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The real exchange rate and real interest differentials: the role of the trend-cycle decomposition.

I. INTRODUCTION

Finding a relationship between real exchange rates and real interest rate differentials is one of the important topics in the field of international macroeconomics. Despite the fact that the theory suggesting such a relationship requires fairly unrestrictive conditions, the vast majority of empirical studies have been unable to discover it. (1)

One of the few exceptions is Baxter (1994), who first considers the decomposition of the real exchange rates in the frequency domain. (2) She then shows: (1) As standard theory suggests that the transitory components of the real exchange rate are correlated with interest rates, if the real exchange rate has a random walk component; (2) The transitory components extracted in several different ways, indeed, produce much stronger empirical connections with real interest differentials than does the real exchange rate itself. In addition to the permanent components in the real exchange rate, subsequent studies, including Nakagawa (2002) and Mark and Moh (2005), develop more sophisticated empirical models that allow for nonlinearity in the time series process of the real exchange rate.

In this paper, however, to move forward with Baxter's (1994) emphasis on the importance of the trend-cycle decomposition, we consider the following questions: What is an exchange rate model that: (1) consists of both permanent and transitory components; (2) has a relationship only between the transitory component of the real exchange rate and real interest differentials; and (3) allows us to decompose the real exchange rate into those two components in certain way(s)? Having such model(s) and real exchange rates detrended in a model-consistent manner, we shall be able to provide a more decisive answer as to whether or not there is a statistically significant link between real exchange rates and real interest differentials.

Using the pricing-to-market with a preset-pricing model, we show that the real exchange rate may have a unit root. If a unit root actually exists, then the Beveridge-Nelson (1981, hereafter BN) decomposition is a model-consistent decomposition; and the BN transitory (cycle) components and the expected future sum of the real interest differentials should be linked. Furthermore, our empirical results confirm most of our models predictions: unit roots in the real exchange rates and stationary real interest differentials. The relationship between the real exchange rate and the real interest differentials is, however, only partially supported.

The paper is structured as follows. Section II reviews the main question together with the appropriate decomposition method. Section III presents our model that is based on Devereux and Engel (2002) with misperception of price-setting firms, showing that we may describe the log of the real exchange rate as an ARIMA(0,1,p) process. Our main results, the BN decomposition and its statistical inference are given in Section IV. Section V concludes.

II. DETRENDING PROBLEM

Theoretically, the relationship between the real interest rate and real interest rate differential is given by the so-called real interest parity:

(1) [E.sub.t][[q.sub.t+1]] - [q.sub.t] = [r.sub.t] - [r.sup.*.sub.t]

where [q.sub.t] is the log of (hereafter variables denoted lower case are logged variables) the real exchange rate; [r.sub.t] and [r.sup.*.sub.t] are real interest rates in Home and Foreign countries respectively; and [E.sub.t] denotes the expectation conditional on information available at time t. A great majority of studies, however, have failed to find this relationship (1). It is important to note that an iterative substitution of Equation (1) becomes

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Most of the previous studies assume [lim.sub.k[right arrow][infinity]] [E.sub.t][q.sub.t+k] = 0, meaning that there is no unit root in [q.sub.t]. Had there been a unit root component in [q.sub.t], the first term on the right-hand side of Equation

(2) would be the trend component of the BN decomposition: (3)

(3) [q.sub.t] = [q.sup.trend.sub.t] + [q.sup.cycle.sub.t],

where [q.sup.trend.sub.t] and [q.sup.cycle.sub.t] are, respectively, the BN trend and cycle components. Hence, when a random walk component exists in the real exchange rate, an appropriate way to search for the real exchange rate-real interest differentials relationship would be: (1) to compute the exact BN decomposition, (4) along with computing the expected sum of future real interest differentials, which is the second term on the right-hand side of Equation (2); and (2) to test the link therein.

III. THE MODEL

Our model is based on that of Devereux and Engel (2002), the pricing-to-market with a preset-pricing model. There are two countries, say, "Home" and "Foreign" on a unit interval. Home lies between 0 and n, while Foreign lies between n and 1. There are: (1) households, (2) firms, (3) governments in both countries; while (4) foreign exchange dealers exist only in Home.

A. Households

Households in each country maximize their (identical) preference:

(4) [E.sub.0] [[infinity].summation over (t=0)] [[beta].sup.t] [[1/[1 - [rho]]] [[C.sup.1 - [rho].sub.t]] + [[chi]] ln [[M.sub.t]/[P.sub.t]] - [[kappa]/2] [L.sup.2.sub.t]]

where [beta] is the intertemporal discount factor; [chi] and [kappa] are constants; [M.sub.t] is nominal money balance; [P.sub.t] is the level of prices in terms of Home currency; [L.sub.t] is the labor supply; and [C.sub.t] is a consumption index defined as

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

[C.sub.ft] = [[[(1 - n).sup.-1/[lambda]] [[integral].sup.1.sub.n] [C.sub.ft] [(i).sup.([lambda]-1)/[lambda]] di].sup.[lambda]/([lambda] - 1)]], (7)

where [lambda] > 1. Variables [C.sub.ht] and [C.sub.ft] represent consumption of goods produced in Home and Foreign, respectively. Let the price index be

[P.sub.t] = [[[[n [P.sup.1-[omega].sub.ht]] + (1 - n) [P.sup.1-[omega].sub.ft]]].sup.1/(1-[omega])]

where the Home currency price of the Home goods is

[P.sub.ht] = [[[[1/n] [[[integral].sup.n.sub.0]] [[P.sub.ht]] [[(i).sup.1-[lambda]]] di].sup.1/(1-[lambda])]],

and the Home currency price of the Foreign goods is

[P.sub.ft] = [[[1/1 - n] [[integral].sup.1.sub.n] [P.sub.ft] [(i).sup.1-[lambda]] di].sup.1/(1-[lambda])].

