THE REAL EXCHANGE RATE AND REAL INTEREST DIFFERENTIALS: THE ROLE OF NONLINEARITIES

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1 THE REAL EXCHANGE RATE AND REAL INTEREST DIFFERENTIALS: THE ROLE OF NONLINEARITIES Nelson C. Mark a;;y and Young-Kyu Moh b a University of Notre Dame and NBER, USA b Tulane University, USA Abstract Recent empirical work has shown the importance of nonlinear adjustment in the dynamics of real exchange rates and real interest di erentials. This work suggests that the tenuous empirical linkage between the real exchange rate and the real interest di erential might be strengthened by explicitly accounting for these nonlinearities. We pursue this strategy by pricing the real exchange rate by real interest parity. The resulting rst-order stochastic di erence equation gives the real exchange rate as the expected present value of future real interest di erentials which we compute numerically for three candidate nonlinear processes. Regressions of the log real US dollar prices of the Canadian dollar, deutschemark, yen, and pound on the fundamental values implied by these nonlinear models are used to evaluate the linkage. The evidence for linkage is stronger when these present values are computed over shorter horizons than for longer horizons. JEL CODE: F30, F31 KEY WORDS: Real exchange rate; real interest di erential; nonlinear adjustment Correspondence to: Nelson C. Mark, Department of Economics and Econometrics, 434 Flanner Hall, University of Notre Dame, Notre Dame, IN 46556, USA. Tel.: ; fax: y address: nmark@nd.edu

2 1 Introduction This paper investigates whether accounting for nonlinear adjustment in the real interest di erential helps to strengthen the evidence linking the real interest di erential to the real exchange rate. Because today s real exchange rate depends on the public s expectations of future real interest di erentials, a potential pitfall in earlier studies lies in misspeci cation of real interest rate dynamics. In our analysis, we price the real exchange rate by real interest parity. This rst-order stochastic di erence equation then gives the real exchange rate as the expected present value of future real interest di erentials with a unit discount factor. We evaluate this expected present value for three candidate nonlinear processes governing the real interest di erential. They are the band threshold autoregression (BTAR), and two smooth-transition variants a cosine smooth transition autoregression (CSTAR) and the exponential smooth transition autoregression (ESTAR) model. Numerically computed expectations of future interest di erentials are used to generate implied fundamental values of the real exchange rate. The link between the real interest di erential and the real exchange rate is assessed by regressing the log real exchange rate on the fundamental values implied by these nonlinear models. Economic theory predicts that the real interest di erential is a key determinant of the real exchange rate [Dornbusch (1976), Obstfeld and Rogo (1995)] but empirical research has struggled to nd a statistically signi cant link. While a variety of econometric methods and modeling strategies have been employed in the empirical literature these analyses have typically centered on linear regressions of real exchange rates on the contemporaneous real interest di erential. Frankel (1979) compares the predictions of a real interest parity model to monetary and Keynesian formulations, Campbell and Clarida (1987) model the ex ante real interest di erential as an AR(1), Meese and Rogo (1988), Edison and Pauls (1993), and Edison and Melick (1999) employ cointegration analysis and Baxter (1994) employs frequency domain decompositions to isolate co-movements at business cycle frequencies. Our paper is motivated by recent research that has detected signi cant nonlinear dynamic adjustment in both real exchange rates and in real interest rate di erentials. Most of this work has studied either the real exchange rate or the real interest di erential separately but not jointly. This body of work does suggest, however, that failure to account for these nonlinearities may underlie the di culties in establishing the linkage between them. In international nance, empirical work investigating nonlinear adjustment in the real exchange rate includes Obstfeld and Taylor (1997), Parsley and Wei (1996), Michael et. al. (1997), Sarno and Taylor (2001), and Bec et. al. (2002) who motivate nonlinear adjustment through transportation costs that de ne a neutral band that de ne the boundary where pro table commodity arbitrage opportunities exist. Kilian and Taylor (2003), on the other hand, obtain nonlinear adjustment from interactions 1

