# Causes of Nonlinearities in Low Order Models of the Real Exchange Rate

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Causes of Nonlinearities in Low Order Models of the Real Exchange Rate By Yamin Ahmad, UW-Whitewater Ming Chien Lo, St. Cloud State University Olena Mykhaylova, University of Richmond Working Paper 12 - 01 University of Wisconsin – Whitewater Department of Economics 4th Floor Hyland Hall 800 W. Main Street Whitewater, WI 53190 Tel: (262) 472 -1361

Causes of nonlinearities in low-order models of the real exchange rate Yamin Ahmad Ming Chien Loy University of Wisconsin - Whitewater St. Cloud State University Olena Mykhaylovaz University of Richmond Abstract This paper investigates the extent to which modern DSGE models, which feature local currency pricing, home bias, nontraded goods and incomplete markets, can generate nonlinear real exchange rate dynamics that are consistent with those found in the time series literature using data from the current ‡oating period. Our key …ndings are as follows. First, if the solution to the DSGE model is approximated to the …rst order, then linearity tests that utilize univariate autoregressions of the real exchange rate su¤er from an omitted variables problem, which leads them to overestimate the true incidence of nonlinearity. Consequently, studies that fail to control for this problem may spuriously …nd evidence of nonlinearities in the data, despite the fact that the data generating process may be linear. Second, we propose a strategy that can largely eliminate this distortion. Finally, we …nd that DSGE models solved using higher order approximations are capable of generating true structural nonlinearities in real exchange rates both asymptotically and in short samples. JEL Classi…cation: C15 C32 F41 F47 Keywords: Simulation, real exchange rate dynamics, nonlinear dynamics, smooth transition estimation, DSGE modeling Corresponding author. Department of Economics, 800 W Main Street, Whitewater, WI 53190 Email: ah- mady@uww.edu, Homepage: http://facsta¤.uww.edu/ahmady/ Tel: (262) 472 5576 y Department of Economics, Stewart Hall 380, St. Cloud State University, 720 Fourth Avenue South, St. Cloud, MN 56301, Email: mclo@stcloudstate.edu z Department of Economics, Robins School of Business, 28 Westhampton Way, University of Richmond, VA 23173, Email: omykhayl@richmond.edu

1 Introduction Understanding the behavior of exchange rates has always been an issue of great importance for central bankers and policy makers, more so in today’s world characterized by the real and …nancial linkages that exist between countries. For example, the large and growing international current account imbalances faced by many developing and developed nations are prompting researchers to look for the causes and e¤ects of these phenomena. However, any investigation into the international linkages that exist between countries, be it international capital ‡ows, optimal currency areas or cross-country trade patterns, clearly hinges upon a proper understanding of the dynamics of the real exchange rate (RER). Over the course of the last decade, dynamic stochastic general equilibrium (DSGE) models have become very popular as a tool among both researchers and policy makers, and have been used extensively to study the implications of these international connections on consumer welfare and policy conduct. Yet, all the policy prescriptions derived using the DSGE framework are conditional on the models’ ability to replicate the observed facts about real and nominal exchange rates. The contribution of our paper pertains to an evaluation of the DSGE framework with regards to its ability to yield real exchange rate dynamics that are consistent with the data, with a speci…c focus on the existence of nonlinearities in the simulated time series. Since the seminal paper by Rogo¤ (1996), researchers have sought to better understand the dy- namics of real exchange rates in order to resolve the purchasing power parity (PPP) puzzle present in empirical work. The open economy literature has taken two distinct approaches to the problem. On the theoretical side, DSGE models were updated with a variety of monetary and real frictions to improve the dynamics of simulated exchange rates. Empirical work, on the other hand, has focused on developing better time-series and panel estimation techniques, in particular suggesting 2

that real exchange rates may exhibit nonlinearities in their behavior.1 Our paper aims to bridge these two distinct strands of literature by studying the time-series properties of real exchange rates generated by the several widely used variants of international DSGE models. To date, the vast majority of the theoretical DSGE literature has primarily focused on studying second moments of the real exchange rate (its variance and autocorrelation, as well as correlations with key macroeconomic indicators).2 In evaluating this approach, Chari, Kehoe and McGrattan (2002), henceforth CKM, were one of the …rst to point out that DSGE models with sticky prices and deviations from the law of one price were unable to replicate the observed volatility and persistence of real exchange rates. Several solutions have been put forth in response to this …nding, although to our knowledge, the characterization of the actual dynamics of real exchange rates produced by DSGE models has received almost no attention. Recent empirical studies, which have focused on the time series properties of real exchange rates, have found evidence of nonlinearities in their behavior. The motivation for this approach arose from the failures of a linear framework: assuming linear dynamics and perfect arbitrage implies that the speed of adjustment is constant at all levels of deviations from PPP. However, several reasons have been put forth which would indicate that perfect arbitrage is not observed in the real world, and consequently nonlinearities would exist in the dynamics of real exchange rates. These reasons range from transactions costs (Michael, Nobay and Peel, 1997; Obstfeld and Taylor, 1997) and heterogeneity of agent’s beliefs (Kilian and Taylor, 2003) to misalignments in the foreign exchange market resulting in a lack of coordination (Reitz and Taylor, 2008; Sarno and Taylor, 2001). These papers are among many which have found that the data on real exchange rates can 1 For recent surveys on the literature which highlight the dynamics of real exchange rates characterized as a nonlinear process, see Ahmad and Glosser (2011), Taylor (2006), and Taylor and Taylor (2004). Engel, Mark and West (2007) provide a thorough overview of the developments in both theoretical and empirical literature. 2 The two exceptions we are aware of are Corsetti, Dedola and Leduc (2008), who estimate exchange rate pass- through coe¢ cients using simulated data, and Rabanal and Rubio-Ramirez (2012), who study the behavior of simu- lated real exchange rates at all frequencies. 3