Then, the demand functions are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Households in Home have the budget constraint:

[P.sub.t][C.sub.t] + [[delta].sub.t] [B.sub.t+1] + [M.sub.t] = [W.sub.t][L.sub.t] + [[PI].sub.t] + [[PI].sup.f.sub.t] + [M.sub.t-1] + [t.sub.t] + [B.sub.t] (8)

where [B.sub.t] is a home currency denominated risk-less bond that pays one unit of Home currency in time t; [[delta].sub.t] is the price of bond [B.sub.t+1]; [W.sub.t] is the wage; [[PI].sub.t] are the profits from firms in Home; [T.sub.t] is the transfer from government; and [[PI].sup.f.sub.t] represents the payments from foreign exchange dealers, which we shall explain in the following section. Having only riskless bonds in this economy, the structure of asset markets is incomplete. Indeed, such a structure opens up the possibility of the nonstationary real exchange rate. (5) The first-order conditions of the Home household include

(9) [[delta].sub.t] = [beta] [E.sub.t] [[P.sub.t][C.sup.[rho].sub.t]/[P.sub.t+1][C.sup.[rho].sub.t+1]]

and

(10) 1 = [chi] [[P.sub.t][C.sup.[rho].sub.t]/[M.sub.t]] + [beta][E.sub.t] [[P.sub.t][C.sup.[rho].sub.t]/[P.sub.t+1][C.sup.[rho].sub.t + 1]].

Denoting variables with asterisk as the foreign variables, Foreign households maximize their utility:

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [C.sup.*.sub.t] is the Foreign composite consumption index and [P.sup.*.sub.t] is the Foreign price level in terms of the Foreign currency.

Similar to Home households they have the budget constraint:

(12) [P.sup.*.sub.t] [C.sup.*.sub.t] + [[delta].sup.*.sub.t] [B.sup.*.sub.ft+1] + [M.sup.*.sub.t] = [W.sup.*.sub.t][L.sup.*.sub.t] + [[PI].sup.*.sub.t] + [M.sup.*.sub.t-1] + [T.sup.*.sub.t] + [[beta].sup.*.sub.ft],

where [B.sup.*.sub.ft] is a foreign currency denominated riskless bond. Note that there are no foreign exchange dealers in Foreign; thus, we assume that their profits do not appear in the Foreign households' budget constraint. The first-order conditions for Foreign households include

(13) [[delta].sup.*.sub.t] = [beta][E.sub.t] [[P.sup.*.sub.t][C.sup.*[rho].sub.t]/[P.sup.*.sub.t+1] [C.sup.*[rho].sub.t+1]].

B. The Governments

The governments of both Home and Foreign issue monies, and pay or receive transfers. The government budget constraints, both Home and Foreign are, therefore, given by

[M.sub.t] = [M.sub.t-1] + [T.sub.t], [M.sup.*.sub.t] = [M.sup.*.sub.t-1] + [T.sup.*.sub.t].

The money supplies follow

(14) [M.sub.t+1]/[M.sub.t] = [[epsilon].sub.t+1], [M.sup.*.sub.t+1]/[M.sup.*.sub.t] = [[epsilon].sup.*.sub.t+1]

where [[epsilon].sub.t+1] and [[epsilon].sup.*.sub.t+1] are lognormally distributed random variables with the same mean and variance: we assume

log [[epsilon].sub.t+1] ~ i.i.d.Normal (0, [[sigma].sup.2.sub.[epsilon]]), log [[epsilon].sup.*.sub.t+1] ~ i.i.d.Normal (0, [[sigma].sup.2.sub.[epsilon]]), and [[sigma].sub.[epsilon]] = [[sigma].sup.*.sub.[epsilon]].

C. Foreign Exchange Dealers

By assumption, Home households are not allowed to buy and sell Foreign currency denominated bonds directly. Instead, foreign exchange dealers do this on behalf of Home households; and they make payments to (receipts from, if foreign exchange dealers suffer losses) Home households. (6) At time t, foreign exchange dealers buy foreign assets by spending [[delta].sup.*.sub.t] [S.sub.t] [B.sup.*.sub.ht+1] worth of Home currency; and in the following period, the bonds will pay [S.sub.t+1] [B.sup.*.sub.ht+1], where [S.sub.t] is the nominal exchange rate (the Home price of Foreign currency); and [B.sup.*.sub.ht+1] is a Foreign currency denominated bond held by Home residents. Thus, together with the discount factor [Z.sub.t] = [beta][P.sub.t][C.sup.[rho].sub.t]/([P.sub.t+1][C.sup.[rho].sub.t+1]), the maximization problem for the foreign exchange dealer is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and its first-order condition is

(15) [E.sub.t][[Z.sub.t] [[S.sub.t+1]/[S.sub.t]] = [[delta].sup.*.sub.t].

Given the assumption of the money supply (14), Equations (13) and (10), and whose Foreign counterpart (15) implies

(16) [E.sub.t] [1/[[epsilon].sup.*.sub.t+1] = [E.sub.t] [[S.sub.t+1]/[[epsilon].sub.t+1][S.sub.t]].

By the log linearlization around the non-stochastic steady state, we have (7)

(17) [E.sub.t] [[s.sub.t+1]] = [s.sub.t].

D. Firms: Price-Setting Rule and Misperception

A monopolistically competitive firm i, which locates Home (between 0 and n) produces one unit of good i using one unit of labor input. Similarly, firm j locating Foreign (between n and 1) produces good j by the same technology. In our model, regardless of the location of a firm, it can sell its products both in Home and Foreign. However, we assume that all firms have to set their prices in terms of the currency that is used in the local market. For example, a Home firm selling its product in the Foreign market sets its price in Foreign currency; and a Foreign firm selling its product in the Home market has to set its price in terms of Home currency. Assuming that prices have to be set one period in advance, firm i sets its price as:

(18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and Foreign firm j sets its price as

(19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In addition to the benchmark model of Devereux and Engel (2002), we allow for firms' "misperception." That is, firms' own expectation about the next period's exchange rate may not be the conditional expectation of the exchange rate. Suppose that firms are not well informed about foreign exchange markets, hence they may assume the existence of "noise traders." (8) Not only is the distribution of disturbances that can be caused by such traders unknown to firms, but also firms at time-t-1 do not know if such noise traders will exist at time t. In this case, the best firms can do when they set their prices is to expect the next period's exchange rate by using the (realized) current and past exchange rates. (9)