3 of market participants with heterogeneous beliefs about exchange rate valuation. Several studies on nonlinear adjustment in domestic interest rates have examined the Markov-switching model. 1 Less work has been done on interest di erentials or their interaction with exchange rates. Nonlinear adjustments in both the real exchange rate and the real interest di erential will be linked if central banks choose the interest rate defense of its currency over direct intervention with foreign currency reserves [e.g., Drazen and Hubrich (2003) and Flood and Jeanne (2000)]. 2 Balke and Wohar (1998) and Peel and Taylor (2002) study threshold e ects in deviations from covered interest parity and Mancuso et. al. (2003) has examined the dynamics of real interest rate di erentials and nd important threshold nonlinearities in their dynamics. Nakagawa (2002) investigated nonlinear regressions of the real exchange rate on the ex post real interest di erential and found stronger statistical evidence of a link between the real interest di erential and the real exchange rate than in earlier studies that worked in a linear environment. Here, we go one step further and ask to what extent the real exchange rate can be priced by the expected future real interest di erentials, which are assumed known to follow nonlinear dynamics. To summarize our results, we nd evidence of the real interest di erential real exchange rate link but they are not uniformly strong. For a particular model of real interest rate dynamics, the statistically signi cant linkage varies across countries and across horizons. The remainder of the paper is as follows. The next section outlines our strategy. Section 3 describes the data used in this study. Estimation of the threshold and smooth transition models of the real interest di erential are reported in section 4. Section 5 reports the results on estimating the linkage between the fundamental value implied by the nonlinear real interest rate models and the real exchange rate. Section 6 concludes. 2 Real Exchange Rate Pricing by Real Interest Parity Let s t be the log spot exchange rate expressed as the home currency price of a unit of foreign currency and i t (i t ) be the 1-period home (foreign) Eurocurrency rate. The uncovered interest parity condition which equates expected nominal yield in terms of 1 Hamilton (1988), Sola and Dri ll (1994), Ang and Bekaert (2002), and Bekaert, Hodrick and Marshall (2001) examine Markov-switching models for interest rates but not for real interest di erentials. 2 Mark and Moh (2003) exploit this idea to explain the forward premium anomaly as resulting from unanticipated central bank foreign exchange interventions that work through interest rates rather than reserves. See also Baillie and Osterberg (2000) who nd that the magnitude of deviations from uncovered interest parity increase during intervention periods. 2

4 the home currency from holding a 1-period domestic Eurocurrency deposit is E t (s t+1 s t ) = (i t i t ): (1) Let q t = s t + p t p t be the log real exchange rate where p t and p t are the log home and foreign price levels respectively and let t be the real foreign-home interest di erential where t rt r t = (i t i t ) E t (p t+1 p t ) (p t+1 p t ). Then real interest parity is obtained by subtracting the expected in ation di erential from both sides of (1), q t = + E t t + E t q t+1 ; (2) which is a stochastic di erence equation in q t. We include a constant to allow for a possible trend in the real exchange rate perhaps induced by Balassa-Samuelson e ects which would not be captured in real interest di erentials. Forward iteration of (2) gives the real exchange rate as the undiscounted present value of expected future interest di erentials and the expected long-run real exchange rate, q t t = ^q t;k + (E t q t+k (t + k)) ; (3)! Xk 1 where ^q t;k E t t+j : (4) A solution based on interest di erentials requires a terminal condition that expected long-run real exchange rate is a constant. To achieve identi cation, we appeal to a variant of what Edison and Melick (1999) refer to as the standard case, and assume that E t (q t+j (t + j)) = E(q t t) = C for all j k: For = 0; this is consistent with long-run implications of standard exchange rate models such as Dornbusch (1976) and Frankel (1979). 3 In (3); ^q t;k is the (real interest di erential) fundamentals value of the real exchange rate. Our empirical work focuses on examining the empirical linkage between the implied fundamentals determinants and the real exchange rate. In the next section, we discuss our empirical measures of this fundamentals value. Then regressing the real exchange rate on this empirical measure, ^q t;k ; we ask if there is a statistically signi cant link between the two variables and if so, whether the choice of k matters. That is, we are interested in assessing how far into the future market participants look in pricing the real 3 Another strategy [e.g., Meese and Rogo (1988)] draws on Dornbusch (1976) to motivate adjustment dynamics E t q t+1 q t+1 = (qt q t ) ; where q is the shadow exible price equilibrium real exchange rate and 0 < < 1. In the absence of real shocks, E t q t+k = qt. Substituting these two expressions into (2) gives q t = ( 1) 1 E t t + q t : j=0 3