be parsimoniously characterized as smooth transition autoregressive (STAR) process, or one of its variants. However, these studies lack rigorous theoretical foundations that could help to explain the existence of nonlinearities in real exchange rate behavior. Drawing on the …ndings of both of these …elds, we build a two-country DSGE model that includes a wide array of frictions commonly used in the theoretical literature to explore whether nonlinearities generated by the model translate into the type of nonlinear real exchange rate dynamics reported in the time series literature. The particular type of nonlinearity we consider is the STAR process, given its success in the empirical literature. Our main …ndings are as follows. First, the results— and their interpretation— of the standard nonlinearity tests like that of Teräsvirta (1994) depend critically on the underlying data-generating process (DGP). In large-scale nominal DSGE models, real exchange rate behavior is driven by many state variables, including technology levels, last-period prices, and beginning-of-the-period capital stocks. When we solve the model using …rst-order approximation (so that the DGP of the simulated RER series is linear in these state variables), linearity tests that express RER as a univariate function of its own lags su¤er from an omitted variables problem, which consequently leads them to overestimate the incidence of nonlinearity in the RER series. Two additional sources of this overestimation arise from the misspeci…cation of the autoregressive order and from distortions in the size of the test itself. We propose a solution to mitigate the e¤ects of the omitted variables problem and demonstrate that including a subset of easily observable state variables as instruments in the linearity test reduces the incidence of nonlinearity down to the size of the test plus a size distortion error. This, in e¤ect, leads us to (correctly) fail to reject the null hypothesis of a linear RER process when the DGP is truly linear. When we solve the model to second order— so that the DGP is in fact nonlinear— we …nd incidences 4

of nonlinearity above and beyond the size of the test (inclusive of the size distortion error) even when we do include the relevant state variables on the right hand side of the test equation. Therefore, we can meaningfully assert that, when solved to second or higher order, some versions of the nominal DSGE models can produce real exchange rates with nonlinear properties described in many recent empirical papers. The corollary of this …nding is that empirical researchers should have a structural model of real exchange rate determination in mind before proceeding with linearity tests, and should include the relevant linear state variables in the test equation to control for the omitted variables problem. Then, any nonlinearity detected beyond the size of the test can be interpreted as structurally meaningful. With this observation in mind, we utilize the solutions that have been put forth in response to the CKM critique to calibrate a large-scale theoretical model to the data and to evaluate the incidence of nonlinearity that is present in the simulated real exchange rate series after controlling for the omitted variables problem. We …nd that the version of the model with incomplete international …nancial markets yields the highest incidence of nonlinearities among the four basic speci…cations we consider, and that the combination of local currency pricing, home bias in consumption and incomplete markets yields the highest incidence of nonlinear dynamics among all the examined convolutions of model extensions. Moreover, the consumption home bias and nontraded goods versions of the model yield real exchange rate dynamics that are consistently characterized as exponential smooth transition autoregressive processes, indicating symmetric adjustment to the PPP norm. In contrast, the local currency price version consistently yields asymmetric adjustment, described by the logistic smooth transition autoregression speci…cation. Finally, our results are robust to sample size: the omitted variables problem is present (and can be eliminated by including state variables in the linearity test), both asymptotically and in shorter 5

samples that more closely correspond to the available post-Bretton Woods RER data. 2 The benchmark model We begin our investigation of real exchange rate nonlinearities by using a simple RBC framework composed of two countries, Home (H) and Foreign (F ), each populated by in…nitely lived house- holds of measure one; there is no migration. Households consume a combination of home and foreign goods, and international asset markets are complete. As a matter of notation, subscripts H and F will refer to a good’s country of origin; asterisks will indicate that it is consumed in country F . For example, CH denotes consumption of country H’s good in country F . The two economies have a similar structure; therefore, below we present only the Home country equations. Since the setup of the benchmark model and its extensions (outlined in Section 4) are well documented in the literature, we keep our description of them short. Solutions to the agents’problems described below, as well as the market clearing conditions, are available for download from the authors’websites. 2.1 Firms A representative …rm rents capital Kt 1 from the domestic households at the rate Rt , hires labor Lt at the rate Wt , and produces the home intermediate good according to 1 YH;t = Zt Kt 1 Lt ; where 0 < < 1, and Zt denotes the level of productivity enjoyed by all home …rms at time t. The …rm maximizes pro…ts max PH;t YH;t Pt (Wt Lt + Rt Kt 1) ; Kt 1 ;Lt 6

where PH;t and Pt are the prices of the intermediate product and the …nal good (de…ned below), respectively, and the …rm takes all prices and input costs as given. The …nal good, used as the numeraire and available for consumption and investment, is aggregated from the home and foreign intermediate goods according to HF 1 HF 1 1 HF 1 HF 1 Yt = HF [YH;t ] HF + (1 ) HF [YF;t ] HF ; (1) where HF measures the elasticity of substitution between home and foreign goods, and captures the degree of home bias in consumption. Since the law of one price (LOP) clearly holds in this speci…cation, the prices of the two …nal goods, which also represent the countries’CPIs, are given by h i 1 1 HF (1 HF ) 1 HF Pt = PH;t + (1 ) PF;t h i 1 (1 HF ) 1 HF 1 HF Pt = PF;t + (1 ) PH;t We de…ne the real exchange rate as Qt Pt =Pt . The only potential source of real exchange rate ‡uctuations in the benchmark model is home bias in consumption ( and/or greater than 0:5). 2.2 Households The representative household chooses consumption, work e¤ort, capital investment and a portfolio of assets to maximize its expected lifetime utility ( ) 1 X j t Cj1 [1 Lj ]1 Ut = Et + (2) 1 1 j=t Households in both countries have access to a complete contingent claims market. Each household faces the following budget constraint: Et [ t;t+1 Dt ] + Pt (Ct + It ) = Pt (Wt Lt + Rt Kt 1) + Dt 1 7

The …rst term on the left-hand side is the price of a portfolio of state-contingent bonds traded internationally, and Dt 1 is the payo¤ of such portfolio in period t. The household’s capital accu- mulation is given by 2 1 It Kt = (1 )Kt 1 + It Kt 1 2 Kt 1 2.3 Calibration Each time period in the model corresponds to one quarter. Since the goal of our paper is to understand how standard DSGE models fare in reproducing the observed nonlinearities in the behavior of real exchange rates, we follow CKM in parameterizing our model. The choice of calibrated and estimated parameters comes from the full model, described in Section 4; we list their values below but defer the discussion of most of these parameters until the later section. We set = 0:99, = 10, = 0:023, and = 0:33, values common in DSGE literature. We set = = 0:8, and assume that home and foreign intermediate goods are substitutes by letting HF = 1:5. The value of the capital adjustment cost parameter is calibrated to replicate the ratio of standard deviations of consumption to output observed in the U.S. data. We set = 5; the high value of the risk aversion parameter is motivated by the need to replicate the observed high volatility of the real exchange rate, which is a function of relative consumptions when asset markets are complete: Qt = [Ct =Ct ] . Productivity in the two countries evolves according to the following autoregressive process: 2 3 ln Zt 60:95 0 7 ln Zt 1 "z;t =6 4 7 5 ln Z + ln Zt t 1 "z;t 0 0:95 Following CKM, the technology shock process is described by V ar("z ) = V ar("z ) = 0:0072 and Cov("z ; "z ) = 1:2 10 5. 8