To simplify this argument, consider the case in which firms expect the time t exchange rate given the information available at t - 1 to be [S.sup.a.sub.t-1], where a is a constant. Owing to the fact that the exchange rate follows a random walk process in our model, a = 1 corresponds to the case in which firms set their prices based upon the conditional expectation. Consider an extreme case, where a = 0. This implies that firms expect the exchange rate to always be at the level of a nonstochastic steady state (S = 1). Because it is reasonable to generalize this assumption so that the firms' expectation about the time t exchange rate relies on exchange rates of current and past p - 1 periods (where p is a fixed number), we denote the logarithm of the firms' expectation regarding the exchange rate at time t as

(20) [s.sup.e.sub.t] = a (L) [s.sub.t-1]

where

a (L) = [a.sub.1] + [a.sub.2]L + ... + [a.sub.p][L.sup.p-1]

and L is the lag operator (i.e., L[X.sub.t] = [X.sub.t-1] for any variable X). Remarkably, it is possible to show that the log of the nominal exchange rate [s.sub.t], still follows the random walk process:

(21) [s.sub.t] = [s.sub.t-1] + [[eta].sub.t]

where [[eta].sub.t] is innovation. (10)

E. The Real Exchange Rate and Real Interest Rates

The Real Exchange Rate. The real exchange rate is defined as

[Q.sub.t] = [S.sub.t] [P.sup.*.sub.t]/[P.sub.t],

and its logarithm is

(22) [q.sub.t] = [s.sub.t] + [p.sup.*.sub.t] - [p.sub.t].

Allowing for the firms' misperception, as an example, when firms set their prices according to their expectations (20), it is possible to show that the log of the real exchange rate is an ARIMA (0,1,p) process:

(23) [DELTA][q.sub.t] = [[eta].sub.t] - [a.sub.1] [[eta].sub.t-1] - [a.sub.2][[eta].sub.t-2] - ... - [a.sub.p][[eta].sub.t-p].

In contrast to Devereux and Engel's (2002) stationary real exchange rate, ours is nonstationary unless a (1) = 1 (without misperception in price-setting firms). (11) This is due to the fact that the firms' misperception prevents prices (Home and Foreign) and the nominal exchange rate from cointegrating. Although both our model and that of Devereux and Engel (2002) have a nonstationary nominal exchange rate, prices in the latter are also nonstationary due to the rational expectations of firms. Therefore, the real exchange rate (the linear combination of those three variables) exhibits stationarity.

The Real Interest Differential. The (gross) real interest rates in Home and Foreign are defined as

[R.sub.t] = [P.sub.t]/[[delta].sub.t][P.sub.t+1], [R.sup.*.sub.t] = [P.sup.*.sub.t]/[[delta].sup.*.sub.t] [P.sup.*.sub.t+1],

respectively. As nominal interest rates in both countries are equal ([[delta].sub.t] = [[delta].sup.*.sub.t]), the ratio of the real interest rates is given by

(24) [R.sub.t]/[R.sup.*.sub.t] = [P.sup.*.sub.t+1] [P.sub.t]/[P.sub.t+1] [P.sup.*.sub.t].

One can show that the log linearization of Equation (24) (i.e., the real interest differential) under the firms' misperception (20) is

(25)

[r.sub.t] - [r.sup.*.sub.t] = -[a.sub.1][[eta].sub.t] - [a.sub.2][[eta].sub.t-1] - ... - [a.sub.p][[eta].sub.t-p+1]

where [r.sub.t] and [r.sup.*.sub.t] are the logarithms of [R.sub.t] and [R.sup.*.sub.t], respectively. (12)

Real Interest Parity. The expected growth rate of the real exchange rate in our preset-pricing model is then

(26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and its log linearization yields real interest parity, stated in Equation (1). It is important to note that even under the firms' misperception (20), Equations (23) and (25) imply real interest parity to hold.

IV. EMPIRICAL RESULTS

Our plan to discover and test the relationship between real exchange rates and real interest differentials is the following. First, we confirm the existence of unit roots in real exchange rates and the nonexistence of unit roots in real interest differentials. Then, the BN cycles of real exchange rates and the corresponding expected sum of future real interest differentials are computed in a way that is consistent with our model. The link between these two variables is then formally tested. (13) The quarterly data from the first quarter of 1973 through the third quarter of 2009 are utilized for Canada (CA), Japan (JA), Switzerland (SW), the United Kingdom (UK), and the United States (US). Our data for exchange rates are from the Federal Reserve Board (for the U.S. and Canadian dollars) and International Financial Statistics (the rest of the exchange rates); data for consumer price indices (CPIs) are from International Financial Statistics; and data for short-term and long-term interest rates are taken from the Federal Reserve Board (secondary market rates of 3-month treasury bills are used for the short-term rate; and the 10-year treasury note yield is used for the long-term rate), the Bank of Canada (short-term interest rates), and the Organisation for Economic Cooperation and Development Economic Outlook (short-term and long-term interest rates are used for the rest). (14)

A. Unit Root Tests

Many empirical studies indicate inconclusive results as to whether real exchange rates have a unit root. More specifically, depending on the unit root test being used, test results vary widely (Edison and Merick 1999). One explanation for such a perplexity can be attributed to finite sample properties of unit root tests. In particular, (1) finite sample properties are largely different across the tests; and (2) finite sample properties depend heavily on the true data generating process (DGP) under the null and alternative hypotheses. Therefore, when researchers have a specific model in mind, it is practically more reasonable to use the most suitable tests for the model, rather than agnostically use as many tests as they can.

In our model, the real exchange rate may have a unit root, together with moving-average (MA) errors. For such a case, it is well known since Schwert (1989) that popular unit root tests, such as the Phillips-Perron test (Phillips and Perron 1988), suffer from large size distortions. Put differently, when the Phillips-Perron test is used for the process that actually has a unit root and MA errors, then the test often mistakenly rejects the null hypothesis of a unit root, thereby indicating that the process being tested is stationary. (15)

To evade the potential problem of erroneous rejection, following Ng and Perron (2001a, 2001b), we implement three types of unit root tests that have less size distortion and higher power than others under the presence of MA terms. Those are: modified Phillips-Perron (MZ[alpha]) tests, the augmented Dickey-Fuller (ADF) test, and the Elliott, Rothenberg, and Stock (1996) feasible point optimal (PT) test. All the series are detrended by the generalized least square (GLS) method with a constant as a regressor; and the length of the lags is selected by the modified Akaike information criteria (MAIC). The results reported in Table 1 indicate a unanimous conclusion: we fail to reject the null hypothesis of the unit root for US-CA, US-JP, US-SW, and US-UK.

