Can the new open economy macroeconomic model explain exchange rate fluctuations?

Journal of International Economics 72 (2007) 381 – 408 www.elsevier.com/locate/econbase

Can the new open economy macroeconomic model explain exchange rate fluctuations? ☆ Yongseung Jung ⁎ Department of Economics, Kyung Hee University, Hoegi-Dong 1, Dóngdaemun Gu, Seoul, South Korea Received 16 November 2001; received in revised form 27 June 2004; accepted 20 April 2006

Abstract This paper explores the successes and failures of the new open economy macroeconomics more critically by addressing the performance of the model at all frequencies along the line of Watson's [Watson, M.W., 1993. Measures of Fit for Calibrated Models, Journal of Political Economy 101, 1011-1041] measures of fit. This paper shows that the NOEM model with either PCP or PTM is not successful in generating the spectral density of the selected variables calculated from the data. In particular, the model cannot generate mass spectra of the exchange rates at low frequencies as in the data. It shows that the NOEM model with either separable preference or incomplete asset market cannot generate the typical hump-shaped spectra of exchange rates. © 2007 Elsevier B.V. All rights reserved. Keywords: Exchange rate volatilities; Measures of fit; New open economy macroeconomics; Spectral density; Taylor rule JEL classification: E52; F31

1. Introduction In recent years, a proliferation of new monetary models that incorporate imperfect competition and nominal rigidities into a dynamic stochastic general equilibrium have surfaced in ☆

I would like to acknowledge Mark Bils, John Campbell, Jang-Og Cho, Mark Spiegel, Alan Stockman, Tack Yun, and the seminar participants at the Asian-Pacific Economic Conference in Seoul, the Federal Reserve Bank at San Francisco, and the UC Riverside for their helpful comments and conversations. Specially, I would like to thank John Campbell for the valuable comments. This paper was completed while I was visiting the Research Department of Federal Reserve Bank at San Francisco. I also thank the Research Department and faculty for their hospitality. All errors are my own. ⁎ Tel.: +82 2 961 0962; fax: +82 2 961 0622. E-mail address: [email protected]. 0022-1996/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jinteco.2006.04.005

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macroeconomics.1 This kind of research program is closely linked to the recent theoretical development of international finance, which has become known as the new open economy macroeconomics. New open economy macroeconomics, embedding imperfect competition and nominal rigidities in a dynamic general equilibrium open economy attempts to explore empirical issues, such as the excessive exchange rate movements and liquidity effects which had been unaccounted for previously. In the theoretical development of international finance, Obstfeld and Rogoff (2000) argue that the assumption of sticky prices in the producer's currency is important for matching the behavior of the terms of trade. Their Redux model assumes no international market segmentation, favoring the producer–currency–pricing (hereafter PCP) approach in the exchange rate fluctuations. However, there is a large body of evidence against the law of one price. In particular, Engel (1999) and Chari, Kehoe, and McGrattan (2000) hereafter Chari et al. (2000)) have documented that the international deviations in tradable prices are responsible for the violation of the law of one price. In line with this empirical evidence, many authors, presuming that international markets for manufacturing goods are sufficiently segmented, have introduced the so-called ‘pricing-to-market (hereafter PTM)’ approach into the new open economy macroeconomic (hereafter NOEM) model. PTM with local-currency sticky prices breaks the link between home and foreign price levels and allows the real exchange rates to fluctuate.2 In particular, Betts and Devereux (1999, 2000) set up a full-fledged PTM model and show that the model outperforms the PCP model in tracking the real exchange rate movements. Notwithstanding these theoretical developments in international finance, relatively little empirical or quantitative studies have been done. Some studies have attempted to evaluate the quantitative importance of the mechanisms emphasized in the NOEM model either via calibration exercises or VAR econometric models. In important quantitative applications of the NOEM model in dynamic general equilibrium settings, Betts and Devereux (1999), Chari et al. (2000), and Kollman (1997) show the potential of the model to replicate international business cycle regularities including the variability of real and nominal exchange rates. In calibration exercises, Chari et al. (2000), and Kollman (1997) have evaluated the NOEM model with PTM by comparing the unconditional moments generated by the model with the unconditional moments observed in the data. In the econometric investigation, Betts and Devereux (1999) have shown that the NOEM model with PTM performs well in matching the stylized facts of the international monetary transmission mechanism as documented by VAR results. In frequency domain, King and Watson (1996), Stock and Watson (1999), and Watson (1993) document interesting stylized facts over business cycles. The selected real macroeconomic variables have common, hump-shaped growth rate spectra. That is, the spectra are relatively low at low frequencies, rise at middle frequencies, and then decline at high frequencies. However, no one in international finance has explored whether the current NOEM models can generate the dynamics of the selected variables at low and high frequencies in addition to business cycle frequencies. Because the height of the spectral density of the selected variable at each frequency indicates the extent of that frequency's contribution to the variance of the corresponding variable, the variance of the corresponding variable occurring between any two frequencies is given by the areas under the spectrum between those two frequencies. Therefore, one cannot argue that the NOEM model performs well in matching the volatile exchange rate movements by comparing 1

See Goodfriend and King (1997) for detailed discussions. Betts and Devereux (1999, 2000) refer to this pricing convention for exports as local currency pricing (LCP). See also Krugman (1987) Lane (2001). In this paper, PTM is used to mean PTM cum LCP. 2

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only the unconditional moments generated by the model with the unconditional moments observed in the data in a specific frequency band. Chari et al. (2000) and Kollman (1997) are no exception. Their quantitative evaluation of the NOEM model is insufficient in obtaining an overall fit of the model because they evaluate the performance of the model only at business cycle frequencies. Even if the spectral density of exchange rates constructed from the model has the inverse shape of the spectral density calculated from the data, the variance of exchange rates generated from the model can be equal to the variance of the exchange rates in the data because the variance is the area under the spectrum between 0.03 = 1/32 and 0.16 = 1/6 cycles per period. Therefore, to address more precisely and critically the NOEM model's performance, it is necessary to use more general measures of fit such as the spectral density and the spectrum of the error required to reconcile the model and the data as in Watson (1993). In this respect, the recent studies by Ellison and Scott (2000), and Jung (2004) deserve attention. Ellison and Scott (2000) examine the performance of a Calvo-type sticky price model with an exogenous monetary policy at both high frequencies and business cycle frequencies. Jung (2004) goes one step further to explore the role of external habit formation in the Calvo-type and Taylor-type sticky price models with an endogenous monetary policy. Both point out that the sticky price model fails because it generates insufficient output fluctuations at business cycle frequencies as well as excessive output volatility at high frequencies. In addition, they argue that it is desirable and necessary to address more critically the quantitative performance of the model with more general measures of fit if one wishes to gain more fruitful intuitions about the model. This paper critically examines the successes and failures of the NOEM model by addressing the performance of the model at all frequencies along the line of Watson (1993)'s measure of fit. For this purpose, I first set up a benchmark quantitative NOEM model with complete asset markets as in Chari et al. (2000). Then I evaluate the performance of the model with either PCP or PTM in terms of second moments over the business cycle frequencies as in Betts and Devereux (1999), and Chari et al. (2000). After that, I extend the evaluation to the overall fit of the model by comparing the spectral densities of the selected variables calculated from the model with those of the data. To address quantitative evaluation of the model more critically, I also discuss the performance of the model using Watson (1993)'s RMSAE. Finally, I perform sensitivity analysis with different assumptions about preference, asset market structure, shocks, and market frictions following Chari et al. (2000), Christiano, Eichenbaum and Evans (2003 hereafter CEE (2003)), and Smets and Wouters (2002) to obtain more robust results. The main findings of this paper can be summarized as follows. First, the NOEM model with either PCP or PTM is not successful in generating the hump-shaped spectral density for the selected variables calculated from the data. Watson (1993)'s RMSAE for output and price is greater than one, showing the poor performance of the model. The most dramatic failure of the model is the business cycle frequency fluctuation in exchange rates. The NOEM model with either PCP or PTM cannot explain the hump-shaped spectral density of exchange rates. Second, the model generates far too much volatility of the selected variables at high frequencies compared to the data. This fact mirrors Ellison and Scott (2000)'s findings in the closed economy model. Third, the introduction of separable preference into the model is not successful in generating the typical hump-shaped spectra of exchange rates whether one assumes either incomplete asset market or other shocks such as uncovered interest parity shocks and preference shocks. Finally, the extended NOEM model with more diverse market frictions such as habit persistence, capital adjustment cost in investment changes, and indexation in prices and wages as in CEE (2003) and Smets and Wouters (2002) is successful in generating the typical hump-shaped spectral density of the some selected variables including consumption. However, the model still fails in replicating