Solution to the model is found using perturbation methods described in Schmitt-Grohé and Uribe (2004) and Collard and Juillard (2001); computer code is written in Dynare (Collard and Juillard, 2003). 3 Theoretical behavior of real exchange rates We now explore the theoretical properties of the simulated real exchange rate series generated by the model above, as well as the ability of existing empirical tests to capture its inherent theoretical nonlinearities. In general, we assume that the simulated log real exchange rate qt may be succinctly captured by a univariate autoregressive process of the form qt = 1 qt 1 + ::: + p qt p + "t ; (3) where "t is white noise. The simulated series is stationary if the sum of the autoregressive coe¢ - P cients, j pi=1 i j, is less than 1.3 3.1 Estimation of the simulated time series In addressing the main question of this paper, we focus our attention on the extent to which the simulated real exchange rates exhibit nonlinear dynamics. Several recent papers have argued that a possible resolution to Rogo¤’s (1996) PPP puzzle is to model the real exchange rate as a nonlinear stationary process. The nonlinearities have taken the form of Markov switching, threshold processes, and variants of smooth transition (STAR) models. Although the nonlinear modeling strategies di¤er across papers, all such studies aim to show that small permanent deviations from PPP are possible due to frictions, whereas large deviations are quickly corrected. Consequently, real exchange rates 3 As a precursor to our analysis, we run unit root tests on the simulated data to verify that the RER series is stationary. In all the trials, where the sample size for each trial is 5000 observations, we are able to reject the null hypothesis of nonstationarity. 9

exhibit mean reverting behavior only when there is a substantial deviation from the level implied by purchasing power parity. As a …rst step to understanding the dynamics of the real exchange rates generated by DSGE models, we limit our attention to the STAR type nonlinearity, given its popularity and success in the literature. The dynamics of the real exchange rate (which can also be thought of as a deviation from PPP) can be described as a STAR process as follows: 0 1 Xp Xp j qt j + @ qt = ~ j qt j A F (qt d; ) + "t ; (4) j=1 j=1 where fqt g is a globally stationary ergodic process, "t iid 0; 2 , and F (:) represents a transition function from one regime to another and determines the degree of mean reversion. Finally, d and represent the delay and speed of adjustment parameters, respectively. According to (4), the dynamics of the real exchange rate are driven by the autoregressive parameters f 1 ; :::; pg in one n o regime, and ~ ~ 1 + 1 ; :::; p + p in the other. A popular approach to characterize this type of nonlinearity involves using an ESTAR (exponential smooth transition) model, where the transition function F (:) takes an exponential form: h i F ( ; qt d) =1 exp (qt d q)2 (5) The function in (5) possesses the following properties: F : (0; 1) ! [0; 1] ; F (0) = 0; limx! 1F (x) = 1, and the function is symmetric around the threshold parameter, q. An alternative speci…cation of F (:) is the logistic transition function, which yields a logistic smooth transition autoregres- sive (LSTAR) model. The LSTAR model captures asymmetric adjustment around the threshold parameter, and in this case, the transition function may be written as: 1 F ( ; qt d) = (6) 1 + exp [ (qt d q)] 10

The ESTAR model has been shown to succinctly capture the dynamics of real exchange rates in the current ‡oating period.4 The popularity of the STAR framework arises from its ability to demonstrate how the real exchange rate may move smoothly from one type of regime to another, depending on how far the RER value is from a particular threshold. Small deviations from PPP are considered by the STAR framework to be persistent, whereas large deviations exhibit mean-reverting dynamics. In the ESTAR variant, positive and negative deviations from the threshold are treated symmetrically, whereas the LSTAR speci…cation is indicative of asymmetric adjustment. Empirical estimation of this class of model proceeds in three steps. Once the order of autoregres- sion, p, has been determined through traditional means (e.g., utilizing information criteria like the AIC/SBC), these steps include: testing for linearity; selecting the value of the delay parameter, d; and choosing between LSTAR and ESTAR speci…cations. We outline the basic elements of the linearity test next, given its relevance for the question being asked in our paper, although we refer the reader to Teräsvirta (1994) or Teräsvirta and Anderson (1992) for additional details regarding the overall estimation methodology. Although there are several ways of de…ning linearity, we assume that nonlinearity is present if the transition function in (4) does not equal zero. For both the ESTAR and LSTAR speci…cations outlined in (5) and (6), this test can be captured by the null hypothesis H0 : = 0: If were known, then it would be possible to proceed using classical inference techniques. However, since and d are typically not known in practice, then (4) is not identi…ed under the null hypothesis, and hence no consistent estimate of either or d exists. This is the essence of the problem outlined by 4 A small sample of papers that …nds that the ESTAR model is able to parsimoniously capture the dynamics of the real exchange rate over the current ‡oating period include Baum et al. (2001), Paya and Peel (2006), Sarantis (1999), and Taylor et al. (2001). We use the dollar-sterling rate during this period as the basis for calibrating our full model in Section 4. 11

Davies (1977). To address this issue, Teräsvirta (1994) follows the suggestion proposed in Davies (1977) and keeps the unidenti…ed values …xed when deriving a Lagrange Multiplier (LM)-type test for linearity. The idea behind the test statistic used to test H0 : = 0 involves taking a third order Taylor expansion of the transition function (4) around = 0. Taking p as given, the researcher estimates the auxiliary regression below for a …xed parameter d: p X 2 3 qt = 00 + 0j qt j + 1j qt j qt d + 2j qt j qt d + 3j qt j qt d + "t ; (7) j=1 and tests the joint null hypothesis that all the coe¢ cients corresponding to the cross products in (7) are zero: H0;LIN = 1j = 2j = 3j =0 (j = 1; :::; p) (8) Nonlinearity is detected if the researcher is able to reject H0;LIN . The test above is also used to determine the delay length, d, by running the linearity test for all plausible values of d and picking the one that minimizes the test’s p-value (as suggested by Tsay, 1989). We follow Teräsvirta’s (1994) and Teräsvirta and Anderson’s (1992) methodology in conducting the linearity tests. We now examine the correspondence between the DGP of the simulated RER series and the spec- i…cations (3) and (4). 3.2 Omitted variables problem A common approach to solving DSGE models is to use linear methods to approximate the policy and transition functions. More speci…cally, most algorithms used to solve DSGE models take an n-th order Taylor expansion around the steady state (see, for example, Collard and Juillard, 2001; 12