5 exchange rate. If events far into the future weigh heavily, we will require large values of k and a focus on current real interest di erentials will tell only a small part of the story about real exchange rate dynamics. On the other hand, if we nd small values of k are more appropriate then it suggests that the di culty in assessing the impact of current news on future prices hinders the evaluation of the exchange rate s fundamental value. 3 The Data Our data set consists of monthly observations from 1978:9 to 2002:4 for Japan (JPN) and from 1975:1 to 2002:4 for Canada (CAN), Germany (GER), United Kingdom (UK), and the United States (USA). 4 1-month Eurocurrency rate data are from Datastream. The US is taken as the numeraire (home) country with exchange rates quoted as dollar prices of the other currencies. Exchange rates and consumer price indices are from the International Financial Statistics. Figure 1 plots the log real exchange rate along with the 1-month real Eurocurrency di erentials in our sample. Some co-movement between the series is visually evident at what appears to be the low-frequency components. There are also some sizable departures between the contemporaneous real interest di erential and the real Canadian dollar at the beginning and end of the sample. Similar departures can be seen for Japan during the early 1990s and for Germany towards the end of the sample. 4 Nonlinear Models of the Interest Di erential We consider three alternative nonlinear mean-reverting models for the real interest di erential t. In all three models, the persistence in t is inversely related to the magnitude of recent values of the real interest di erential. The economics behind such an adjustment pattern are straight forward to motivate. International mobility of real capital creates a long-run constraint on international real interest rate divergence whereas international monetary policy coordination provides a short-to-medium run constraint. If policy interventions tend to occur when the interest di erential is large but not when it is small, the interest di erential will tend to behave unsystematically as in a random walk process when the di erential is small. Policy interventions drive t towards its mean value when the di erential becomes large. These considerations suggest that the interest di erential will be mean reverting where the speed of adjustment is directly related to the size of the di erential. 4 Following the adoption of the euro, we use imputed values of the deutschemark. 4

6 4.1 Models Band-Threshold Autoregression (BTAR) In the BTAR model, the real interest di erential follows a driftless random walk when it lies between the xed bands [c 2 ; c 1 ]: It follows a band-reverting autoregression when it lies outside the bands. The BTAR model is speci ed as, 8 < c 1 (1 1 ) + 1 t 1 + v t if t 1 > c 1 t = : t 1 + v t if c 2 < t 1 < c 1 : (5) c 2 (1 2 ) + 2 t 1 + v t if t 1 < c 2 In the BTAR model, the transition from the autoregression to the random walk occurs abruptly at the bands. We estimate the BTAR model by sequential conditional least squares. This involves doing a grid search over the bands and delay parameter (in our case it is 1). For given values of [c 2 ; c 1 ] and delay parameter, we estimate (5) by least squares and the nal estimates are those that minimize the overall sum of squared errors. 5 As a diagnostic test, we employ Hansen s (1999) test of the null hypothesis that t follows a linear AR(1) against the alternative hypothesis of the nonlinear BTAR model. The test statistic is H = (s L s N ) =s N where s L is the sum of squared errors from the linear AR(1) model and s N is the sum of squared errors from the nonlinear BTAR model. The test statistic H does not have an asymptotic chi-square distribution because it depends on the particular speci cation under the null hypothesis (in our case, the AR(1) model) and on the particular alternative speci cation. In addition, the threshold parameters (c 1 ; c 2 ) and delay parameter are not identi ed under the null hypothesis. 6 As a result, we follow Hansen s suggestion for obtaining the distribution of H via the parametric bootstrap. The p-value of the test is obtained by a relative frequency count of the values of bootstrapped H statistics that exceed the value of H obtained from the data. The BTAR estimation and test results are presented in Table 1. Here, it can be seen that estimates of the autoregressive parameters ( 1 ; 2 ) are highly statistically signi cant. The negative estimates of 1 for Canada, Japan, and the UK indicate very rapid reversion 5 We do this over the range lower 5th c 1 lower 100th observation, upper 100th c 2 upper 5th observation. 6 Our model is the BTAR model with threshold parameter = (c 1 ; c 2 ) and delay parameter d = 1. Consider a general BTAR model of a form Y t = 0 1X t 1 I 1t (; d) + ::: + 0 mx t 1 I mt (; d) + e t ; (6) where X t 1 = (1 Y t 1 Y t 2 ::: Y t p ) and = ( 1 ; :::; m ) with 1 < 2 < ::: < m. I jt (; d) = I( j 1 < Y t d j ) where I( _ ) is the indicator function, is threshold parameter, and d is delay parameter with 1 d p. Thus in the general BTAR framework another nuisance parameter is the delay parameter. In our case, the delay parameter is 1. 5