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the hump-shaped spectral density of exchange rates. Even if the uncovered interest parity condition holds, the monetary policy shock cannot lead to a hump-shaped response of exchange rate as long as the elements that disconnect the tight relation between interest differentials and expected exchange rate depreciation do not dominate the effect on interest rate differentials on the expected exchange rate depreciation. This paper is composed as follows. Section 2 discusses the features of the data, focusing on the fluctuations of exchange rates. Section 3 specifies the benchmark NOEM model with PCP as well as the NOEM model with PTM, and discusses the properties of the equilibrium. Section 4 discusses regarding the quantitative implications of the model. Section 5 performs a sensitivity analysis with the extended NOEM model and Section 6 contains concluding remarks. 2. Features of the data In this section, features of international business cycles are examined, focusing on the time series relationship between gross domestic product, exchange rate, interest rates, and prices. The statistical relationships presented in this section will be used to evaluate the performance of the NOEM model with PTM as well as with PCP. 2.1. Power spectrum of selected variables The power spectra of growth rates have important implications for the overall nature of business cycles. King and Watson (1996), and Watson (1993) present the power spectra of growth rates of selected macroeconomic variables, and discuss implications of the typical spectral shape

Fig. 1. A. Domestic output, B. Domestic price, C. Nominal exchange rate, D. Domestic interest rate.

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Table 1 Moments of the data Variable

SD

Y C I N P Y⁎ P⁎ r S ϵ

1.69 1.31 4.67 1.12 1.56 1.06 1.34 1.59 8.23 7.72

Cross autocorr.

Xt with GDP Yt+k

k=−4

−3

−2

−1

0

1

2

3

4

0.32 0.17 0.36 0.51 0.12 0.49 − 0.24 0.48 0.40 0.36

0.56 0.39 0.60 0.69 − 0.02 0.59 − 0.34 0.54 0.29 0.26

0.79 0.60 0.80 0.85 − 0.18 0.65 − 0.45 0.58 0.18 0.15

0.94 0.79 0.93 0.94 −0.34 0.66 −0.55 0.54 0.07 0.05

1.00 0.90 0.96 0.94 − 0.50 0.61 − 0.61 0.41 − 0.01 − 0.02

0.94 0.92 0.89 0.86 − 0.62 0.48 − 0.65 0.22 − 0.06 − 0.05

0.79 0.86 0.74 0.71 −0.71 0.33 −0.66 −0.02 −0.09 −0.06

0.56 0.75 0.54 0.51 − 0.77 0.20 − 0.62 − 0.27 − 0.10 − 0.06

0.32 0.60 0.33 0.31 − 0.79 0.09 − 0.54 − 0.48 − 0.11 − 0.05

(corr(Xt, Yt+k))

Note: The statics are based on logged and band-pass filtered quarterly data for 1972:1-2000:1. The statistics for Europe are trade-weighted aggregates of countries in the table. r is the federal fund rate.

of growth rates for business cycles.3 Before presenting the broad features of the selected variables, I review basic elements of time series in the frequency domain. One can decompose a covariance stationary variable xt into an integral of period components. Z p xt ¼ xt ðxÞdx: 0

Then the variance of the corresponding variable can be decomposed as: Z p zðxÞdx; varðxt Þ ¼ 2

ð1Þ

0

where the power spectrum z(ω) is the contribution to variance at frequency ω It is possible to interpret the typical spectral shape of growth rates in terms of the variance. That is, the height of x the spectra in Fig. 1 at cycles per period 2p shows the extent of that frequency's contribution to the variance of the corresponding variable's growth rate.4 King and Watson (1996) and Stock and Watson (1999) present the empirical result that the growth rate spectrum of the selected variables is relatively low at low frequencies, rises to a peak at a cycle length about 20–40 quarters and then declines at very high frequencies. Moreover, the business cycle interval contains the peak as well as the bulk of the variance of the growth rates of the selected variables as the spectra presented in the short dashed lines in Fig. 1 below show. The power spectrum of exchange rates, using bilateral exchange rates between the US and a European aggregate is no exception to this pattern.5 The estimated spectra of the selected variables have the following characteristics in relation to business cycles. First, exchange rates as well as output show the typical spectral shape of growth rates: There is a common, hump-shaped spectrum in output and exchange rates. Because the 3

The typical spectral shape of growth rates has played an important role in the discussion of the nature of business cycles. See Watson (1993), King and Watson (1996), and Stock and Watson (1999) for more details. 4 For each selected variable in Fig. 1, solid lines and long dashed lines show the spectrum of a model and the spectrum of the data. Dotted lines are the spectrum of the error required to reconcile the models and the data. 5 A European aggregate consists of nine countries as in Charie et al. (2000): Australia, Finland, France, West Germany, Italy, Norway, Spain, Switzerland, and the UK.

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height of the spectrum of each variable reflects the relative volatility, the volatility of exchange rates is about four times larger than that of output. Second, the spectrum of interest rates shows that variations in the monetary policy may be an important source of business cycles as noted by King and Watson (1996). Third, the spectrum of inflation rates has a peak at a business frequency as in the spectrum of other variables, which implies a role for nominal rigidity in the business cycle. 2.2. Business cycle comovements Features of business cycles in terms of cross autocorrelations are also useful to examine. In Table 1, various moments of selected variables calculated from the estimated spectral density matrix with only the business cycle (6–32 quarter) frequencies of the U.S. and Europe over the sample period 1972:1 through 2000:1 are shown. Three key features are evident in Table 2. First, the nominal (real) exchange rate is very volatile. The standard deviation of the nominal (real) exchange rate between the US and Europe is 8.23 (7.72), about 4.8 (4.6) times the volatility of the U.S. output. Second, there are systematic movements of nominal interest rates in relation to output. The correlation between nominal interest rates and future output are negative (corr(rt, yt+4) = − 0.48). Third, the correlations between prices and future output are also negative (corr( pt, yt+4) = − 0.79). These cross correlations imply that nominal interest rates and prices serve as countercyclical leading indicators in the business cycle. The model's prediction for cross correlation properties of selected variables are evaluated in light of the evidence provided above. 3. The new open economy macroeconomic model 3.1. The benchmark model Consider a world economy with two-countries, two composite goods, and a flexible exchange rate between the two moneys. The home (foreign) country is populated by infinitely-lived agents Table 2 The calibrated parameters Parameter

Value

Description of parameters

g sn δ r ϵC (σ−1) ψ μp μw αp αw ϵ nk ηq γp γw ηs b

1.004 0.65 0.025 0.016 1/6, 1/2 3/2 1.1 1.1 0.75, 0.9 0.75 1 2, 4 0, 1 0, 1 1 0.65

Steady state quarterly growth rate of technology Steady state labor share Rate of depreciation of capital stock Steady state rate of return Intertemporal elasticity of consumption Intratemporal elasticity of consumption Steady state price markup Steady state wage markup Degree of nominal price rigidity Degree of nominal price rigidity Elasticity of substitution between capital and labor Elasticity of i / k (or it / it−1) to Tobin's q Indexation degree of previous inflation rate to price Indexation degree of previous inflation rate to price Capital utilization adjustment cost Degree of habit persistence in consumption