Kim et al., 2008; and Schmitt-Grohe and Uribe, 2004). In Dynare, the solution approximated to the …rst order expresses the current value of the endogenous variables as a function of the previous state of the model and the realization of shocks at the beginning of the current period: ut = Au + Bu xt 1 + Cu "t (9a) xt = Ax + Bx xt 1 + Cx "t (9b) where ut is a ku 1 vector of non-predetermined variables (controls); xt is a kx 1 vector of predetermined endogenous variables (states); "t is a k" 1 vector of predetermined exogenous variables (shocks); and Ai , Bi and Ci , i 2 fu; xg are appropriately-dimensioned coe¢ cient matrices. Given that the focus of our paper is on the real exchange rate, assume, for illustrative purposes and without loss of generality, that ku = 1 so that ut = qt . We demonstrate in Appendix A that the RER series generated by the process (9) cannot be re-written as a univariate AR(p) equation (3), except for the simple case of kx = 1. Instead, if we were to express qt as a function of its own lags, we would instead obtain an ARMAX process of the following form: p X qt = 0 + j qt j + '0x xt 2 + '0" "t 1 + Cu "t (10) j=1 where 'x and '" are functions of the coe¢ cients of Ai ; Bi ; Ci ; i 2 fu; xg and of the sum of the Pp autoregressive coe¢ cients, j=1 j. What we may infer from this exercise is that if the theoretical model presented in Section 2 appropriately characterizes the economy, and subject to the accuracy of the model’s solution (from the …rst order Taylor expansion), then (10) represents the optimal characterization of the DGP to a …rst order approximation. Hence, failure to account for the presence of the lagged state variables on the right hand side of (10) when estimating the DGP would lead to an omitted variables problem. Consequently, estimates of the autoregressive coe¢ cients used to represent the dynamics of the real exchange rate may be biased. Conceivably, a parsimonious nonlinear process may capture the dynamics of the data quite well in the presence of omitted 13

variables under a linear approximation to the model’s equations. A simple approach to resolve this issue would be to include some subset of relevant lagged state variables on the right hand side of the auxiliary equation (7) used in the linearity test. We would then expect to see a reduction in the incidence of nonlinearity down to the size (plus any size distortions) of the linearity test. We investigate this claim in the next section. Another important point to consider is that if the underlying DGP is nonlinear, no matter how complex, it may still be approximated arbitrarily well by an n-th order Taylor expansion. Dynare has the ability to compute model solutions to second order and third order. Second-order ap- proximation to the model’s equations introduces nonlinearities into the dynamics of endogenous variables as follows: ut = Au + Bu xt 1 + Cu "t + Du (xt 1 xt 1) + Eu ("t "t ) + Gu (xt 1 "t ) (11a) xt = Ax + Bx xt 1 + Cx "t + Dx (xt 1 xt 1) + Ex ("t "t ) + Gx (xt 1 "t ) (11b) The matrices Au and Ax now include a constant capturing the variance of future shocks. Suppose again that ut = qt . Analogous to the logic presented above for the linear approximation, if one were able to include all the right hand side variables from (11a), then testing the null of linearity for qt should yield an incidence of nonlinearity that would equal the size of the test. In this case, however, the result is in some sense meaningless, given that we do know that the true DGP is, in fact, nonlinear. Instead, if we were to estimate the equation for qt as an autoregressive process including only the linear terms: p X qt = 0+ j qt j + '0x xt 2 + '0" "t 1 + Cu "t ; j=1 then incidences of nonlinearity beyond the size of the test would indicate the presence of the higher order terms in the RER DGP. This is the approach we take later in the paper: when running the linearity tests for higher order approximations, we include the linear lagged state variables as 14

instruments in the auxiliary equation (7). If any of the higher order terms in (7) are found to be signi…cant, and if we are able to reject the null hypothesis (8), then we take it as evidence of nonlinear behavior in the dynamics of the simulated real exchange rate. 3.3 Results of nonlinearity tests We now turn our attention to the time series properties of the real exchange rates simulated using the benchmark model outlined in Section 2. We generate 1000 trials (each 5000 observations long) of the (log) real exchange rate and then examine the properties of the simulated data in order to infer the underlying dynamics embedded in the series. It should be noted that since the time series literature examining the dynamics of real exchange rates utilizes raw (un…ltered) data, we proceed to examine the dynamics of the simulated data without HP-…ltering it. The properties of the actual data are summarized in Table 1 for the various G7 countries in our sample. For the 1970Q1–2010Q1 period, we are unable to reject the null hypothesis of a unit root in the real exchange rate for the majority of the countries. Although this might be a point of concern, since the DSGE model above produces stationary real exchange rates, it is important to note that other studies were also unable to reject a unit root in post-Bretton Woods data: see, for example, Lothian and Taylor (1996) and Ahmad and Craighead (2011). This result is unsurprising, given the notoriously low power of unit root tests. In fact, several papers have found that once longer spanning data is utilized, the behavior of the real exchange rate appears to be stationary (Lothian and Taylor, 1996 and 2008; Ahmad and Craighead, 2011).5 Moreover, when …tting an autoregressive structure to the real exchange rate, we …nd that ; the sum of the autoregressive coe¢ cients in (3), is less than one in absolute value for all countries, which indicates to us the stationarity of the underlying series. Hence we proceed to utilize the simulated RER series generated from the model, 5 See Taylor, Peel and Sarno (2001, pg. 1016) for more discussion of this issue. 15