7 towards the upper band (large positive values of t = r t r t ). 7 Hansen s test rejects the null hypothesis of linearity at very small signi cance levels for each of the four real interest di erential pairs. Smooth Transition Models We begin our development of smooth transition models by testing the null hypothesis that the interest di erential follows a linear AR(p) model against alternative of smooth transition AR(p) model. The test, suggested by Escribano and Jordá (1999), proceeds by estimating the auxiliary regression, y t = b 0 x t + b 1 x t z t + b 2 x t z 2 t + b 3 x t z 3 t + b 4 x t z 4 t + e t ; (7) where y t = t = t t 1, x t = ( t 1 ; :::; t p ) and z t = t d is the transition variable. 8 The test statistic is F = (s L s A )=4p s A where s =(T 5p) L is the sum of squared errors from the linear AR(12) model and s A is the sum of squared errors from the auxiliary regression. Under the null hypothesis of linearity H 0 : b 1 = b 2 = b 3 = b 4, the test statistic has an approximate F distribution with 4p and T 5p degrees of freedom. 9 We obtain the following results: F 48;268 = 196:7; (Canada), F 48;268 = 348:0 (Germany), F 48;225 = 165:6 (Japan), and F 48;268 = 267:4 (UK). The Escribano-Jordá test rejects the linear AR(12) model at very small levels of signi cance. The rst of two smooth transition models that we consider uses the cosine function to parameterize the transition coe cient. We call it the cosine smooth threshold autoregressive model (CSTAR) 0 v u t = 1` `X j=1 2 t j 1 A t 1 +! px j t j + v t : (8) The cosine term is a time-varying transition coe cient that depends on an ` period moving window measure of t volatility which modulates the speed at which t makes its transition back to 0. This term is close to 1 when t has been near 0 during the recent past so that the process behaves locally as a unit-root process. When t exhibits large deviations from its mean, the cosine term shrinks and the process behaves locally like a stationary autoregression. After extensive experimentation, we set ` = p = 12: We estimate the CSTAR model by nonlinear least squares and report the results in Table 7 The asymmetry in the bands does not con ict with our requirement that the unconditional mean of the interest di erential be zero since the proportion of time that t spends in each of the regions is di erent. 8 The auxiliary regression can be obtained by applying Taylor approximation to the transition function of smooth transition models. The auxiliary regression (7) can be used to test linearity null against smooth transition autoregression alternatives. See van Dijk et. al. (2002). 9 In our sample we have T = 328 for Canada, Germany, and UK and T = 285 for Japan. j=1 6