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and is completely specialized in the production of its own goods, Y (Y⁎).6 Foreign values of the corresponding domestic variables will be denoted by an asterisk (⁎). The goods production is subject to production shocks, A and A⁎ respectively. The monetary policy rules r and r⁎ are also subject to interest rate shocks, εr and ε⁎r . In this section, I present a pricing-to-market model as well as a producer–currency–pricing model based on Betts and Devereux (1999). In particular, I assume that there is a continuum of goods varieties in each country of measure 1. In each country, a fraction of s of goods varieties are invoiced in the currency of the buyer, while the remaining 1 − s goods varieties are priced in the currency of the seller as in Betts and Devereux (1999). 3.1.1. Household's problem Suppose that the utility function of the representative domestic household takes the form:7 " # l X i Et b uðCtþi ; Ltþi Þ ; 0 b b b 1; ð2Þ i¼0

where Ct is a composite consumption index defined by   w−1 w−1 w 1 1 Ct ¼ hw Chtw þ ð1−hÞw Cftw w−1 ; w N 0:

ð3Þ

Here Cht and Cft are indices of domestic and foreign consumption goods, and θ and 1 − θ represent the share of domestic consumption allocated to domestic goods, and imported goods. The indices are given by the following CES aggregator of the quantities consumed of each variety of good: Z 1  Z 1  /−1 / /−1 / Cht Cht ð jÞ / dj /−1 ; Cft ¼ Cft ð jÞ / dj /−1 ; /N1: ð4Þ 0

0

Here ψ and ϕ measure the elasticity of substitution between domestic and foreign goods, and the elasticity of substitution among goods within each category. β is the household's discount factor, and Et denotes the expectation operator conditioned on the information available in period t. Lt represents the domestic household's leisure in period t. The state of the economy, zt evolves according to a Markov process described by a density function f(zt+1, zt). The household faces a time constraint such that ¯ Lt þ Ht V H;

ð5Þ

where Ht and H¯ denote the hours worked and time endowment of the home resident respectively. Since the monetary policy is specified in terms of an interest rate rule, money is not introduced in the model.8 The optimal allocation for each differentiated good yields the demand functions: Cht ð jÞ ¼ 6

   ⁎ −/ Pht ð jÞ −/ P ð jÞ Cht ; ; Cft ð jÞ ¼ ht ⁎ Cft ; Pht Pht

ð6Þ

The measure of the total population in each country is normalized to one. When the financial market is complete, each household's behavior can be rewritten in the same way as in Woodford (1996). For notational simplicity, I suppress the index j ∈ [0, 1] which represents the variety as well as the identity of the household in discussing the household's problem. 8 Money plays the role of a unit of account only. 7

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R 1 1 1 ⁎ ¼ ðR 1 P⁎ ð jÞ1−/ djÞ1−/ for all j ∈ [0, 1], where Pht ¼ 0 Pht ð jÞ1−/ dj 1−/ and Pht are the price 0 ht indexes for domestic and foreign goods, both expressed in home currency. The optimal allocation of expenditures between domestic and foreign goods implies:  −w  ⁎ −w Pht P Cht ¼ h Ct ; Cft ¼ ð1−hÞ ht Ct ; Pt Pt

ð7Þ

where the consumer price index (CPI) is given by 1 1−w þ sð1−hÞðPh⁎t Þ1−w þ ð1−hÞð1−sÞSt ðPf⁎t Þ1−w 1−w ; Pt ¼ ½hPht

ð8Þ

Here s = 1 for PTM and s = 0 and for PCP, and Pft⁎ and St are price indexes for foreign goods, expressed in foreign currency, and the nominal exchange rate in period t respectively. The domestic household starts with nominal wealth Θt carried over from period t − 1 and receives lump-sum transfers of home currency, Tt before the asset market opens. There exists a complete asset market in the economy. In particular, I assume that there is a domestic currencydenominated contingent one-period bond market as in Betts and Devereux (1999) and Chari et al. (2000). That is, I assume that the representative household chooses one-period nominal contingent home currency bond, Bt+1 at the asset market. Bt+1 pays one dollar if zt+1 state occurs next period and 0 otherwise, and Qt,t+1 denotes the price of such a bond in units of home currency in period t and state zt.9 The riskless one-period nominal domestic interest rate rt is given by 1 + rt ≡ [EtQt,t+1]− 1. The domestic household's budget constraint at the beginning of the period t is given by Pt ðCt þ It Þ þ Et ½Btþ1 Qt;tþ1  V Bt−1 þ Wt Ht þ Vt Kt þ Pt þ Tt :

ð9Þ

Here Πt, Wt, and Vt denote the home country firm's nominal profits, nominal wages and nominal rental rate for capital stock given to the home residents respectively. Moreover, each country's household owns capital stock that it rents to firms and there is no firm specific capital stock. Since we do not empirically observe large discrete capital stock adjustments, it is reasonable to introduce an adjustment cost in capital stock installments. If there are costs of installing capital, the capital stock will move more sluggishly. I assume that there are deadweight costs of installing capital stock. To simplify the model structure as far as possible, I will adopt the Uzawa–Lucas–Prescott form of investment adjustment costs. Ktþ1 ¼ uðIt =Kt ÞKt þ ð1−dk ÞKt ;

ð10Þ

where φ(It / Kt) is a positive, concave function, and It is the composite investment of the home resident at period t, and Kt is the composite capital stock of the home resident at period t. Under the assumption of complete asset markets, optimal risk sharing implies that the marginal utility of consumption of foreign residents is proportional to that of domestic residents multiplied by the real exchange rate (ϵt), i.e. u1 ðCt⁎ ; L⁎t Þ ¼ kϵt ; u1 ðCt ; Lt Þ where k is a constant that depends on initial conditions. 9

zt = (z0, z1, …, zt) represents the history of events up through period t.

ð11Þ

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3.1.2. Firms Labor and capital services are free to move between firms and factor prices are assumed to be fully flexible. Individual firms therefore have an identical cost function for producing output, so long as they have a constant return to scale technology. In this section, I describe a representative firm's cost function. Demands for capital and labor are determined by a cost minimization CðWt ; Vt ; Yt ð jÞ; Ho ; zt Þu minHt ð jÞ;Kt ð jÞ fVt Kt ð jÞ þ Wt Ht ð jÞg s:t: Yt ð jÞ V At FðKt ð jÞ; zt ðHt ð jÞ−Ho ÞÞ:

ð12Þ

Here Ho, zt and At are the home country resident's fixed overhead cost in units of labor hours, labor augmenting permanent technology progress, and transitory technology process at period t. Yt( j)10 and Ht( j) are the output and total labor input of the jth firm in the home country respectively. I assume that the technology shock follows an AR(1) process. The permanent zt changes in the total factor productivity, zt are taken as growing deterministically, i.e. g ¼ zt−1 for all t as in King, Plosser and Rebelo (1988). log At ¼ ð1−qA Þ þ qA log At−1 þ nAt ; 0 b qA b 1: where E(ξAt) = 0 and ξAt is uncorrelated over time. 3.1.3. Calvo-type staggered price setting with PCP Firms cannot segment their markets by country and must set prices in the home currency. In each period, a fraction of (1 − αp) of the domestic goods producing firms is allowed to set a new price, while the other fraction of firms, αp, set its price by multiplying the average inflation rate (ω) by its previous price level as in the Calvo (1983) model. Hence, the profit maximization problem of the firms resetting prices in period t is given by ( ) l X k Ktþk Pt maxd Et ðap bÞ ½Pht;tþk ðYht;tþk þ Yft;tþk Þ − MCtþk ðYht;tþk þ Yft;tþk Þ ; ð13Þ Kt Ptþk k¼0 where Λt and Λt+k are the marginal utility of wealth in period t and t + k and k = 0, 1, 2…∞.11 Since the Law of One Price holds in PCP, the price of home goods in foreign currency (Pft,t) equals the price of home goods in home currency (Pht,t) divided by the nominal exchange rate (St): Pft;t ¼

Pht;t : St

ð14Þ

Next, the price level at period t under the Calvo-type staggered price-setting can be written as the recursive form: 1−/ 1−/ 1−/ Pht ¼ ð1−ap ÞPht;t þ ap x1−/ Pht−1 ;

ð15Þ

1−/ 1−/ Pft1−/ ¼ ð1−ap ÞPft;t þ ap x1−/ Pft−1 :

ð16Þ

10 Yt( j) = AtF(Kt( j),zt(Ht( j) − Ho( j))) is strictly concave, twice continuously differentiable, and CRS in Kt( j) − Ho( j) and H t( j) − Ho( j) but it is IRS in Kt( j) and Ht( j).  −/1 P 11 The demand Yt,t given the price Pt,t is determined by the demand curve facing each firms Yt;t ¼ Pt;tt Yt as Eq. (A.7).