which are stationary in nature. It is important to note that the empirical literature that estimates STAR type models typically tests for the presence of a unit root in the data, since several recent papers examining the properties of the linearity tests suggest that the size of Teräsvirta and Anderson (1992) and Teräsvirta (1994) tests can be distorted.6 When the test is applied to a linear but nonstationary or highly persistent DGP, the null hypothesis of linearity is rejected too often. Sandberg (2008), for example, …nds that the null is rejected 31 percent of the time for a test with 5 percent level of signi…cance. As such, the literature has adopted an approach that tests for the presence of a linear unit root process before proceeding to nonlinearity tests.7 We follow this approach for each trial, and …nd that we are able to reject the unit root null in all the trials where the sample size is 5000 observations long. We then proceed to test for STAR-type nonlinearity vis-à-vis a linear speci…cation for each trial using the methods suggested by Teräsvirta and Anderson (1992) and Teräsvirta (1994), described in Section 3.1. We set the signi…cance level of the linearity test at 5 percent; rejections of the null hy- pothesis in (8) indicate the presence of nonlinear dynamics that may be captured as ESTAR/LSTAR type processes. Our goal here is to determine the number of times that nonlinearity is detected, and, conditional on this detection, to compute the number of times that an ESTAR/LSTAR speci- …cation is chosen. We do this under two scenarios: in the …rst, we force the order of autoregression, p; to equal one, corresponding to the case where the researcher knows that the model solution and, therefore, the data generated by Dynare, is based on an AR(1) processes of the form (9). In the second scenario, we determine p using the Schwartz criterion (SBC), representing the case where the researcher does not know anything about the DGP and attempts to infer it from the data itself. 6 The list includes, but is not limited to, Kiliç (2004) and Sandberg (2008). 7 Kapetanios, Shin and Snell (2003) and Rothe and Sibbertsen (2006) propose several tests which have higher power than the ADF test when the alternative (a nonlinear but globally stationary DGP) is true. 16

Our results for the linearity test are reported in panel A of Table 2. When we do not restrict p to equal one, and instead …t it based on the SBC criterion, we …nd that a maximum autoregressive order of p = 2 is selected in all instances. Out of the 1000 trials in the benchmark case, p is chosen to equal one, 989 times; in the remaining cases, p is selected to equal two. Therefore, we do not report any results for p > 2:8 The nominal size of the linearity test, derived from the Monte Carlo results, is approximately 5 percent, although it is an increasing function of the delay parameter, d. At d = 4; linearity is rejected 8.4 percent of the times, which could be due to either the omitted variables problem discussed above, or to the test size distortion (although it is much lower than reported in Sandberg, 2008). We detangle these two mechanisms below, when we add the lagged state variables to the test equation. Consequently, the results in the benchmark version of the model are consistent with what we would expect. Given that home bias is the only source of RER movement, and the model is simulated to the …rst order, we would not expect to detect any signi…cant deviations from PPP. We …nd that the simulated data is approximated very well by a linear function, at least to the extent of the size of the linearity tests that were employed. The right side of Panel A reports the fraction of times that an ESTAR model with symmetric adjustment is selected in those cases where the null hypothesis of linearity is rejected. Based on our …ndings, we conclude that the RER in the benchmark version of the model is best described by the LSTAR speci…cation, although it is important to keep in mind that this result comes from only 4–8 percent of the trials where linearity was rejected. An additional point to note from panel A is the very small increase, of 0.1 percent, in the incidence of nonlinearity when we go from a scenario where the autoregressive order is known (p = 1) to one 8 In the full model and the extensions explored later, the number of times p equals one or two di¤ers, although the maximum value of p is never selected to be larger than 2 across all the trials. 17

where it is inferred from the simulated data (p = 2) for values of d > 1: Although its magnitude is small, even smaller than the size distortion, its presence indicates that there may be a bias arising from the selection of the autoregressive order. As such, it could be a potential factor that contributes to the incidence of nonlinearity observed in the models that we examine in Section 5. Finally, panel A demonstrates that the estimated RER dynamics can sometimes be captured by a high order linear process when the model solution is approximated to …rst order. Thus, when we test for the existence of nonlinear dynamics using the methodology advocated by Teräsvirta’s (1994) and Teräsvirta and Anderson’s (1992), the presence of the omitted variables problem discussed in Section 3.2 can manifest itself in the form of nonlinearities detected by the linearity tests. This could help to explain the higher incidences of nonlinearities (beyond 5 percent) for values of d > 1: A large portion of the empirical literature has estimated the dynamics of real exchange rates as univariate processes. A natural question that arises based on our …ndings is whether the nonlinear models (like the popular STAR framework) are truly capturing theoretical nonlinearities, or whether they are simply misspeci…ed. We address this question below by incorporating the missing state variables into the linearity tests, and by considering the results of higher order approximations. 3.4 Real exchange rate dynamics as a multivariate process The solution we consider for mitigating the e¤ects of the omitted variables problem on the incidence of nonlinearity is to include the lagged state variables on the right hand side of (3). Dynare solution to the model of Section 2 includes four state variables: Kt 1, Zt 1, and the corresponding foreign quantities. The results of estimating the RER dynamics with these variables included in the test are reported in panel B of Table 2. We …nd that the size distortion in the linearity test remains present for larger values of d: When we 18

force the autoregressive order to equal 1, the inclusion of the lagged state variables either maintains or decreases the incidence of nonlinearity for d = f1; 2; 3g, just as expected; for d > 3; we detect a slight increase in the incidence of nonlinearity. When we do not specify the AR order and instead select it using the SBC criterion, we note a small increase in the incidence of nonlinearity compared to the counterpart values for p = 2 in panel A, ranging from 0.1 percent for d = 2 to 1.2 percent when d = 4. We attribute this increase to the misspeci…cation of the autoregressive order. Overall, the absence of large incidences of nonlinearity in the benchmark case, above and beyond the size of the linearity test, indicates that a linear speci…cation is appropriate given that the benchmark model’s solution was computed to a …rst order approximation. Nonetheless, our conjecture that the inclusion of lagged states should lower the incidence of nonlinearity in con…rmed when the autoregressive order is speci…ed correctly. 3.5 Higher order approximations As our next step, we examine the results of linearity tests when the theoretical model is solved using higher order approximations. Just as before, we generate 1000 trials of the RER data, each 5000 observations long, based on the second and third order approximations to the model’s solution. Again, we …nd that the RER series is largely linear; the results are reported in panels A and B of Table 2. Panel A, which does not include any state variables, indicates that moving from …rst to second (and third) order approximation actually results in a very small (0.1 percent) decrease in the incidence of nonlinearity. However, when state variables are included (panel B), we observe a marginal increase in the incidence of nonlinearity (between 0.1 to 0.2 percent) for p = 1. When we select the autoregressive order based on the SBC information criterion, we once again see a small reduction in the incidence of nonlinearity in going from …rst to second or third order. 19