Having size corrected, we focus on the power of the tests. Our simulations (Appendix S1-S4) reveal that when the DGP is an MA(1) process without a unit root, as we model for real interest differentials, the size-corrected unit root tests have relatively low power. Instead, the KPSS test (Kwiatkowski, Phillips, Schmidt and Shin, 1992), with no unit root as a null, behaves considerably well under the null hypothesis of a stationary MA(1) (i.e., it has good size) as well as under the alternative of an ARIMA(0,1,1) (i.e., it has good power). After the power of the unit root tests is taken into account, the stationarity of the Switzerland-US real exchange rate is doubted; therefore, we exclude it from further investigation.

Table 2 reports the results of the unit root tests for real interest differentials. In our model, the real interest differentials are stationary, and they follow MA processes. The unit root tests for both the short-term and long-term real interest rate differentials, in general, fail to reject the null hypothesis. However, not being able to reject the null hypothesis does not necessarily mean the test's null hypothesis is correct: this is because of the fact that the test has low power, as Table S1 in the Supporting Information shows. Moreover, the KPSS test--which appears to do a good job for our purposes, from Table A1 panels (a) and (b)--does not reject the null hypothesis of stationarity for all the short-term and long-term real interest differentials. (16)

What is the conclusion of the unit root tests? On the basis of the tests we have conducted, it is meaningful to compute the BN decomposition for the real exchange rates of US-CA, US-JP, and US-UK pairs. Then, it is meaningful to compare the BN cycle component of those real exchange rates with their corresponding real interest differentials, in order to investigate the link predicted by our model.

B. BN Cycles of the Real Exchange Rate and the Expected Sum of Future Real Interest Differentials

The BN decomposition of the real exchange rate and future real interest differentials are obtained as follows. First, we choose the length of lags, p, for the real exchange rate, using the Akaike Information Criterion (AIC) and Schwartz Bayesian Information Criterion (BIC) (17) of the estimated ARIMA(0,1,p) process. Then, given the estimated parameters in the ARIMA(0,1,p) and its fitted residuals, the BN cycle of the real exchange rate is computed exactly. (18) Similarly, an ARIMA(0,0,[p.sup.*]) process of the real interest differentials is estimated, (19) where [p.sup.*] is chosen by AIC, BIC, or is set to p-1. Finally, the expected sum of future real interest differentials, [D.sub.t] [equivalent to] [E.sub.t] [[summation].sup.[infinity].sub.j=0] ([r.sub.t+j] - [r.sup.*.sub.t+j]), are computed.

Tables 3 and 4 report the MA estimation of the real exchange rates and real interest differentials, respectively. The first-order MA coefficients and the estimated standard deviations of the errors are all statistically significant. Given the fact that the MA roots are far from -1, and the fact that the estimated standard deviations of the errors are significantly different from zero, the estimated results are consistent with the conclusions from the unit roots tests, as well as with our model.

Selected p and [p.sup.*] are summarized in Table 5. It is interesting that for the US-UK pair, the BIC chooses [p.sup.*] = 4 for the short-term real interest differentials. This number coincides with p - 1, when the AIC chooses the length of the lags for the same pair's real exchange rate. In fact, this is the very case that we assume in our model.

Figures 1-7 compare the BN cycle components of the real exchange rate and the expected sum of future real interest differentials. (20) Keep in mind from Equations (2) and (3) that [q.sup.cycle.sub.t] and [D.sub.t] should be negatively correlated, although the figures do not explicitly show that they are negatively correlated. Still, the BN-detrended real exchange rates and the corresponding real exchange differentials are moving more closely together, compared to the growth rates of the real exchange rates ([DELTA][q.sub.t]) and the real interest differentials ([r.sub.t] - [r.sup.*.sub.t]) shown in Figure 8. Considering the fact that earlier studies of this literature tried to connect those two variables (to be more precise, [DELTA][q.sub.t+1] and [r.sub.t] - [r.sup.*.sub.t]), the detrending substantially improves the resemblance of the two variables.

Table 6 measures sample standard deviations of real exchange rates and real interest differentials, together with that of the BN-detrended real exchange rates and the corresponding real interest differentials. From this, it is quite obvious that the real exchange rates are almost five times more volatile than the real interest rate differentials (first three columns from the left). Thanks to the BN decomposition and computed expected future sums of real exchange differentials, this difference in the volatilities becomes insignificant, or even negative. For example, the standard deviation of the BN-detrended real exchange rate for the US-UK pair is 2.03 when the AIC is used. This is strikingly smaller than the standard deviation of the first difference of the real exchange rate, 5.03. Also, when the expected future sum is computed with the BIC (or p - 1) chosen number of lags, the real interest differentials for this pair is increased to 2.12 from 1.17. The standard deviation of interest differentials that is almost four times smaller than that of the real exchange rate turns out to be larger than the standard deviation of the real exchange rate, due to the model-consistent decomposition.

C. Statistical Inference

Under the null hypothesis, which states that the cyclical components of [q.sub.t] and the real interest differentials, [D.sub.t] are not correlated, the estimated cross-autocorrelation function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has an asymptotic distribution (Brockwell and Davis 1991 (21)) of

(27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [??] denotes convergence in distribution; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[rho].sub.D] (h) are autocorrelation functions of the cyclical components of [q.sub.t] and [D.sub.t], respectively.