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3.1.4. Calvo-type staggered price setting with PTM Firms can segment their markets by country and set prices in the currency of the buyer in the segmented home and foreign markets to avoid the arbitrage opportunity that is implied by the Law of One Price. Specifically, suppose that only a fraction (1 − αp) of home firms sets the new price, Pht,t to home consumers, and Pft,t to foreign consumers and the other fraction of firms, αp sets its price by multiplying the average inflation rate by its previous price level. Then the firm's maximization problem can be written as follows. ( maxd Et

l X k¼0

) Ktþk Pt ðap bÞ ½Pht;tþk Yht;tþk þ Stþk Pft;tþk Yft;tþk − MCtþk ðYht;tþk þ Yft;tþk Þ ; Kt Ptþk k

ð17Þ where Pft,t+k = ωkPft,t. If the price level is flexible, then the markup — the ratio of price to marginal cost — is constant at each period, while it responds to monetary and real shocks when prices are predetermined. 3.1.5. Monetary authority There has been extensive debate over the most appropriate way to model monetary policy in the U.S. and other countries. It concerns whether the money supply rule is more appropriate than the interest rule to evaluate the effect of monetary policy in the actual economy. Recently, many leading macroeconomists follow Taylor's recommendation of using a simple interest rule or a variant such as an interest smoothing policy to evaluate the effect of the monetary policy. In this paper, I employ the interest rate smoothing rule to evaluate the model. I assume that the nominal interest rate rt is set according to a generalized Taylor rule as in Clarida, Gali, and Gertler (1999 hereafter Clarida et al. (1999)): rt ¼ qr rt−1 þ ð1−qÞ½by yt þ bp Et ptþ1  þ nrt ;

ð18Þ

where πt+1 is the inflation rate between t + 1 and t, and ξrt is uncorrelated over time with mean-zero. 3.2. Equilibrium The equilibrium condition for the home goods market is given by "   #   Pht −w Pht −w ⁎ ⁎ Yt ¼ ð1−sÞ h ðCt þ It Þ þ ð1−hÞ ðCt þ It Þ Pt St Pt⁎ "   #  −w Pft Pht −w ⁎ ⁎ ðCt þ It Þ þ ð1−hÞ ⁎ ðCt þ It Þ ; þs h Pt P

ð19Þ

t

where Yt = At F(Kt, zt (Ht − Ho)). Because I will focus on the symmetric equilibrium in which all agents in the same country make the same decisions, I will define a symmetric equilibrium. The symmetric equilibrium conditions consist of the efficiency conditions and the budget constraint of the home consumers, and firms and foreign consumers and firms, and the optimal risk sharing condition and equilibrium conditions of each goods market, capital rental market, labor market, money, and

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bond market in each country. Specifically, a symmetric equilibrium is an allocation of home ⁎ ⁎ ∞ agents {Ct, Ct⁎, Kt+1, K⁎ t +1, It, It , Ht, Ht }t=0, a sequence of prices and costate variables for the ⁎ , P⁎ ⁎ ⁎ ⁎ ⁎ ⁎ home country {Pht,t, Pht, Pft,t, Pft, P⁎ht,t , Pht ft,t , Pft , Pt, Pt , Vt, Λt, Wt, MCt, qt, Vt , Λt , Wt , ∞ ∞ and a sequence of exchange rate {S } such that (1) the households decision MCt⁎, qt⁎, rt, r⁎ } t t=0 t t=0 rules solve their optimization problem given the states and the prices; (2) the demands for labor and capital solve each firm's cost minimization problem and price setting rules solve its present value maximization problem given the states and the prices; (3) each goods market, labor market, bond market, and money market are cleared at the corresponding prices, given the initial ∞ ⁎ }t=0 conditions for the state variables and the exogenous stochastic processes {ξrt, ξ⁎rt , ξAt, ξAt . 4. Quantitative evaluation of the model 4.1. Parameter values To find the quantitative implications of the model, I will utilize the following CES subutility function which satisfies the condition of a balanced growth path

uðCt ; Lt Þ ¼

8 1−r < ½C q L1−q −1 t  :

; r N 0; r p 1; 1−r logCt þ vðLt Þ; r ¼ 1:

ð20Þ

Here σ− 1 is the intertemporal elasticity of substitution. With this temporal utility function, I can determine the parameter values which will be used in the simulation. As stated above, this paper assumes a two country world with identical features as in Betts and Devereux (1999), and Chari et al. (2000). With this reason, I will use the same parameter values of the U.S. economy for the home country as well as for the foreign country. All parameter values used in this paper are reported in Table 2. The benchmark model of this paper takes a value of intertemporal elasticity of substitution, ϵC = 1/2, i.e. σ = 2. Although little is known about the intratemporal elasticity of substitution ψ, the benchmark value of the parameter is set to 1.5 as in Backus, Kehoe, and Kydland (1994). As an interest rate smoothing rule, Clarida et al. (1999)'s estimate for the Fed's monetary reaction function, as shown in the simulation below, is utilized. rt ¼ 0:67rt−1 þ 0:33ð1:97Et ptþ1 þ 0:07yt Þ þ ert Though I need not specify the functional form for the capital stock adjustment cost function, φ, I should specify three parameters which describe the behavior around the steady state. First, I must specify the steady state value of Tobin's q and the share of investment in national product. Since the steady state value of Tobin's q is 1.0, I also set the value of this variable to 1.0 in the steady state. And I will take the same investment share in the steady state as in a model without adjustment cost. Next, I have to specify the parameter which determines the elasticity of the marginal adjustment cost function. The value of elasticity of I / K with respect to Tobin's q, ηq is the adjustment cost elasticity which reflects the volatility of investment. Though many studies have estimated this adjustment cost parameter, there is still a lot of uncertainty regarding the size of the adjustments cost. I will choose 4.0 as the benchmark parameter value as in Bernanke et al.

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(1999).12 When both the value of the elasticity and the degree of nominal rigidity are very high, output as well as employment becomes volatile because investment responds more to shocks.13 The nominal rigidity parameter value αp is also uncertain because the empirical value of this parameter changes depending on the period of interest and the estimation method. However, most economists report the estimates of this parameter in the range from 0.75 to 0.8. For this reason, I will report the simulation results when the nominal rigidity value equals 0.75.14 Finally, I will set the benchmark average size of markup, μ at 1.1. Though this value is much lower than the value that many sources of evidence suggest, it is consistent with the average markup estimates in Basu and Fernald (1993). 4.2. Relative mean square approximation error To evaluate the goodness of fit of the models, the minimum approximation error representation developed by Watson (1993) is applied. Following Watson (1993), consider the error ut defined by ut ¼ y t − x t ;

ð21Þ

where xt is the evolution of n × 1 vector coming from the economic model, and yt is the empirical counterparts of xt. Suppose that xt and yt are transformed to be jointly covariance stationary. Then the autocovariance generating function (ACGF) of ut, Au(z) is given by Au ðzÞ ¼ Ay ðzÞ þ Ax ðzÞ − Axy ðzÞ − Ayx ðzÞ;

ð22Þ

where Ax(z) is the ACGF of xt, Axy(z) is the cross ACGF between xt and yt and so forth. Under certain assumptions,15 Watson (1993) suggested a bound on the relative mean square approximation error (RMSAE) for the economic model — the bound analogous to a lower bound on 1 − R2 from a regression as following Rj ðxÞ ¼

½Au ðzÞjj ½Ay ðzÞjj

; z ¼ e−ix ;

ð23Þ

where [Au(z)] jj, [Ay(z)] jj are the jth diagonal elements of Au(z), Ay(z), respectively. Because Rj(ω) is the variance of the error relative to the variance of the data for each frequency, it tells us how well the economic model fits the data over different frequencies. Because the spectrum of the data, yt is not known, it must be estimated. In this paper, the spectrum of y was calculated by estimating an unrestricted VAR as in Betts and Devereux (1999), 12 Baxter and Crucini (1993) used the elasticity of 15 as a benchmark parameter value. But most empirical studies suggest a lower value than this one. King and Wolman (1996) set ηq = 2, while Bernake et al. (1999) set ηq = 4. See Chirinko (1993) for more details. 13 In Chari et al. (1997), the standard deviation of output is 13% when firms preset prices for 6 quarters and the capital adjustment cost is low in a Taylor-type sticky price model. 14 There is a lot of uncertainty in the degree of price rigidities. The range of empirical values for the degree of price rigidities(α) are estimated around 0.5 or 0.85. Yun (1996) set α = 0.82 for his endogenous money supply model in his paper. King and Watson (1996) use 0.9 as a benchmark parameter value, while King and Wolman (1996) use 0.75 to consider the optimal monetary policy in a Calvo-style sticky price model. Chari et al. (2000) set α = 0.75 in the benchmark model. 15 See Watson (1993) for more details.