Moreover, going from second to third order of approximation does not a¤ect the computed incidence of nonlinearity, since the simulated data generated by the third order approximation is equal to that generated by the second order approximation to at least six decimal places. Consequently, we do not continue to investigate the dynamics of the real exchange rate simulated using the third order approximation.9 To summarize our results so far, we conclude (albeit based on numerically small di¤erences) that (a) small size distortions exist in the linearity test we employed; (b) inclusion of state variables in the linearity test mitigates the omitted variables problem when the researcher correctly speci…es the autoregressive order of the RER series; (c) the misspeci…cation of the autoregressive order can lead to an increase in the detected incidence of nonlinearity; and (d) tests of RER series generated using second or higher order approximations can uncover true structural nonlinearities. The results presented in Section 5 con…rm these …ndings (with much larger magnitudes) for large-scale versions of nominal DSGE models. 4 Real exchange rates in nominal DSGE models Within the international DSGE framework, real exchange rate is typically determined through the PPP equation as a function of nominal exchange rate and international relative prices. In order to generate dynamic behavior of real exchange rates, we must specify the source(s) of deviations from the purchasing power parity. The recent literature has identi…ed several of these:10 presence of nontraded goods in the consumer basket (Corsetti, Dedola and Leduc, 2008), home bias in consumer preferences (CKM; Faia and Monacelli, 2008; Steinsson, 2008), sticky prices (CKM; 9 It has been noted in Kim et al. (2008) that high order approximations of DSGE models often result in explosive solutions. This issue can be solved by "pruning out" the extraneous high-order terms in the Taylor series expansion of the model equations; however, pruning for third order approximations has not yet been implemented in Dynare. While the simple RBC framework of Section 2 does not exhibit explosive solutions, we were unable to solve its large scale extensions (discussed in the next section) to orders higher than two. 10 The following references comprise a non-exhaustive list of papers that have used one or more of these features. 20

Bergin and Feenstra, 2001; Kollmann, 2001), and local currency pricing (CKM; Corsetti, Dedola and Leduc, 2008; Benigno, 2004; Steinsson, 2008). We add these features one at a time to an otherwise basic two-country DSGE model and study the resulting time-series properties of the simulated real exchange rate. We also consider the case of incomplete international asset markets, in which the uncovered interest parity (UIP) condition does not hold, and nominal (and real) exchange rates become decoupled from the relative interest rate movements. The choice of shocks included in the model also has non-trivial consequences for the behavior of exchange rates. Monetary policy, which requires the presence of nominal rigidities to a¤ect real variables, must itself be quite persistent to generate the observed half-lives of real exchange rates. On the other hand, several studies found that real shocks, which can include (but are not limited to) government spending, technology, price markups, and the cost of international borrowing, are important ingredients of the simulated real exchange rate behavior (Steinsson, 2008; Adolfson et al., 2007; Christiano et al., 2011). We include all of these disturbances in our model to increase the chances of matching simulated time series to the data. Finally, to capture the potential di¤erences in the economic capacities of the two countries, we assume that they are populated by in…nitely lived households of measure M at home and M abroad. Therefore, aggregate and per-capita quantities will be di¤erent in equilibrium: aggregate consumption is a multiple of the per-capita value. 4.1 Firms Each country has a continuum of …rms that produce tradable goods indexed by f on the unit interval. At time t, each Home …rm hires capital and labor from domestic households to produce 21

one of the varieties of the domestic good according to: YH;t (f ) = M [KT;t 1 (f )] [ZT;t LT;t (f )]1 We use the subscript T to di¤erentiate between tradable and non-tradable goods (introduced in Section 4.4). Several papers have recently pointed out the importance of technology speci…cation (levels or growth rate) for the dynamics of DSGE models,11 especially when examining un…ltered simulated variables. To capture this channel, we rewrite the technology process in the vector error correction form as follows ln ZT;t ln Z1 "Tz;t = + ln ZT;t 1 ln ZT;t 1 + ln ZT;t ln Z2 "Tz;t All goods varieties are then bundled into a composite home good, available for consumption and investment, using the Dixit-Stiglitz aggregator: Z 1 T;t 1 T;t T;t 1 YH;t = [YH;t (f )] T;t df ; 0 where T;t > 1 is the time-varying price markup. This composite good can be used for public and private consumption or private investment. As in Calvo (1983), …rms reset their prices each period with a constant probability (1 !); otherwise, the old prices remain in e¤ect. If a …rm f gets to announce a new price in period t, it chooses P~H;t (f ) to maximize its expected discounted future pro…ts 1 X n h i o Et t;j ! j t P~H;t (f ) YH;j d d (f ) + YH;j (f ) d T Cj YH;j d (f ) + YH;j (f ) j=t Here t;j captures the stochastic discount factor of households, the two terms in the brackets denote the aggregate demands for the home goods bundle from domestic and foreign agents, respectively, d (f ) + Y d (f ) = Y and T C (:) represents total costs. Notice that in equilibrium, YH;j H;j H;t (f ). 11 See, for example, Christiano et al. (2011) and Rabanal, Rubio-Ramirez and Tuesta (2011). 22

4.2 Households A representative household’s expected lifetime utility (2) is updated with the variables CT;t and LT;t , where CT;t is aggregated from home and foreign bundles using the CES aggregator (1). Analogously to Section 2, the countries’CPIs are given by h i 1 1 HF (1 HF ) 1 HF PT;t = PH;t + (1 ) St PF;t (12a) h i 1 (1 HF ) (1 HF ) 1 PT;t = PF;t + (1 ) St 1 PH;t HF (12b) Here St is the nominal exchange rate, expressed in units of Home currency per one unit of Foreign currency. We rede…ne the real exchange rate as Qt St PT;t =PT;t . Each household faces the following budget constraint: Et [ t;t+1 Dt ] + PT;t [CT;t + IT;t TT;t ] = Wt LT;t + Dt 1 + Rt KT;t 1 + T;t Households receive transfers PT;t TT;t from their government (which can be negative in the event of lump-sum taxation). T;t represents per-capita pro…ts of all domestic …rms. Finally, the house- hold’s capital accumulation is given by 2 1 IT;t KT;t = (1 )KT;t 1 + IT;t ( + Z1 1) KT;t 1 2 KT;t 1 4.3 The government The Home monetary policy is a variant of the Taylor rule, in which the short-term nominal interest rate responds (with inertia) to deviations of in‡ation and output from their targets: gap it = (1 i) i + i it 1 + (1 i) t + y yt + "i;t (13) 23

