.25 5.18 3.06

0.51 0.27 4.75 5.53

1/ Optimal coefficients are obtained by minimizing the weighted sum of variances for inflation, nominal interest rates and output, with equal weights on inflation and interest rates, but varying the relative weight on output. As in Tables 3 and 5, the variances are obtained by simulation. The two modes are reported in Table 6.

Figure 1: Optimal Coefficient on Exchange Rate as Weights Vary

Australia

0.2

0.1

0 1

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

weight output

weight nominal interest

Canada

0.2

0.1

0 1

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

weight output

weight nominal interest

New Zealand

0.2

0.1

0 1

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

weight output

weight nominal interest

Figure 2: Impulse Responses to Import Cost?Push Shock as Coefficient on Exchange Rate Varies

Inflation all goods 0.3 0.2 0.1 0 0 1 2 Output 0 0.15 0.1 0.05 0 ?0.05 0 0.8 0.6 0.4 0.2 0 0 1 2 3 4 1 2 3 4 0 1 2 LOP ?3 ?3.2 ?3.4 ?3.6 0 1 2 Inflation Imports 0.8 0.6 0.4 0.2 0 0 1 2 Real interest rate 0 ?0.1 ?0.2 0 1 2 3

0 0.20 0.40

Nominal interest rate 0 ?0.05 ?0.1 3 4 0 1 2 3 4 Nominal exchange rate growth

?1 ?2 3 4

Terms of trade growth

3

4

Inflation home goods

0.1 0.05 0 3 4

0

1

2

3

4

4

For Australia using optimal coefficients when weight on output is 0.5 (table 3) Optimal coefficient on exchange rate: solid ; counterfactually increased to 0.2: longer dash; further counterfactual increase to 0.4: shorter dash

Figure 3: Impulse Responses to Risk Premium Shock as Coefficient on Exchange Rate Varies

Inflation all goods 0.15 0.1 0.05 0 -0.05 0 0 -0.1 -0.2 -0.3 0 1 2 3 4 Terms of trade growth 1 0.5 0.5 0 0 -0.05 1 -0.1 -0.15 -0.2 0 1 2 Real interest rate 0.3 0.25 0.2 0.15 0.1 0 1 2 3 4

0 0.20 0.40

Nominal interest rate 0.25 0.2 0.15 0.1 0.05

1

2 Output

3

4 3 2 1 0

0

1

2

3

4

Nominal exchange rate growth

0

1

2 LOP

3

4

1.5 1

1

2

3

4

0

1

2 Inflation Imports

3

4

Inflation home goods

0.5 0 3 4 0 1 2 3 4

For Australia using optimal coefficients when weight on output is 0.5 (table 3) Optimal coefficient on exchange rate: solid; counterfactually increased to 0.2: longer dash; further counterfactual increase to 0.4: shorter dash