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Table 3 Moments of a benchmark model with PCP and PTM Variable

SD

PCP Y C I N P Y⁎ P⁎ r S ϵ PTM Y C I N P Y⁎ P⁎ r S ϵ

αp = 3/4 1.61 0.89 2.54 1.44 1.07 1.61 1.07 1.05 0.99 0.00 αp = 3/4 1.80 1.51 3.76 1.98 1.12 1.80 1.12 2.12 2.85 2.31

Cross autocorr.

Xt with GDP Yt+k

k=−4

−3

−2

Nonseparable preference ηq = 4 0.06 0.33 0.64 0.03 0.25 0.52 − 0.08 0.16 0.48 0.18 0.36 0.57 0.07 0.14 0.21 − 0.13 − 0.10 − 0.02 − 0.03 − 0.06 − 0.11 0.11 − 0.07 − 0.32 0.05 0.11 0.17 0.00 0.00 0.00 Nonseparable preference ηq = 4 − 0.04 0.23 0.58 − 0.10 0.08 0.34 − 0.14 0.05 0.33 0.00 0.21 0.49 0.04 0.10 0.13 − 0.21 − 0.09 0.11 0.25 0.30 0.30 0.14 − 0.02 − 0.49 − 0.12 − 0.12 − 0.10 − 0.05 − 0.05 − 0.04

(corr(Xt, Yt+k)) −1

0

1

2

3

4

0.90 0.74 0.77 0.70 0.22 0.06 − 0.15 − 0.55 0.20 0.00

1.00 0.80 0.91 0.68 0.17 0.09 −0.16 −0.63 0.18 0.00

0.90 0.66 0.85 0.49 0.07 0.06 −0.13 −0.54 0.11 0.00

0.64 0.39 0.61 0.21 −0.05 −0.02 −0.07 −0.31 0.01 0.00

0.33 0.09 0.32 − 0.06 − 0.13 − 0.10 0.00 − 0.06 − 0.07 0.00

0.06 − 0.12 0.07 − 0.23 − 0.15 − 0.13 0.06 − 0.11 − 0.11 0.00

0.88 0.58 0.59 0.72 0.09 0.31 0.22 − 0.60 − 0.07 − 0.03

1.00 0.66 0.72 0.75 0.00 0.39 0.07 −0.53 −0.03 −0.01

0.88 0.55 0.68 0.57 −0.11 0.31 −0.11 −0.34 0.01 0.01

0.58 0.31 0.48 0.25 −0.19 0.11 −0.25 −0.11 0.04 0.02

0.23 0.04 0.24 − 0.08 − 0.20 − 0.09 − 0.32 0.05 0.07 0.03

− 0.04 − 0.14 0.04 − 0.28 − 0.14 − 0.21 − 0.31 0.27 0.09 0.03

Clarida and Gali (1994), and Eichenbaum and Evans (1995). The VAR was specified as the regression of st = (rt, Δpt, Δp⁎t , Δyt, Δyt⁎, Δϵt) on a constant and four lags of st.16 4.3. Implications of the model In this subsection, I first compare the second moments of the benchmark model with those of the actual data, and then review the main goal of this paper to evaluate the performance of the benchmark model with Watson (1993)'s measure of fit. 4.3.1. Cross correlations Can a NOEM model explain the stylized facts about business cycles? For the purpose of the evaluation, I first compare volatilities and cross correlations of the real variables in the benchmark model with those of the data to examine the performance of the model in business cycle frequencies. The column labeled ‘Data’ in Table 2 reports composite data moments of six countries (Canada, France, Germany, Japan, United Kingdom, and United States). Here moments are calculated for actual time series that have been band-pass filtered. After that, I compare the power spectrum of the model with the power spectrum calculated from the data drawn from these major industrial economies. First, Table 3 provides the volatilities and serial correlations of the selected variables in the PCP model when one fourth of the firms in the economy adjust their prices optimally per period. 16

All variables except the nominal interest rate rt are in natural logarithms.

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When one fourth of the firms in the economy adjust their prices optimally every period, the relative volatility of consumption, output, and investment well match that of the data. However, nominal exchange rates are much less volatile than those of the data and real exchange rates do not move at all because of the law of one price. Nominal interest rates move countercyclically, while they move procyclically in the data. This has been expected because output does not display a hump-shaped response to a negative shock on interest rates. However, the correlations between nominal interest rates and future output are negative (corr(rt, yt+4) = − 0.11) as in the data, which shows that nominal interest rates serve as countercyclical leading indicators in the business cycle. The correlations between prices ( pt) and future output are slightly negative (corr( pt, yt+2) = − 0.05) as in the data, while the contemporaneous correlation between output and prices is positive contrary to the data. Next, Table 3 also provides the volatilities and serial correlations of the selected variables in the PTM model when one fourth of the firms in the economy adjust their prices optimally per period. The real and nominal exchange rates of the model with PTM are more volatile than those of the model with PCP, but they are much less volatile than those of the data. The real and nominal exchange rates move slightly countercyclically to the current output as in the data. In regard to contemporaneous correlations of output and interest rates, nominal interest rates move countercyclically in the model, while they move procyclically in the data. Moreover, the correlations between nominal interest rates and future output are positive, contrary to the data, which shows that nominal interest rates do not serve as countercyclical leading indicators over the business cycle in the model. The correlations between prices and future output are negative (corr( pt, yt+2) = −0.19) as in the data, while the contemporaneous correlation between output and prices is negligible. In sum, the benchmark NOEM model with PTM can generate more volatile exchange rates than the model with PCP. However, the second moment shows that neither is successful in explaining the movement of exchange rates and interest rates over business cycle frequencies as in Chari et al. (2000). 4.3.2. Relative mean square approximation error One can see the successes and failures of the benchmark model more critically with the spectral density of the selected variables and Watson (1993)'s RMSAE than with the second moments. As Ellison and Scott (2000) and Watson (1993) put forth, the spectral density of the selected variables evidently show the strengths and weaknesses of the calibrated models including an RBC model and a sticky price model. For this reason, I next examine whether the spectra of the growth rates of the selected variables calculated from the model correspond to the spectra implied by the data. For each selected variable, Fig. 1 shows the spectrum of a benchmark NOEM model with PCP (solid lines), the spectrum of the data (long dashed lines), and the spectrum of the error required to reconcile the models and the data (dotted lines) when αp = 3/4, i.e. when one fourth of the firms in the economy set their optimal prices per period.17 The figure shows that there are significant differences between the model and the data. The model has mass spectra at high frequencies, while the data has mass spectra around the business cycle frequencies. The hump-shaped spectral density of exchange rate changes calculated from the data may reflect the mean reversion of exchange rates.18 The NOEM model with PCP generates too little variation of exchange rates at all frequencies. 17

The error process was chosen to minimize the unweighted trace of the error spectral density matrix, subject to the constraint as in Watson (1993). 18 King and Watson (1996) notice that the mean reversion of output in the data generates the hump-shape spectral density of output.

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Fig. 2. A. Domestic output, B. Domestic price, C. Nominal exchange rate, D. Domestic interest rate.