1 Here i = 1 is the steady state level of the interest rate, t is the growth rate of CPI, and ytgap denotes output gap, which we measure as the deviation of output from its steady state level.12 The Foreign economy pursues independent monetary policy analogous to rule (13). Government consumption is described by an autoregressive process ln GT;t = (1 g ) ln GT + g ln GT;t 1 + "g;t ; where "g;t is a white noise process, and GT denotes the steady state level of government spending. Both governments balance their budgets every period, so GT;t = TT;t and GT;t = TT;t . 4.4 Extensions The benchmark speci…cation, in which PPP holds at all times, generates virtually no real exchange rate movement and therefore serves as a stepping stone to richer frameworks with more frictions. Below we outline four widely used extensions of the basic DSGE model that break the absolute PPP or the UIP relationships and thus introduce richer dynamics of the real exchange rate. We add them one at a time, and also examine several combinations of the extensions in the next section. 4.4.1 Home bias in consumption The existence of home bias in consumption has been well documented in the trade literature (for example, in Obstfeld and Rogo¤, 2000); to introduce it into our model, we require that + 6= 1. By allowing for a di¤erent composition of Home and Foreign consumption baskets, we decouple movements in the two countries’CPI levels (12). To see the impact of this assumption on the RER dynamics, we substitute (12) into the de…nition of the real exchange rate and linearize around the 12 Generally, output gap can be modeled as deviation of current output from either (a) its steady state level; (b) ‡exible-price level; or (c) last period level. 24

symmetric steady state to obtain qt = [ + 1] t; where lowercase letters indicate log deviations from steady state, and t pF;t +st pH;t represents the terms of trade. As the economies move away from the symmetric case ( = = 0:5 or, more generally, =1 ), real exchange rate movements grow in magnitude in response to ‡uctuations in the terms of trade. 4.4.2 Local currency pricing (LCP) Empirical evidence points to a rather low degree of pass-through from exchange rates to import prices, which of course is inconsistent with our benchmark assumption that the LOP holds in the individual, as well as the aggregate price level.13 The literature has identi…ed several potential causes of this phenomenon: local currency pricing by …rms with some degree of monopoly power, the presence of nontraded goods either in the product distribution network or directly in the consumer basket, or price stickiness at the consumer level. We begin by adding international price discrimination by …rms, which may optimally choose to charge di¤erent prices, P~H;t (f ) at home and P~H;t (f ) abroad, to maximize the present discounted value of future pro…ts: 1 X n h io Et t;j ! j t P~H;t (f ) YH;j d (f ) + Sj P~H;t (f ) YH;j d (f ) d T Cj YH;j d (f ) + YH;j (f ) j=t The presence of price rigidities implies that …rms have to take into account the entire path of future expected nominal exchange rate St when setting foreign prices for their products; thus, the LOP 13 For example, Engel (1993) …nds empirical evidence that the volatility of the price of a good relative to a similar good within a country is lower than the volatility of the price of a good relative to the price of the same good in a di¤erent country. Engel and Rogers (1996) …nd that the "border e¤ect" introduces signi…cant variation in the price of a good sold in di¤erent countries. 25

need not hold. The aggregate price indices of the Home goods at home and abroad are given by 1 1 1 PH;t T = (1 !)P~H;t T + !PH;t T 1 (14a) (1 T) (1 T) (1 T) PH;t = (1 !)P~H;t + !PH;t 1 (14b) In addition to the home bias mechanism described above, the real exchange rate now responds to international price di¤erences of the same good. Linearizing the de…nition of the real exchange rate around the steady state yields qt = [ + 1] t + "H t + "Ft ; where "H F t and "t measure deviations from the LOP that arise due to local currency pricing coupled with nominal rigidities: "Ft pF;t + st pF;t and "H t pH;t + st pH;t . 4.4.3 Nontraded goods The second way of lowering the degree of pass-through from exchange rates to in‡ation is to introduce nontraded goods. To do so, we add a continuum of …rms, indexed by n 2 [0; 1], which, similar to the producers of the tradables above, hire capital and labor from local households to produce one of the varieties of non-tradable goods: YN;t (n) = M [KN;t 1 (n)] N [ZN;t LN;t (n)]1 N We assume that there are no correlations in the productivity processes across sectors. All the varieties of nontraded goods are bundled together analogously to the tradable output: Z 1 N;t 1 N;t N;t 1 YN;t = [YN;t (n)] N;t dn 0 Each …rm in the nontraded sector chooses a price P~N;t (n) to maximize pro…ts; this price can be reset each period with a probability (1 !). For simplicity and symmetry, we extend the speci…cation 26

of the technological processes as follows: ln ZT;t = ln Z1 + ln ZT;t 1 ln ZT;t 1 + (ln ZT;t 1 ln ZN;t 1) + "Tz;t (15a) ln ZT;t = ln Z2 ln ZT;t 1 ln ZT;t 1 + ln ZT;t 1 ln ZN;t 1 + "Tz;t (15b) ln ZN;t = ln Z1 (ln ZT;t 1 ln ZN;t 1) + "N z;t (15c) ln ZN;t = ln Z2 ln ZT;t 1 ln ZN;t 1 + "N z;t (15d) The consumption aggregate in the household’s utility function (2) is now given by NT 1 NT 1 1 NT 1 NT 1 Ct = NT [CN;t ] NT + (1 ) NT [CT;t ] NT (16) Here NT denotes the elasticity of substitution between tradables and non-tradables, and 0 <