8 2. Here, our main interest rests with the speed of adjustment parameter ; so we do not report estimates of the lagged coe cient values ^ j : As can be seen from the table, the estimates of are correctly signed and are highly signi cant. Our second smooth-transition model takes the transition coe cient as an exponential function of the volatility estimate, 0 t = v u t 1` `X j=1 2 t j 1 A t 1 +! px j t j + v t ; (9) iid where v t N(0; 2 v). This model has been employed by Kilian and Taylor (2003) to model the real exchange rate. Here, the time-varying transition coe cient is speci ed as exponential term that depends on the ` period moving window of t volatility. As with the CSTAR model, we set ` = p = 12: Our nonlinear least squares estimates of the ESTAR speed of adjustment parameter are reported in Table 3. Again, the estimates are all highly statistically signi cant. The mean squared errors from ESTAR and CSTAR are roughly equivalent. j=1 4.2 Comparing Fundamentals Measures We now turn to estimation of ^q t;k which proceeds as follows. At time t 1 and conditional on t 1 ; simulate the k future values according to the particular nonlinear process (BTAR, CSTAR, or ESTAR). Call the simulated sequence t ; t+1; : : : ; t+k and use them to form the sum P k j=0 t+j. Repeat this 50; 000 times. The mean value of these sums gives our estimate ^q t;k : Next, advance the time subscript and repeat the procedure conditional on t to get ^q t+1;k : Continue in this fashion through t = T; to obtain the T t=1 : sequence ^q t;k Having obtained estimates of the expected partial sum of future real interest di erentials, we look for evidence that the real exchange rate is priced by the sum of the expected future real interest di erentials as implied by real interest parity. We examine three alternative values of k = 12; 120; and 1000 which correspond to forecast horizons of 1; 10; and 83 years respectively. Taking k = 1000 emphasizes the long-horizon features of the data as in Alexius (2001), and Chinn and Meredith (2004) who nd evidence that long-horizon linkages between the nominal exchange rate and nominal interest di erentials are stronger than they are at short horizons. To explore some basic properties of the alternative ^q t;k ; Table 4 reports the standard deviation and rst two autocorrelations. In the CSTAR and ESTAR models, the volatility of ^q t;k increases quite dramatically with k: This contrasts sharply to the BTAR model where the rapid reversion to the bands limits range and volatility of the ^q t;k : The volatil- 7

9 ity of ^q t;k implied by BTAR is fairly stable across our choices of k and is substantially smaller than volatility implied by either of the smooth-transition models. The BTAR ^q t;k are also generally a bit more persistent than those from the smooth transition models. For example, the rst-order autocorrelation coe cient for the German BTAR ^q t;12 is 0.38 whereas the largest rst-order autocorrelation from a smooth-transition model is 0.09, for Germany s ESTAR speci cation of ^q t;1000: Table 5 shows the correlation matrix of the alternative ^q t;k for each country where it is seen that the BTAR ^q t;k are highly correlated across horizons whereas the CSTAR and ESTAR ^q t;k are largely uncorrelated across k: For a given horizon, the BTAR ^q t;k are moderately correlated with the CSTAR and ESTAR ^q t;k for Germany (average correlation 0.21). For the other countries, the BTAR measures of ^q t;k are uncorrelated with either CSTAR or ESTAR measures. For a given horizon, the average correlation between the CSTAR and ESTAR ^q t;12 is a rather high value of 0:81. This average correlation drops to 0:34 for k = 1000: Although the CSTAR and ESTAR models appear to have quite similar in sample t, the information content for long-run appears to be somewhat di erent. 5 Expected Future Real Interest Di erentials and the Real Exchange Rate We are now ready to investigate the extent to which expected future real interest di erentials are priced into the real exchange rate. We begin with the regression q t = t + ^q t;k: We perform system estimation generalized method of moments using the current value and three lags of ^q t;k across each of the four currencies as instrumental variables. We also report constrained GMM estimates for those cases where homogeneity restrictions on the slope coe cient across currencies are not rejected at the 10 percent level. Robust asymptotic standard errors computed with VAR(1) pre-whitening and with Newey-West (1994) automatic lag-length selection are used to form the asymptotic t-ratios. Table 6 reports estimation results both with and without trend. The regressions yield slope coe cients that are correctly signed in every case except one that of the yen for the BTAR ^q t;1000: The estimated coe cients under BTAR without trend are signi cant (at the 10 percent level) at the medium and long horizons (k = 120; 1000). At the short horizon (k = 12) without trend, the slope is signi cant only for Germany while the constrained slope estimate is highly signi cant. When a trend is added, the estimated slope is signi cant both for Canada and for Germany. 8