The NOEM model with PTM also fails to replicate this typical power spectrum of the data. Fig. 2 shows the spectrum of a benchmark NOEM model with PTM, the spectrum of the data, and the spectrum of the error when αp = 3/4 The spectral density of output and nominal exchange rates calculated from the benchmark model with PTM as well as the benchmark model with PCP display a flat spectrum for exchange rates without a noticeable business cycle peak, while the spectral density of output and nominal exchange rates of the data displays a peak at business cycle frequencies. In particular, the model displays a negligible spectrum of exchange rates at low and business cycle frequencies, while the data displays a peak as well as mass spectrum of exchange rates at the corresponding frequencies. The largest differences occur at a frequency corresponding to approximately 20–40 quarters. This is the most dramatic failure of the NOEM model with either PTM or PCP. The failure of the model can be seen more clearly in the spectrum of the error required to reconcile the model and the data. The error needed to reconcile the model and the data for the exchange rate are almost the same as the power spectrum of the data at low frequencies, which implies that the RMSAE for nominal exchange rates is not much different from one. This failure has been expected because the NOEM model has difficulty in generating the hump-shape response of exchange rates to monetary policy shocks. These substantial differences between the model and the data at low and business cycle frequencies indicate that we need to find elements, which generate both the hump-shape spectral density of exchange rate changes and the humpshape response of exchange rates to monetary shocks. In the case of prices, the NOEM model with either PCP or PTM does not show a hump-shape at business cycle frequencies, while the data has a peak at the corresponding frequencies. There is simply too great a power at very low frequencies in the NOEM model. In the case of nominal

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Table 4 Relative mean square approximation error of a benchmark model Variable

Y P r S

Y P r S

Model with PCP

Model with PTM

Business cycle frequencies:

6–32 Quarters

1.86 1.62 0.30 0.84

2.29 1.84 0.15 0.83

Hodrick–Prescott detrended levels:

All frequencies

2.00 1.62 0.28 0.84

2.67 1.77 0.41 0.77

Note: Relative mean square approximation error is the lower bound of the relative bound of the variance of the approximation error derived by the variance of the series.

interest rates, the model has mass spectra at high frequencies irrespective of the price setting rule, while the data has mass spectra at low and business cycle frequencies. However, the difference between the power spectrum calculated from the data and that constructed from the benchmark model at low and business cycle frequencies is relatively small. This implies that a small error is needed to reconcile the model and the data for the interest rate as in Figs. 1 and 2. Table 4 provides a summary of the relative mean square approximation error (RMSAE) for the levels of the series integrated over business cycle frequencies (6–32 quarters) and those detrended by HP filter integrated across all frequencies when the unweighted trace of the spectrum is minimized. The figures in Table 4 confirm the findings in the spectral density. In terms of the RMSAE for the selected variables, the performance of a NOEM model with either PCP or PTM is very poor. The RMSAE for domestic output and domestic price is in excess of one, either using only business cycle frequencies or HP filter integrated across all frequencies. The RMSAE for exchange rates in the benchmark model is about 0.8. This high RMSAE for exchange rates reflects the fact that the NOEM model with either PTM or PCP generates insufficient exchange rate fluctuations at low and business cycle frequencies compared to the data. Summing up, the benchmark NOEM model with either PCP or PTM is not successful in generating the spectral density of the selected variables calculated from the data. More precisely, the model cannot generate mass spectra of exchange rate changes at low and business cycle frequencies as in the data. 5. Sensitivity analysis Taking into account the NOEM model's weakness in generating volatile exchange rate movements, many authors such as Chari et al. (2000), and Kollman (2001) suggest that one should to consider alternative models with different asset market structures or other sources of shocks. In this section, I examine the findings of the paper with different assumptions about preference, asset market structure, and other sources of shocks. First, I will explore whether a separable utility function can improve the performance of the model at the corresponding frequencies. Second, I will take into account an incomplete asset market and examine the explanatory power of the model. Finally, I will add other shocks and see whether they can improve the performance of the model.

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5.1. Separable preference Chari et al. (2000) argue that the failure of the benchmark PTM model in generating volatile real exchange rates lies in nonseparable preferences. When a household has nonseparable preferences, a decrease in the domestic interest rate increases not only domestic consumption, but also domestic employment. Because the increase in employment decreases the marginal utility of domestic consumption, it works to offset the depreciation of the real exchange rate. For this reason, they suggest that one needs to assume a separable preference if he wants to have volatile exchange rate movements as in the data. In particular, Chari et al. (2000) propose a separable utility function of the form u¼

Ct1−r −1 ð1−Lt Þ1−m þj : 1−r 1−m

ð24Þ

The optimal risk sharing condition makes it clear why one needs the role of separable preferences. The optimal risk sharing condition implies that a high curvature parameter σ, can generate the high volatility of real exchange rates. To see this clearly, I log-linearize the optimal risk sharing condition which leads to ϵˆ t ¼ rðˆct − cˆ t⁎ Þ;

ð25Þ

where ˆxt is the log-linearized value of Xt around its steady state value. Thus, the relative standard deviation of the real exchange rate to output is roughly equal to the relative standard deviation for consumption difference to output sdðˆct −ˆc⁎t Þ sdðˆϵt Þ ¼r : sdð yˆ t Þ sdð yˆ t Þ The real exchange rate movements in the model can be as volatile as in the data if the degree of intertemporal elasticity of substitution is sufficiently low, i.e. if σ, is sufficiently high. Following Chari et al. (2000), I take the same value for separable preferences and see whether the separable preference can improve the model's performance at explaining exchange rates at low and high frequencies. According to Chari et al. (2000), ν = 3/2, κ = 10, and σ = 6 are sufficient to generate volatile exchange rates relative to output as in the data. Of course, there should be a sufficient interest rate shock to generate a large response of exchange rates because the expected consumption growth rate is proportional to the degree of intertemporal elasticity of substitution. That is, households with a high curvature parameter value adjust their consumption level smoother to the exogenous shock than households with a low curvature parameter value. Thus, even if the real exchange rate responds σ times more to the monetary shock than consumption difference, the volatility of real exchange rates may be incongruous to the data. When the standard deviation of monetary shocks are chosen to give the same volatility for output as in the U.S. data as in Chari et al. (2000), the model can generate nominal and real exchange rate volatilities that match the data. Table 5 shows that the band-pass filtered second moments of the exchange rates are as volatile as the data. From this result, one can ascertain that the NOEM model with separable preference can generate volatile exchange rate movements as in the data. However, the variance of exchange rates constructed from the model can be the same as that calculated from the data, even though the spectral density of exchange rates calculated from the model may have an inverse hump-shape or flat shape contrary to the typical shape calculated from

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Table 5 Moments of the NOEM model with separable preferences Variable

SD

Cross autocorr.

Xt with GDP Yt+k

PTM

αp = 3/4

k=−4

Separable preference ηq = 4

−3

Y C I N P Y⁎ P⁎ S ϵ

1.43 0.61 4.05 1.73 2.08 1.43 2.08 6.33 5.22

0.03 − 0.14 − 0.15 0.11 0.16 − 0.26 0.32 − 0.09 − 0.05

0.29 − 0.01 0.01 0.28 0.22 − 0.23 0.36 − 0.10 − 0.06

(corr(Xt, Yt+k))

−2

−1

0

1

2

3

4

0.62 0.20 0.24 0.48 0.24 − 0.13 0.35 − 0.09 − 0.06

0.89 0.39 0.47 0.62 0.19 − 0.02 0.26 − 0.06 − 0.05

1.00 0.48 0.57 0.61 0.08 0.03 0.09 −0.03 −0.04

0.89 0.41 0.52 0.43 −0.06 −0.02 −0.11 0.00 −0.01

0.62 0.24 0.34 0.16 −0.18 −0.13 −0.28 0.04 0.01

0.29 0.04 0.13 − 0.10 − 0.23 − 0.23 − 0.37 0.07 0.02

0.03 − 0.10 − 0.03 − 0.27 − 0.21 − 0.26 − 0.38 0.09 0.04

the data. Fig. 3 and Table 6 show that there are significant differences between the power spectrum constructed from the model and that calculated from the data. The NOEM model with separable preferences cannot generate the typical hump-shaped spectral density of exchange rates. The model has mass spectra at high frequencies, while the data has mass spectra around business cycle frequencies. The NOEM model with separable preferences still displays a negligible spectrum of exchange rates at lower frequencies, while the data displays a peak as well as a mass spectrum of exchange rates at the business cycle frequencies. The introduction of separable preferences into the model increases only spectra at high frequencies without noticeable change at

Fig. 3. A. Spectrum for nominal exchange rate, B. Nominal exchange rate.