the relative consumption growth rates is directly linked to real exchange rate movements. By combining the international risk-sharing condition and households’…rst order conditions, we …nd that the correlation between the real exchange rate and the growth of relative consumption is 1: qt+1 = ct+1 ct+1 , where as it is only 0:07 in the data. This observation has motivated many researchers to examine the incomplete international …nancial markets speci…cation, which has the potential to lower the theoretical correlation to the empirically observed levels. We continue assuming (for model tractability) that all consumers can perfectly share risks within a country; additionally, a risk-free bond At , issued by the foreign country, can be traded inter- nationally. The bond is denominated in foreign currency and o¤ers nominal interest rate it . We furthermore assume that when borrowing from abroad, home households must pay a risk premium t. We follow Christiano et al. (2011) in assuming that t depends on the home country’s position in the international asset market and on the international interest rate di¤erential: N F Dt t = exp d NFD r (it it ) + " ;t 4PH;t YH;t The …rst term in parentheses captures the deviation of the home consumers’ net foreign debt to (annual) GDP ratio from its steady state level, and the parameter d measures the responsiveness of risk premium to changes in this ratio.14 The risk premium payments are distributed evenly to foreign households. The home household’s budget constraint becomes Et [ t;t+1 Dt ] + St At + PT;t [CT;t + IT;t TT;t ] = = Wt LT;t + Dt 1 + 1 + it 1 t 1 S t At 1 + Rt KT;t 1 + T;t 14 Equilibrium dynamics of a small open economy with incomplete asset markets generally include a random walk component; the transaction cost modi…cation guarantees stationary of the model. See Schmitt-Grohé and Uribe (2003) for explicit treatment of the problem. 28

4.5 Calibration Parameter values used in the full model and its extensions are listed in Table 3. Several of them are quite common in the literature and have been discussed in Section 2; we describe the rest below. Since we use the data on the U.S.-UK real exchange rate as a benchmark for model performance, we set M = 6 and M = 1, and let = = 0:94. We follow Corsetti, Dedola and Leduc (2008) and the sources referenced therein to calibrate the nontraded sector. We set the elasticity of substitution between tradables and nontradables, NT , to 0.74; assume that the share of nontradables in consumption is 60%, and the share of capital in the production of nontradables is N = 0:44. We let T = N = 7, resulting in a 17 percent markup. Setting ! = 0:75 causes price contracts in both sectors to last for four quarters on average. As described in CKM, our utility function speci…cation is consistent with the existence of a balanced growth path. We stay close to Rabanal, Rubio-Ramirez and Tuesta (2011) in calibrating the VECM process for technology: = 0:007, Z1 = Z2 = 0:001, and the standard deviations of innovations "Tz = "Tz = 0:0105. For simplicity, we assume that the shocks in the nontraded sector have the same covariance matrix as those in the traded sector. Based on U.S. data, we set the ratio of government debt to GDP to 20 percent, and let g = 0:97. Standard deviation of the government spending shock is set equal to 0:01. We borrow the values for the monetary policy parameters from CKM. However, since the authors do not report a value for the policy shock (and instead calibrate it to match certain moments of the data), we set its standard deviation to 0.016, a value well within the range of U.S. estimates.15 As in CKM, we calibrate Cov ("i ; "i ) to match the cross-correlation of U.S. and UK outputs. 15 See, for example, Adolfson et al. (2007), Canzoneri et al. (2006) and Christiano et al. (2011). 29

We follow Adolfson et al. (2007) in calibrating the process for the time-varying price markup. To this end, we assume j;t = 0:1 j + 0:90 j;t 1 + "j ;t ; j 2 fT; N g We depart from the authors’estimation somewhat by setting V ar "T = V ar "N = 2:6 10 4, equal to the monetary policy shock described above.16 We set N F D = 0, r = 1:10, and V ar (" ) = 3:9 10 5, as in Christiano et al. (2011). We calibrate the elasticity of the risk premium to the level of international debt, d, to replicate the volatility of current account to GDP ratio observed in the U.S. data. 4.6 Sources of real exchange rate dynamics At this point, it perhaps useful to summarize the various mechanisms that contribute to the dy- namics of the real exchange rate. To this end, equation (18) presents the disaggregated de…nition of the real exchange rate in the full model: ( ) 1 h i1 NT 1 NT 1 NT 1 HF 1 HF 1 HF PN;t + (1 ) PH;t + (1 ) PF;t Qt = St 8 9 1 1 < 1 NT 1 HF 1 HF 1 NT HF =1 NT PN;t + (1 ) PF;t + (1 ) PH;t : ; (18) where the prices of tradables evolve according to (14) and its foreign economy equivalent. The real exchange rate ‡uctuates due to (a) movements of the nominal exchange rate St ; (b) presence of nontraded goods ( > 0); (c) di¤erences in the composition of the tradable baskets ( 6= 1 ); (d) international price discrimination (PH;t 6= St PH;t ); and (e) nominal rigidities (! 6= 0), which force …rms to set prices based in part on expectations about future shocks. Finally, the incomplete markets speci…cation breaks the UIP relationship between real interest rates and exchange rates 16 The model estimated in Adolfson et al. (2007) includes many more frictions and shocks than our framework. If we followed their calibration exactly, price markup movements would dominate our model dynamics. 30

and also decouples the movements of the latter from the dynamics of relative consumptions. It is also useful to note that (18) clearly demonstrates the existence of structural nonlinearities in RER dynamics arising from the microfoundations of the DSGE models. In Section 5, we explore whether these micro-level nonlinearities translate into the type of macro-level nonlinearities that can be identi…ed using time series methods. If the answer is yes, then it provides an external validity to both literatures: DSGE framework and RER dynamics represented by time series models. 4.7 Business cycle properties of the models Table 4 summarizes several relevant moments of the simulated data. In addition to examining the benchmark model and its four extensions, we also study several combinations of the structural primitives. Thus, column 8 refers to the speci…cation of the model that includes both LCP and home bias (we dub this version CKM since Chari, Kehoe and McGrattan, 2002 used it as the starting point in their analysis). Columns 8 and 9 add incomplete markets (CKM_IM) and (separately) nontraded goods (CKM_NT) to the CKM model. The interaction between nominal rigidities and the presence of home bias in consumption (column 4) produces much more volatile real exchange rates compared to the other basic model extensions, matching the data very closely. We …nd that the various combination of the basic extensions (CKM, column 8; CKM_IM, column 9; and CKM_NT, column 10) overshoot the real exchange rate volatility by 40–90%. Turning to persistence, we …nd that all speci…cations, except the ones involving incomplete asset markets, do a good job in matching the empirical autocorrelation of the real exchange rates. As is often found in DSGE literature, other moments are less satisfactory. For example, all versions of the model understate the persistence of GDP, consumption and investment, and none of the models studied in this paper are able to replicate the empirical correlation between 31

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