In order to compute [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which are unknowns, we utilize the estimated ARIMA errors of [q.sub.t] and ([r.sub.t]- [r.sup.*.sub.t). (22) Clearly, our test encompasses the one for the contemporaneous correlation, namely, testing whether or not [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Figures 9-15 show the cross-autocorrelations together with the 95% bounds based on the distribution (27). In particular, it is worth mentioning that the cross-autocorrelations around j = 0, which correspond to the contemporaneous correlation, are negative and significant for the US-UK pair. The cross-autocorrelation at j = 1 for "uk q114" in Figure 14 (the real exchange rate and real interest differentials are estimated by an ARIMA (0,1,1) and an MA(4), respectively) is -0.1829, and the 95% bound is -0.166. This result indicates that the BN cycle of the real exchange rate and the expected future real interest differentials may not be independent, and are, indeed, negatively correlated (with one lag), as our model predicts. Other results, however, are still puzzling. For instance, as Figures 10-13 present, the US-JP pair has a positive and significant cross-autocorrelation at j = 0 or its vicinity.

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Having found only a weak link between the real exchange rate and real interest differentials (for the US-UK pair), our next question is whether the relationship has remained stable over time. In other words, can we find subsamples, for which the link between the two variables is stronger than that in the full sample? To answer this question, Figure 16 demonstrates rolling 10-year moving-average estimates of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (0) together with its 95% bounds (box). Once again, the US-UK pair of an ARIMA(0,1,5) real exchange rate and an MA(4) real interest differentials are consistent with our model, in the sense that the BIC chosen lag length is indeed p - 1: one lag less than the MA components in the real exchange rate ("uk q5s4" and "uk q514" in Figure 16). Including such model-consistent combinations, the 10-year windows centered in 1990 and earlier indicate significant negative relationships for both the short-term and long-term real interest differentials, regardless of the lengths of lags that are used to compute [q.sup.cycle] and [D.sub.t] (there is no significant link after 1990, however). (23)

V. CONCLUSION

Following Baxter's (1994) influential paper, we focused on the trend-cycle decomposition issue in the real exchange rate. In particular, we developed a model which allows for a more flexible time-series process of the real exchange rate, namely the nonstationary nature of a process that is widely supported by a vast majority of empirical studies. It is important to note that the nonstationarity of the real exchange rate in our model comes from the presetting price rule of monopolistically competitive firms, together with their possible misperceptions. Thanks to the well-specified time-series model, we can accurately compute the cycle components of the BN decomposition that theoretically correspond to the sum of the future expected values of real interest differentials.

However, the model presented in this paper is by no means the only one that causes a non-stationary real exchange rate. For example, as argued by Mark (2001), the Balassa-Samuelson model (Balassa 1964; Samuelson 1964) with nonstationary productivity shocks predicts a nonstationary real exchange rate. More recently, a seminal paper by Engel and West (2005) predicts a "random walk like" real exchange rate, when the value of the discount factor is near unity.

Our model is partly empirically supported. First, after taking into account the time-series process of real exchange rates, the most appropriate statistical tests are in favor of the existence of unit root components in real exchange rates, with one seeming exception, for which a unit root is strongly doubted. As for the real interest differentials, in accordance with our model, all of them are found to be stationary. Secondly, as theory implies, a rigorous statistical test constructed from the asymptotic process of the cross-autocorrelation function confirms the link between the BN cycle of the real exchange rate and the expected future sum of real interest differentials for the pair of US-UK (especially before 1990). Yet, some cases, including US-Japan, remain as a puzzle: not only is an insignificant correlation found, but a significant positive relationship is also found.

A possible extension of this paper may include developing a model in which the misperception parameter, a (1) changes over time so that the real exchange rate sometimes follows the random walk process, while at other times it exhibits a stationary variable. Such a model would reconcile the debate over nonstationarity or nonlinearity of the real exchange rate.

doi: 10.1111/j.1465-7295.2011.00387.x

ABBREVIATIONS

ADF: Augmented Dickey-Fuller

AIC: Akaike Information Criterion

BIC: Bayesian Information Criterion

CPI: Consumer Price Indices

DGP: Data Generating Process

GLS: Generalized Least Square

MA: Moving Average

MAIC: Modified Akaike Information Criteria

OLS: Ordinary Least Squares

PPP: Purchasing Power Parity

SUPPORTING INFORMATION

Additional Supporting Information may be found in the online version of this article:

APPENDIX S1. The Beveridge-Nelson decomposition.

APPENDIX S2. The expected future sum of the real interest differentials.

APPENDIX S3. OLS and the t-test as not desirable method.

APPENDIX S4. Simulations.

Table S1. Rejection rates of unit root tests.

Figure S1. The estimated bias [[??].sub.2] and the misperception parameter a(1).

Please note: Wiley-Blackwell are not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article.

REFERENCES

Asea, P. K., and C. M. Reinhart. "Le Prix de l'Argent: How (Not) to Deal with Capital Inflows." Journal of African Economies, 5(3, Supplement I), 1996, 231-71.

Balassa, B. "The Purchasing Power Parity: A Reappraisal." Journal of Political Economy, 72, 1964, 584-96.

Baxter, M. "Real Exchange Rates and Real Interest Differentials: Have We Missed the Business Cycle Relationship?" Journal of Monetary Economics, 33, 1994, 5-37.

Baxter, M., and R. G. King. "Measuring Business Cycles: Approximate Band-Pass Filter for Economic Time Series." The Review of Economics and Statistics, 79, 1999, 551-63.

Beveridge, S., and C. R. Nelson. "A New Approach to Decomposition of Economic Time Series into Permanent and Transitory Components with Particular Attention to Measurement of the 'Business Cycle'." Journal of Monetary Economics, 7, 1981, 151-74.

Brockwell, P. J., and R. A. Davis. Time Series: Theory and Methods. New York: Springer, 1991.

Campbell, J. Y., and R. H. Clarida. "The Dollar and Real Interest Rates." Carnegie-Rochester Conference Series on Public Policy, 27, 1987, 103-40.

Devereux, M. B., and C. Engel. "Exchange Rate Pass-through, Exchange Rate Volatility, and Exchange Rate Disconnect." Journal of Monetary Economics, 49, 2002, 913-40.

Edison, H. J., and W. R. Merick. "Alternative Approaches to Real Exchange Rates and Real Interest Rates: Three Up and Three Down." International Journal of Finance and Economics, 4, 1999, 93-111.

Elliott, G., T.J. Rothenberg, and J. H. Stock. "Efficient Tests for an Autoregressive Unit Root." Econometrica, 64, 1996, 813-36.

Engel, C., and K. D. West. "'Exchange Rates and Fundamentals." Journal of Political Economy, 113, 2005, 485-517.