10 Under the CSTAR fundamentals, we obtain signi cant slope estimates only at the short horizon k = 12: These results are robust to whether or not a time trend is included in the regressions. We note that the magnitude of the slope coe cient point estimates declines as the horizon increases. This occurs on account of the higher volatility of ^q t;k as k increases under the smooth transition models. Turning now to the regressions on ESTAR fundamentals, we nd at the short horizon the slope coe cients are signi cant for the deutschemark and the pound and these results are robust to whether or not a time trend is included. The constrained estimate with trend is also marginally signi cant. At the medium horizon k = 120, the estimates are signi cant at the 10 percent level for the Canadian dollar, deutschemark and the yen without trend, but is only signi cant for the yen when a trend is included. At the long horizon, the constrained estimates are signi cant both with and without trend. We included a time trend to account for potential Balassa-Samuelson e ects that would not be re ected in the real interest di erentials. However, if such an e ect is present and the productivity di erentials for the countries in our sample evolve with a stochastic trend, then the regression should be run using rst-di erences in the (log) real exchange rate rather than including a time trend. To explore this possibility, we report the GMM estimation results from regressions of q t on ^q t;k in Table 7. Here, under BTAR, all point estimates have the predicted sign. Individual coe cient estimates are signi cant for the deutschemark and pound for all three horizons as are the constrained estimates. Under the smooth transition models, statistical evidence of linkage is mixed. Here, we obtain some point estimates that have the wrong sign but they are not signi cant. Coe cient estimates for the pound are signi cant both for k = 12 and k = 1000; and are signi cant for the yen at k = 120: 6 Conclusions Nonlinear adjustment in log real exchange rates and real interest di erentials are a signi cant and prominent feature of the data. Statistical evidence that the real exchange rate is linked to the real interest di erential is tenuous. In this paper, we look for evidence of a link when a standard pricing relationship in which the real interest di erential is assumed to follow a nonlinear process. The evidence, while not uniformly strong, does suggest the existence of such a link when the pricing relationship is evaluated over the relatively short horizon of one year. 9

11 Acknowledements Mark thanks the National Science Foundation for support under grant no Moh gratefully acknowledges the COR fellowship from Tulane University. 10

12 References [1] Alexius A Uncovered interest parity revisited. Review of International Economics 3: [2] Ang A, Bekaert G Regime switches in interest rates. Journal of Business and Economic Statistics 20: [3] Baillie RT, Osterberg WP Deviations from daily uncovered interest rate parity and the role of intervention. Journal of International Financial Markets, Institutions and Money 10, [4] Balke N, Wohar M Nonlinear dynamics and covered interest rate parity. Empirical Economics 23: [5] Baxter M Real exchange rates and real interest di erentials: have we missed the business-cycle relationship? Journal of Monetary Economics 33: [6] Bec F, Ben Salem M, Collard F Asymmetries in monetary policy reaction function: evidence for U.S. French and German central bank. Studies in Nonlinear Dynamics and Econometrics 6: issue 2. [7] Bekaert G, Hodrick R, Marshall D Peso problem explanations for term structure anomalies. Journal of Monetary Economics 48: [8] Campbell J, Clarida R The dollar and real interest rates. Carnegie-Rochester Conference Series on Public Policy 27: [9] Chinn M, Meredith G Monetary policy and long horizon uncovered interest parity. International Monetary Fund Sta Papers, forthcoming. [10] Dornbusch R Expectations and exchange rate dynamics. Journal of Political Economy 84: [11] Drazen A, Hubrich S Mixed signals in defending the exchange rate: what do the data say? CEPR Working Paper DP4050. [12] Edison H, Pauls D A re-assessment of the relationship between real exchange rates and real interest rates: Journal of Monetary Economics 31: [13] Edison H, Melick W Alternative approaches to real exchange rates and real interest rates: three up, three down. International Journal of Finance and Economics 4: [14] Escribano Á, Jordá O Improved testing, speci cation of smooth transition regression models. In Nonlinear Time Series Analysis of Economic and Financial Data, Rothman P (ed). Kluwer, Boston;