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Table 6 Relative mean square approximation error of alternative models Variable

CM with separable preference

IM with other shocks

Business cycle frequencies:

6–32 Quarters

Y P r S

1.88 3.44 1.51 0.89

1.77 1.68 0.15 0.79

Hodrick–Prescott detrended levels:

All frequencies

Y P r S

2.18 3.28 3.10 0.85

2.10 1.62 0.41 0.74

Note : CM and IM denote complete asset market and incomplete asset market, respectively. Other shocks mean preference shock and UIP shock.

low and business cycle frequencies. The figure also shows that a large error is required to reconcile the model with separable preferences and the data for nominal exchange rates. In view of this large error, separable preferences do not seem to improve the model's explanatory power at business cycles using either band-pass frequencies or Hodrick–Prescott detrended levels. These findings can be seen more clearly in the plot of fitted values of the model and that of the data. Fig. 3 shows that there are significant differences between the NOEM model and the data. In particular, the NOEM model cannot generate major exchange rate swings that occurred from 1982 to 1990. Overall, the NOEM model with separable preferences fails in explaining exchange rate movements at low and business cycle frequencies, even though it can generate volatile exchange rate movements with mass spectra at high frequencies. 5.2. Incomplete market Obstfeld and Rogoff (1995) argue that the assumption of complete asset markets is incongruous alongside the sticky price model, and the bond-only formulation is the more natural one. Taking their arguments into account, I will explore the role of the incomplete asset market structure in exchange rate movements. Suppose that only a single uncontingent nominal bond is traded across countries. That is, assume that the household can choose a domestic currencydenominated one-period nominal bond, Bt, in the asset market as in Betts and Devereux (1999), and Obstfeld and Rogoff (1995). Then the home household's budget constraint is replaced by Pt ðCt þ It Þ þ Bt R−1 t V Ht þ T t ;

ð26Þ

where Rt = 1 + rt If the household buys one unit of Bt at the price of Rt− 1, then it can have one unit of the home currency in all states zt+1 that can occur at t + 1. The foreign household's budget constraint is modified similarly. The first-order condition for bond holding in the home country is now given by " #    Ctþ1 ð1−rÞq−1 Ltþ1 ð1−rÞð1−qÞ Pt bRt Et ¼ 1; ð27Þ Ct Lt Ptþ1

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while that in the foreign country is given by " # ⁎ ð1−rÞq−1 L⁎ ð1−rÞð1−qÞ P⁎ S Ctþ1 t tþ1 t bRt Et ⁎ Stþ1 ¼ 1: Ct⁎ L⁎t Ptþ1

ð28Þ

After equating Eqs. (27) and (28), the log-linearizing equation the results, I get the following equation: Et ½ˆϵtþ1 − ϵˆ t  ¼ ðð1−rÞq−1ÞEt ½ˆctþ1 − cˆ t −ðð1−rÞq −1ÞEt ½ˆc⁎tþ1 −ˆc⁎t  þ ð1−rÞð1−qÞEt ½lˆtþ1 −lˆt −ð1−rÞð1−qÞEt ½lˆ⁎ −lˆ⁎  tþ1

t

ð29Þ

In the case of incomplete asset market structures, the difference of the expected domestic consumption growth rate and the expected foreign consumption growth rate, not the difference of the domestic and foreign consumption, matter to the expected depreciation rate of the real exchange rate. As Betts and Devereux (1999) and Chari et al. (2000) note, however, the asset market structure does not change the fundamental features of the NOEM model. Fig. 4 shows the spectral density for selected variables for an incomplete asset market model which has the same parameter values as the benchmark economy. The NOEM model with incomplete asset markets cannot generate the typical hump-shaped spectral density; instead it generates a flat spectral density of exchange rates contrary to the data. The RMSAE for the selected variables in the incomplete asset market is comparable to that in the complete asset market as in Table 6.

Fig. 4. A. Domestic output, B. Domestic price, C. Nominal exchange rate, D. Domestic interest rate.

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5.3. Other shocks In this subsection, I add other shocks to the model with incomplete asset markets such as shocks to preference and to uncovered interest rate parity shocks to see whether they can improve the performance of the model at all frequencies. For the preference shock, I have included a multiplicative shock in consumption as in Amato and Laubach (2001), and Kollman (2001): uðCt ; Lt Þ ¼

1−r ½Ctq expðnCt ÞL1−q −1 t  : 1−r

ð30Þ

Here it is assumed that the log of preference shock, ξCt follows an AR(1) process as nCt ¼ ð1−qnC Þ þ qnC nCt−1 þ enCt ; 0 V qn b 1; where εξCt is independent white noise with standard deviation ,σξC In the quantitative evaluation of the model, I set ρξC = 0.9, and σξ = 0.01 as the technology shock in the model. For the uncovered interest rate parity shock (UIP shock, hereafter), I follow Kollman (2001). For the UIP shock, I assume that the household holds nominal one-period bonds denominated in domestic and foreign currency. The Euler condition for foreign household's bond holding is disturbed by a stationary exogenous stochastic random variable, ξUt whose unconditional mean is

Fig. 5. A. Domestic output, B. Domestic price, C. Nominal exchange rate, D. Domestic interest rate.

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unity as in Kollman (2001). To take into account the incomplete asset market with the UIP shock, I rewrite the first order conditions of domestic and foreign household's bond holdings as 

 Pt Kt ¼ bEt Rt Ktþ1 ; Ptþ1

ð31Þ

  Pt⁎ St ⁎ ⁎ ⁎ K Kt ¼ expðnUt ÞbEt Rt ⁎ ; Ptþ1 Stþ1 tþ1

ð32Þ

where R⁎t is the foreign nominal interest rate in period t. Here it is assumed that the UIP shock follows an AR(1) process: nUt ¼ ð1−qnU Þ þ qnU nUt−1 þ enUt ; 0 V qnU b 1; where εξUt is an independent white noise with a standard deviation σξU. In the quantitative evaluation of the model, I set ρξU = 0.5 as in Kollman (2001). Fig. 5 and Table 6 show the results for the model with other shocks such as the preference and the UIP shocks. The addition of other shocks to the model does not improve the model's performance of explaining exchange rates at low and business cycle frequencies. The model with other shocks cannot generate the typical hump-shaped spectral density of exchange rates and output.

Fig. 6. A. Domestic output, B. Domestic price, C. Nominal exchange rate, D. Domestic interest rate.

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Table 7 Relative mean square approximation error of alternative models Variable

Y P r S

Y P r S

CM with αp = 0.9 ηq = 2

IM with αp = 0.9 ηq = 2

Business cycle frequencies:

6–32 Quarters

2.00 1.11 0.30 0.90

1.59 1.04 0.32 0.82

Hodrick–Prescott detrended levels:

All frequencies

2.30 1.09 0.69 0.84

1.87 1.03 0.71 0.76

Finally, I have performed a sensitivity analysis with different values for important parameters. Fig. 6 and Table 7 report the results for the model with a separable preference by varying the important parameter values as with King and Watson (1996). The spectral density and RMSAE, however, show that the NOEM model with different parameter values fails in generating the mass spectra at low and business cycle frequencies as in the data. This possibly reflects the failure of the sticky price model in generating the hump-shaped response of output and consumption to monetary shocks. That is, because the real exchange rate in the model is a function of domestic and foreign consumption ratio, the model cannot generate a hump-shaped response of exchange rates to monetary shocks. 5.4. Other nominal and real rigidities In recent, Christiano, Eichenbaum and Evans (2003) and Smets and Wouters (2002) have extended the canonical new Keynesian model and shown that the hybrid model is successful in replicating the empirical facts such as hump-shaped response of the endogenous variables to a monetary shock. The specific frictions embedded in the hybrid model are slower adjustment of wage rates, habit persistence in consumption, capital adjustment cost in investment change rate, indexation in price and wage setting, and variable capital utilization rate. In this subsection, I extend the benchmark NOEM model by incorporating these features and see whether the model can generate the hump-shaped spectral density of output and exchange rates. The representative household is supposed to derive utility from the level of consumption relative to a time-varying subsistence or habit level: u¼