Kwiatkowski, D., P. C. B. Phillips, P. Schmidt, and Y. Shin. "Testing the Null Hypothesis of Stationary against the Alternative of a Unit Root." Journal of Econometrics, 54, 1992, 159-78.

Maddala, G. S., and I.-M. Kim. Unit Roots, Cointegration, and Structural Change. Cambridge, UK: Cambridge University Press, 1998.

Mark, N. C. International Macroeconomics and Finance. Malden, MA: Blackwell Publishing, 2001.

Mark, N. C., and Y.-K. Moh. "The Real Exchange Rate and Real Interest Differentials: The Role of Nonlinearities." International Journal of Finance and Economics, 10, 2005, 323-35.

Meese, R. A., and K. Rogoff. "Was It Real? The Exchange Rate Interest Rate Relation." Journal of Finance, 43, 1988, 933-48.

Morley, J. C. "A State-Space Approach to Calculating the Beveridge Nelson Decomposition." Economics Letters, 75, 2002, 123-27.

Nakagawa, H. "Real Exchange Rates and Real Interest Differentials: Implication of Nonlinear Adjustment in Real Exchange Rate." Journal of Monetary Economics, 49, 2002, 629-49.

Newbold, P. "Precise and Efficient Computation of the Beveridge-Nelson Decomposition of Economic Time Series." Journal of Monetary Economics, 26, 1990, 453-57.

Ng, S., and P. Perron. "Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power." Econometrica, 69, 2001a, 1519-54.

--. "PPP May Not Hold After All: Further Investigation." Annals and Economics and Finance, 3, 2001b, 43-64.

Phillips, P. C. B., and P. Perron. "Testing for a Unit Root in Time Series Regression." Biometrika, 75, 1988, 335-46.

Samuelson, P. A. "Theoretical Notes on Trade Problems." Review of Economics and Statistics, 46, 1964, 145-54.

Schwert, G. W. "Tests for Unit Roots." Journal of Business and Economis Statistics, 7, 1989, 147-59.

Stock, J. H. "Unit Roots, Structural Breaks, and Trends," in Handbook of Econometrics, Vol. IV, Chapter 46, edited by R. Engle and D. McFadden. Amsterdam, The Netherlands: Elsevier, 1994, 2740-843.

TATSU MA WADA *

* I am grateful to two anonymous referees, Michael Day, Anthony Landry, Erwan Quintin, Robert Rossana, and seminar participants at Oakland University, the 14th International Conference on Computing in Economics and Finance, and the 2009 Midwest Econometrics Study Group Meeting for their useful comments. I thank Simon Gilchrist for the conversations in the very early stages of this paper.

Wada: Assistant Professor, Department of Economics, Wayne State University, Faculty/Administration Building, 656 W. Kirby St., Detroit, MI 48202. Phone 313-577-3001, Fax 313-577-9564, E-mail tatsuma.wada@wayne.edu

(1.) In the earlier stage of this study, neither Campbell and Clarida (1987) nor Meese and Rogoff (1988) found a statistically significant link. See Edison and Merick (1999) for different empirical approaches to this problem.

(2.) The business cycle frequency components of the data are extracted by the Baxter and King (1999) Band-pass filter.

(3.) Baxter (1994) also points this out. Another example is Asea and Reinhart (1996), who examine the link in African coutries' data by utilizing the BN decomposition. They, however, compute the BN decomposition without modeling the real exchange rate and real interest differentials, as opposed to our paper whose model is fully described below. In effect, Asea and Reinhart (1996) estimate more general ARIMA(q,1,p) models to find the BN cycle of the real exchange rate; whereas our model predicts (as we shall see later) ARIMA(0,1,p) models are the model-consistent time-series process for the real exchange rate.

(4.) The BN decomposition is computed exactly when the time-series process of qt, such as an ARIMA (p,1,q), is known (Morley 2002). Otherwise, approximate BN decomposition can be computed in several different ways (Newbold 1990).

(5.) If markets were complete, the real exchange rate would be

[S.sub.t][P.sup.*.sub.t]/[P.sub.t] = [[P.sup.*.sub.0] [C.sup.*[rho].sub.0] [S.sub.0]/[P.sub.0][C.sup.[rho].sub.0]] [C.sup.[rho].sub.t]/[C.sup.*[rho].sub.t].

(6.) Unlike Devereux and Engel (2002), the assumption that foreign exchange dealers are only in Home country is not a crucial assumption for our model.

(7.) [s.sub.t] = ln [S.sub.t]: Lower case letters are log deviations from the steady state.

(8.) Devereux and Engel (2002) also assume noise traders, but their assumption is that foreign exchange dealers take into account the effect of noise traders when foreign exchange dealers maximize their objective function.

(9.) We assume that the degrees of misperception by Home and Foreign firms are the same.

(10.) This innovation is a linear combination of monetary shocks in Home and Foreign (the natural logs of [[epsilon].sub.t] and [[epsilon].sup.*.sub.t]). By assumption (in Equation 14), two shocks are normally distributed: mutually and serially uncorrelated. Therefore, we have

[[eta].sub.t] ~ i.i.d.Normal (0, [[sigma].sup.2.sub.[eta]])

(11.) Note that the ex ante purchasing power parity (PPP) holds in this case.

(12.) Owing to the preset prices, there is no distinction between ex ante and ex post real interest rates in our model.

(13.) From Equations (23) and (25), one may think that we can test the link between those two variables by ordinary least squares (OLS). However, as the Appendix S1-S4 (Supporting Information) explains, OLS is an inappropriate method to measure the link.

(14.) Real interests are constructed as follows. First we compute the (ex post quarterly) inflation rate:

[[pi].sub.t] = ([p.sub.t] - [p.sub.t-1])

where [p.sub.t] is the (logarithm of) CPI (quarterly) at t. Then, the real interest rate for Home is computed as

[r.sub.t] = [i.sub.t] - [[pi].sub.t+1];

where [i.sub.t] is the nominal interest rate for one quarter (i.e., the annual nominal interest rate is divided by 4).

(15.) Size distortions are more serious when errors follow a negative MA process. See Maddala and Kim (1998) and Stock (1994), for example.