13 [15] Flood R, Jeanne O An interest defense of a xed exchange rates? mimeo, International Monetary Fund. [16] Frankel J On the mark: a theory of oating exchange rates based on real interest di erentials. American Economic Review 69: [17] Froot K, Thaler R Anomalies: foreign exchange. Journal of Economic Perspectives 4: [18] Hamilton J Rational-expectations econometric analysis of changes in regime. Journal of Economic Dynamics, Control 12: [19] Hansen BE Testing for linearity. Journal of Economic Surveys 13: [20] Kilian L, Taylor MP Why is it so di cult to beat the random walk forecast of exchange rates? Journal of International Economics 60: [21] Mancuso A, Goodwin B, Grennes T Nonlinear aspects of capital market integration and real interest rate equalization. International Journal of Economics and Finance 12: [22] Mark NC, Moh Y O cial interventions and occasional violations of uncovered interest parity in the dollar-dm market. National Bureau of Economic Research Working Paper: 9948, August. [23] Meese R, Rogo K was it real? the exchange rate-interest di erential relation over the modern oating rate period. Journal of Finance 43: [24] Michael P, Nobay R, Peel D Transactions costs and nonlinear adjustment in real exchange rates: an empirical investigation. Journal of Political Economy 105: [25] Nakagawa, H Real exchange rates and real interest rate di erentials: implications of nonlinear adjustment in real exchange rates. Journal of Monetary Economics 49: [26] Newey W, West K Automatic lag selection in covariance matrix estimation. Review of Economic Studies 61: [27] Obstfeld M, Rogo K Exchange rate dynamics redux. Journal of Political Economy 103: [28] Obstfeld M, Taylor A Nonlinear aspects of goods-market arbitrage and adjustment: Heckscher s commodity points revisited. Journal of the Japanese and International Economics 11: [29] Parsley D, Wei S-J Convergence to the law of one price without trade barriers or currency uctuations. Quarterly Journal of Economics 111:

14 [30] Peel D, Taylor MP Covered interest rate arbitrage in the interwar period and the Keynes-Einzig conjecture. Journal of Money, Credit and Banking 34: [31] Sarno L, Taylor MP Nonlinear mean reversion in real exchange rates: towards a solution to the purchasing power parity puzzles. International Economic Review 42: [32] Sola M, Dri ll J Testing the term structure of interest rates using a stationary vector autoregression with regime switching. Journal of Economic Dynamics, Control 18: [33] van Dijk D, Teräsvirta T, Franses PH transition autoregressive models a survey of recent developments. Econometric Reviews 21:

15 14 Figure 1: Log real exchange rate and contemporaneous real interest di erential.

16 15 Figure 2: Standardized values of qt and ^q t;12 from the BTAR model.

17 16 Figure 3: Standardized values of qt and ^q t;12 implied by the CSTAR model.

18 17 Figure 4: Standardized values of qt and ^q t;12 implied by the ESTAR model.

19 Table 1: BTAR interest rate di erential CAN GER JPN UK (-1.979) (1.124) (-4.511) (-2.805) (-3.705) (1.131) (-0.262) (3.013) c c MSE H p-value Note: Asymptotic t-ratios are in parenthesis and MSE is the mean squared error of the model. For Japan, sample period is 1978:9-2002:4. H is the test statistic for the linearity test and p-value is the marginal signi cance level from 1,000 bootstrap replications of Hansen s (1999) linearity test. Table 2: CSTAR interest rate di erential CSTAR CAN GER JPN UK (8.793) (12.149) (8.267) (11.221) MSE Notes: Asymptotic t-ratios are in parenthesis. MSE is the mean squared error of the tted model. Table 3: ESTAR interest rate di erential ESTAR CAN GER JPN UK (2.909) (3.115) (2.753) (3.917) MSE Notes: Asymptotic t-ratios are in parenthesis. MSE is the mean squared error of the tted model. 18

20 Table 4: Basic features of alternative ^q t;k BTAR CSTAR ESTAR Sample Country Statistic Short Med Long Short Med Long Short Med Long 2/76-4/02 CAN s.d /76-4/02 GER s.d /79-4/02 JPN s.d /76-4/02 GBR s.d Note: s.d. is the standard deviation. j is the j th order autocorrelation (j = 1; 2). 19