ðCt −bCt−1 Þ1−r −1 ð1−Lt Þ1− g þj ; 1− r 1− g

ð33Þ

where b allows for habit formation in consumption when it is greater than zero. The capital adjustment cost is assumed to depend on the investment rate, not on the investment level: Ktþ1 ¼ ð1− wðIt =It −1 ÞÞIt þ ð1−dk ÞKt :

ð34Þ

It is also assumed that households act as price setters in the labor market as in CEE (2003). In particular, it is assumed that only the fraction (1 − αw) of the households sets the new wage, Wt,t, while

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Fig. 7. A. Domestic output, B. Domestic price, C. Nominal exchange rate, D. Domestic interest rate.

the fraction of households, αw sets its price by indexing the previous inflation rate  other g Pt−1 w ð0 Vg w V 1Þ to their previous wage level. Firms in the product market that do not set their Pt−2 optimal prices are also assumed to set their prices by indexing the previous inflation rate  gp Pt−1 ð0 V g V 1Þ to their previous price levels. p Pt−2 I explore whether the extended NOEM model can generate the typical hump-shaped spectral density for output and exchange rates as in the data. The extended model is successful in replicating the typical hump-shaped impulse response of output and the hump-shaped spectral density of output. However, it fails to generate the hump-shaped response of exchange rates and thus the typical humpTable 8 Relative mean square approximation error of alternative models Variable Business cycle frequencies: 6–32 Quarters Y P r S Hodrick–Prescott detrended levels: all frequencies Y P r S

Hybrid model with αp = 0.75 ηq = 2 1.11 1.00 0.39 0.84 1.18 1.00 0.81 0.79

Note: Hybrid model includes slower adjustment of wage rates, habit persistence in consumption, capital adjustment cost in investment change rate, indexation in price and wage setting, and variable capital utilization rate as in CEE (2003).

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405

shaped spectral density of exchange rates. In the extended NOEM model with habit persistence in consumption, the domestic monetary shock delays not only domestic consumption but also the foreign consumption. However, the uncovered interest parity condition implies that the extended model cannot generate the hump-shaped response of exchange rates (See Fig. 7). Table 8 shows that the incorporation of other nominal and real rigidities into the model improves the performance of the model in terms of explaining the variables such as output. However, the extended model fails in generating the mass spectra of exchange rates at low and business cycle frequencies as in the data. The extended NOEM model can generate neither the hump-shaped response of exchange rates to a monetary shock nor the hump-shaped spectral density of exchange rates because the model cannot disconnect the tight relation between interest rate difference and the expected exchange rate depreciation. In sum, as long as the uncovered interest parity condition holds, the monetary policy shock cannot lead to a hump-shaped response of exchange rate. Even if the tight relation between interest differential and expected exchange rate depreciation is muted by the assumption that the domestic interest rate is an increasing function of net foreign asset as in Benigno (2003), the close relation between interest rate differential and expected exchange rate depreciation still dominates the effect of incomplete asset on exchange rates. Therefore, some elements that dominate the effect of interest rate differential on exchange rate differentials are necessary to be incorporated into the model to have a hump-shaped impulse response of exchange rates to a monetary shock as well as a hump-shaped spectral density of exchange rate. 6. Concluding remarks This paper investigates whether the NOEM model with either PCP or PTM can generate the typical hump-shaped spectra of exchange rate changes as in the data. The paper shows that the NOEM model is not successful in generating the spectral density of the selected variables calculated from the sticky price model, though NOEM model with separable preferences does generate volatile exchange rate movements as in the data. Specifically, the model cannot generate mass spectra of exchange rates at low and business cycle frequencies as in the data. The NOEM model displays a flat spectrum for exchange rates and output without a noticeable business cycle peak, while the spectral density of output and nominal exchange rates of the data displays a peak at business cycle frequencies. These are the dramatic failures of the NOEM model. The NOEM model either with other shocks or an incomplete asset market fails to produce the typical humpshaped spectral density of output and exchange rates at low and business cycle frequencies. The extended NOEM model with the market frictions suggested by CEE (2003) and Smets and Wouters (2002) generates the typical hump-shaped spectral density of output, but it still fails in generating the hump-shaped spectral density of exchange rates. In future research, it would be desirable to explore elements that can improve the performance of the model at low and business cycle frequencies. Essentially, it would be necessary to find elements that endogenously disconnect the close relation between the interest rate differentials and the expected exchange rate depreciation, i.e. the uncovered interest parity. Appendix A This appendix summarizes the optimizing behavior of the representative household and firms. The representative home household maximizes the lifetime expected utility function subject to the budget constraint and time constraint.

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First order conditions for the domestic household can be summarized as follows. u1 ðCt ; Lt Þ ¼ Kt ;

ðA  1Þ

u2 ðCt ; Lt Þ ¼ Kt wt ;

ðA  2Þ

  Pt Ktþ1 ; Kt Qt;tþ1 ¼ b Ptþ1

ðA  3Þ

  −1 It uV ; Kt

ðA  4Þ

qt ¼

Ktþ1 ¼ uðIt =Kt ÞKt þ ð1−dk ÞKt ;

ðA  5Þ

Et ½ð1 þ rtþ1 ÞKt qt  ¼ Et ½Ktþ1 ðqtþ1 ð1− dk Þ þ Atþ1 F1 ðKtþ1 ; Htþ1 − Ho ÞÞ;

ðA  6Þ

8   > Pht ð jÞ −/ > > Cht ; < Cht ð jÞ ¼ Pht  −/ > Pht ð jÞ > > : Iht ð jÞ ¼ Iht ; Pht

ðA  7Þ

 ⁎ −/ Pht ð jÞ Cft ð jÞ ¼ Cft ; P⁎  ⁎ ht −/ Pht ð jÞ Ift ð jÞ ¼ Ift ; ⁎ Pht

and the time constraint Eq. (5), and the budget constraint Eq. (9). Eq. (A.1), which is the first order condition for consumption good, says that the marginal utility of consumption good equals the marginal utility of wealth. Eq. (A.2) relates the marginal utility of leisure to the marginal utility of the real wage rate. Eq. (A.3) refers to the intertemporal decision of the domestic household, that is, the decision of bond holdings. Eq. (A.4) which is the first order condition with respect to the home representative household's investment represents that Tobin's q equals the inverse of the investment/capital adjustment function derivative. Eq. (A.6) represents the relationship between the rent paid to a unit of capital in t + 1 and the expected return to holding a unit of capital from t to t + 1 and thus the evolution of Tobin's q over time. Eq. (A.7) states that the demand for jth consumption goods and investment goods are determined by the cost minimization demands when the composite demands are given. Next, the firm's cost minimization conditions are given by Vt ¼ MCt At F1 ðKt ðjÞ; zt Ht ð jÞ− Ho Þ;

ðA  8Þ

Wt ¼ MCt At zt F2 ðKt ð jÞ; zt Ht ð jÞ− Ho Þ;

ðA  9Þ

where MCt is the marginal cost of the firm in period t Moreover, CRS production function implies H0 0 that Ht Kð jÞ− ¼ HtK−H for all j and thus the cost minimization conditions specified in the above t ð jÞ t equations hold for aggregate quantities.

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Finally, the home firm's optimal price setting equations are given by

Pht;t ¼

Pft;t ¼

/Et

hP

l k Ktþk k¼0 ðap bÞ Ptþk

ð/−1ÞEt

/Et

hP l

k Ktþk k¼0 ðap bxÞ Ptþk

hP l

ð/−1ÞEt

i Yht;tþk MCtþk

k Ktþk k¼0 ðap bÞ Ptþk

hP

Yht;tþk

i ; for home markets;

ðA  10Þ

i Yft;tþk MCtþk

l k Ktþk k¼0 ðap bxÞ Ptþk

i for foreign markets in PTM

ðA  11Þ

Stþk Yft;tþk

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