(16.) Another justification for using the KPSS test for real interest differentials is that our null hypothesis should be an MA process without a unit root, as our model predicts.

(17.) An ARIMA(0,1,p) is estimated by maximum likelihood estimation method. The likelihood function is computed exactly by the Kalman filter.

(18.) See Appendix S1-S4 for a detailed computation.

(19.) Equation (25) is estimated by an MA([p.sup.*]) model:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

See Appendix $2.

(20.) Both series are multiplied by 100 to represent percentage deviations from the BN trend (real exchange rates) and percentage deviation from the mean (real interest differentials).

(21.) Let two series be [X.sub.1t] = [[summation].sup.[infinity].sub.j=-[infinity]] [a.sub.j][z.sub.1j] and [X.sub.2t] = [[summation].sup.[infinity].sub.j=-[infinity]] [b.sub.j][z.sub.2j] where {[z.sub.1j]} and {[z.sub.2j]} are independent. Then, the asymptotic distribution of the estimated cross-autocorrelation function is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Normal (0, [[summation].sup.[infinity].sub.h=-[infinity]] [[rho].sub.11] (h) [[rho].sub.22] (h)) where [[rho].sub.lm] (h) = E [([X.sub.lt] - E [[X.sub.lt]]) ([X.sub.mt-h] - E [[.sub.Xmt-h]])], l, m = 1, 2.

(22.) Brockwell and Davis (1991) recommend the use of the fitted residual of [X.sub.1t] and [X.sub.2t] in calculating [[rho].sub.11] (j) and [[rho].sub.22] (j).

(23.) Although not reported here, rolling 10 years moving-average estimates of [??] (0) for the US-CA and US-JP pairs are not promising at all for any 10-year windows.
TABLE 1
Unit Root Tests: Real Exchange Rates

 MZa ADF MPT KPSS

US-CA -3.45 -1.28 7.10 0.548 *
US-JP -2.95 -1.05 8.09 0.598 *
US-SW -3.69 -1.17 6.70 0.313
US-UK -7.77 -1.88 3.39 0.572 *
5% critical value -8.10 -1.98 3.17 0.463
1% critical value -13.8 -2.58 1.78 0.739

Notes: The lag length is chosen by the modified Akaike
information criteria (MAIC) with [k.sub.max] = 12[(T/100).sup.1/4]
where T is the sample size. GLS detrending uses [bar.[alpha]] =
1 - [7/T] as an autoregressive parameter.

* One can reject the null hypothesis at a 5% level. For
more details, see Appendix S1-S4.

TABLE 2
Unit Root Tests: Real Interest Differentials

 MZa ADF MPT KPSS

US-CA (Short) -1.64 -1.06 12.50 0.175
 (Long) -9.18 * -2.65 ** 2.96 * 0.187
US-JP (Short) 0.37 -0.57 46.23 0.092
 (Long) 0.14 -0.35 40.64 0.101
US-UK (Short) -4.05 -2.64 ** 6.05 0.450
 (Long) -11.45 * -3.64 ** 2.16 * 0.263
5% critical value -8.10 -1.98 3.17 0.463
1% critical value -13.8 -2.58 1.78 0.739

Notes: The lag length is chosen by the modified Akaike
information criteria (MAIC) with [k.sub.max] = 12[(T/100).sup.1/4]
where T is the sample size. GLS detrending uses [bar.[alpha]] = 1 -
7/T as an autoregressive parameter. "Long" and "Short" are
long-term and short-term interest rate differentials, respectively.
For more details, see Appendix S1-S4.

*, ** One can reject the null hypothesis at the 5% and
1% levels, respectively.

TABLE 3
MA Estimation of Real Exchange Rates

 [a.sub.1] [a.sub.2] [a.sub.3] [a.sub.4] [[a.sub.5]

US-CA 0.446 **
 (0.085)
US-JP 0.310 ** -0.023 0.174 0.236 **
 (0.082) (0.087) (0.101) (0.093)
US-JP 0.305 **
 (0.089)
US-UK 0.262 ** 0.012 -0.067 0.089 -0.252
 (0.080) (0.090) (0.080) (0.098) (0.l09)
US-UK 0.242 **
 (0.08l)

 [sigma] [mu] LL Criterion

US-CA 0.024 ** -0.001 337.180 AIC, BIC
 (0.00l) (0.003)
US-JP 0.049 ** 0.004 232.927 AIC
 (0.003) (0.007)
US-JP 0.050 ** 0.003 228.919 BIC
 (0.003) (0.005)
US-UK 0.047 ** 0.002 238.688 AIC
 (0.003) (0.004)
US-UK 0.049 ** 0.002 233.537 BIC
 (0.003) (0.005)

Notes: LL stands for the value of the log likelihood. a is the standard
deviation of the error term. Standard errors are given in parentheses.

** The parameter is significant at a 1% level.

TABLE 4
MA Estimation of Real Interest Rate Differentials

 [[theta].sub.1] [[theta].sub.2] [[theta].sub.3]

US-CA(S) 0.427 **
 (0.067)
US-CA(S) 0.432 ** 0.128 0.173 *
 (0.082) (0.090) (0.088)
US-CA(L) 0.368 **
 (0.076)
US-CA(L) 0.365 ** 0.038 0.137
 (0.084) (0.098) (0.089)
US-JP(S) 0.027 0.270 ** 0.098
 (0.095) (0.062) (0.l02)
US-JP(S) 0.096 0.312 ** 0.200 *
 (0.l48) (0.086) (0.080)
US-JP(S) 0.l68 * 0.295 ** 0.254 **
 (0.082) (0.089) (0.094)
US-JP(L) 0.001 0.261 ** 0.145
 (0.l39) (0.063) (0.138)
US-JP(L) 0.108 0.292 ** 0.221 **
 (0.121) (0.084) (0.080)
US-JP(L) 0.146 0.269 ** 0.254 **
 (0.08l) (0.086) (0.086)
US-UK(S) 0.263 ** 0.153 -0.009
 (0.084) (0.092) (0.083)
US-UK(S) 0.l97 * 0.273 ** 0.085
 (0.083) (0.082) (0.084)
US-UK(L) 0.154 0.045 -0.073
 (0.082) (0.088) (0.077)
US-UK(L) 0.111 0.127 -0.085
 (0.081) (0.082) (0.085)