21 Table 5: Correlation matrices ^q t;k BTAR12 BTAR120 BTAR1000 CSTAR12 CSTAR120 CSTAR1000 ESTAR12 ESTAR120 US-Canada BTAR BTAR BTAR CSTAR CSTAR CSTAR ESTAR ESTAR ESTAR US-Germany BTAR BTAR BTAR CSTAR CSTAR CSTAR ESTAR ESTAR ESTAR

22 (Table 5 continued) BTAR12 BTAR120 BTAR1000 CSTAR12 CSTAR120 CSTAR1000 ESTAR12 ESTAR120 US-Japan BTAR BTAR BTAR CSTAR CSTAR CSTAR ESTAR ESTAR ESTAR US-UK BTAR BTAR BTAR CSTAR CSTAR CSTAR ESTAR ESTAR ESTAR

23 Table 6: Real exchange rate and interest rate fundamentals: q t = t + ^q t;k CAN GER JPN UK Constrained slope slope slope slope Wald slope Model trend (asy-t) (asy-t) (asy-t) (asy-t) (p-value) (asy-t) BTAR 12 no (0:538) (2:693) (0:145) (0:224) (0:972) (3:383) yes (2:684) (3:293) (0:955) (0:950) (0:035) BTAR 120 no (4.400) (5.530) (2.201) (2.203) (0.000) yes (1.054) (1.140) (0.336) (0.031) (0.946) (0.621) BTAR 1000 no (5.452) (6.146) (1.904) (3.633) (0.000) yes (1.580) (0.138) (-0.213) (0.080) (0.951) (0.118) CSTAR 12 no (3.654) (4.951) (1.229) (3.866) (0.001) yes (2.159) (4.067) (2.794) (2.621) (0.001) CSTAR 120 no (-0.119) (0.296) (0.830) (0.557) (0.832) (0.394) yes (-0.192) (0.055) (0.429) (0.570) (0.942) (0.950) CSTAR 1000 no (0.687) (1.940) (2.364) (-0.442) (0.027) yes (0.349) (0.904) (1.695) (0.273) (0.336) (1.392) ESTAR 12 no (1.020) (4.298) (1.155) (3.757) (0.000) yes (0.893) (2.126) (0.610) (2.077) 0.155) (1.949) ESTAR 120 no (1.760) (2.248) (2.278) (0.836) (0.030) yes (0.761) (0.913) (1.890) (0.372) (0.255) (1.380) ESTAR 1000 no (0.870) (0.592) (1.952) (0.867) (0.289) (1.852) yes (2.311) (0.788) (1.232) (0.111) (0.089) (1.849) Note: Wald is the test statistic to test the 22homogeneity restriction across currencies. Bold face indicates that estimates are signi cant at the 10 percent level.

24 Table 7: Real exchange rate depreciation and interest rate fundamentals: q t = 0 + ^q t;k CAN GER JPN UK Constrained slope slope slope slope Wald slope Model (asy-t) (asy-t) (asy-t) (asy-t) (p-value) (asy-t) BTAR (0.474) (4.785) (1.171) (2.506) (0.108) (4.439) BTAR (0.535) (4.789) (1.572) (2.662) (0.078) (4.598) BTAR (0.497) (4.316) (2.056) (2.418) (0.060) (4.545) CSTAR (-1.594) (-0.525) (-0.071) (3.467) (0.000) CSTAR (1.673) (-0.922) (2.048) (1.362) (0.300) (2.508) CSTAR (-0.306) (-1.200) (1.463) (2.125) (0.038) ESTAR (-0.513) (0.493) (0.786) (3.118) (0.024) ESTAR (-0.685) (0.942) (2.059) (0.109) (0.174) (1.407) ESTAR (0.690) (1.977) (-0.897) (2.762) (0.074) (1.590) Note: Wald is the test statistic to test the homogeneity restriction across currencies. Bold face indicates that estimates are signi cant at the 10 percent level. 23