 [[theta].sub.4] [[theta].sub.5] [[theta].sub.6]

US-CA(S)

US-CA(S) 0.195 **
 (0.072)
US-CA(L)

US-CA(L) 0.l76 *
 (0.074)
US-JP(S)

US-JP(S) 0.335 **
 (0.085)
US-JP(S) 0.247 * 0.038 0.157
 (0.l13) (0.l16) (0.100)
US-JP(L)

US-JP(L) 0.387 **
 (0.084)
US-JP(L) 0.297 ** 0.060 0.l94 *
 (0.l06) (0.106) (0.09l)
US-UK(S) 0.350 **
 (0.069)
US-UK(S) 0.516 ** 0.212 * 0.019
 (0.085) (0.097) (0.095)
US-UK(L) 0.372 **
 (0.070)
US-UK(L) 0.392 -0.232 ** 0.044
 (0.087) (0.090) (0.093)

 [[theta].sub.7] [[theta].sub.8] [[theta].sub.9]

US-CA(S)

US-CA(S)

US-CA(L)

US-CA(L)

US-JP(S)

US-JP(S)

US-JP(S) -0.206 * 0.265 **
 (0.095) (0.099)
US-JP(L)

US-JP(L)

US-JP(L) -0.086 0.368 **
 (0.114) (0.102)
US-UK(S)

US-UK(S) 0.135 0.394 ** 0.037
 (0.086) (0.077) (0.082)
US-UK(L)

US-UK(L) 0.047 0.165 -0.156
 (0.083) (0.095) (0.092)

 [[theta].sub.10] [sigma] [mu] LL

US-CA(S) 0.006 ** -0.003 ** 545.496
 (0.000) (0.001)
US-CA(S) 0.006 ** -0.003 ** 550.248
 (0.000) (0.00l)
US-CA(L) 0.005 ** -0.002 * 553.650
 (0.000) (0.00l)
US-CA(L) 0.005 ** -0.002 * 556.732
 (0.000) (0.001)
US-JP(S) 0.009 ** 0.000 480.391
 (0.00l) (0.00l)
US-JP(S) 0.009 ** 0.000 487.963
 (0.000) (0.001)
US-JP(S) 0.008 ** 0.000 493.509
 (0.000) (0.00l)
US-JP(L) 0.009 ** 0.002 * 481.749
 (0.001) (0.001)
US-JP(L) 0.008 ** 0.002 492.164
 (0.001) (0.001)
US-JP(L) 0.008 ** 0.002 499.385
 (0.000) (0.002)
US-UK(S) 0.010 ** -0.003 ** 459.780
 (0.00l) (0.00l)
US-UK(S) -0.217 * 0.010 ** -0.004 469.908
 (0.090) (0.00l) (0.085)
US-UK(L) 0.010 ** 0.000 470.541
 (0.00l) (0.00l)
US-UK(L) -0.212 * 0.009 ** 0.000 478.776
 (0.085) (0.00l) (0.00l)

 Criterion

US-CA(S) BIC

US-CA(S) AIC

US-CA(L) BIC

US-CA(L) BIC

US-JP(S) p - 1

US-JP(S) BIC

US-JP(S) AIC

US-JP(L) p - 1

US-JP(L) BIC

US-JP(L) AIC

US-UK(S) BIC, p - 1

US-UK(S) AIC

US-UK(L) BIC, p - 1

US-UK(L) AIC

Notes: LL stands for the value of the log likelihood; [sigma] is the
standard deviation of the error term. Standard errors are given in
parentheses. Entries below Criterion are criteria used to choose the
length of lags. p - 1 means that the length of lags is p - 1, where p is
the length of lags used to compute the BN decomposition of the real
exchange rate. (p for the real exchange rate is selected by the AIC.)

* The parameter is significant at a 5% level; ** the parameter is
significant at a 1% level.

TABLE 5
Summary of the Selected Length of Lags

Real Exchange Rates Interest Differentials
Real

 AIC BIC AIC BIC p - 1

US-CA 1 1 (Short) 4 1 0
 (Long) 4 1 0
US-JP 4 1 (Short) 8 4 3,0
 (Long) 8 4 3,0
US-UK 5 1 (Short) 10 4 4,0
 (Long) 10 4 4,0

Notes: p - 1 means that the length of lags is p - 1,
where p is the length of lags used to compute the BN
decomposition of the real exchange rate. (p for the real
exchange rate is selected by the AIC.)

TABLE 6
Standard Deviations

 q.sup.cycle.
 [DELTA]q [r.sub.short] [r.sub.long] sub.AIC]

US-CA 2.56 0.64 0.59 1.07
US-JP 5.25 0.93 0.93 4.42
US-UK 5.03 1.17 1.07 2.03

 [q.sup.cycle. [D.sup.short. [D.sup.short. [D.sub.p - 1],
 sub.BIC] sub.AIC] sub.BIC] [AIC.sup.short]

US-CA 1.07 1.26 0.87 0.64
US-JP 1.55 2.21 1.95 1.35
US-UK 1.09 2.72 2.12 2.12

 [D.sub.p - 1], [D.sup.long. [D.sup.long. [D.sub.p - 1],
 [BIC.sup.short] sub.AIC] sub.BIC] [AIC.sup.long]

US-CA 0.64 1.04 0.78 0.59
US-JP 0.93 2.63 2.01 1.39
US-UK 1.17 1.25 1.83 1.83

 [D.sub.p - 1],
 [BIC.sup.long]

US-CA 0.59
US-JP 0.93
US-UK 1.07

Notes: [DELTA]q is the first difference of the real exchange rate;
[r.sub.short] and [r.sub.long] are short-term and long-term real
interest differentials.

AIC, BIC, p - 1 indicate the criterion used to choose the length of
lags. For example, [D.sup.short.sub.AIC], p - 1 means the expected
future sum of short-term real interest differentials, computed from an
MA model with the lags of p - 1, where p is the length of lags used to
compute the BN decomposition of the real exchange rate; and p is
selected by AIC.

Standard deviations are multiplied by 100, so that they indicate
percentage deviations.
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