The UIP: An Unbiased and Efficient Estimator Abstract

The UIP: An Unbiased and Efficient Estimator

W. A. Razzak1

Abstract This paper examines the uncovered interest rate parity condition at long and short run horizons (low and high frequency). An asymptotically unbiased and an efficient nonlinear dynamic least square estimator is used to estimate the long-run coefficients. Conditional on cointegration, short-run relationships over the cycles are then examined. There is evidence in favor of UIP. [JEL # F31, F41, C13]

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The author is an advisor at the Economics Department of the Reserve Bank of New Zealand. Contact address is Reserve Bank of New Zealand, P. O. Box 2498, Wellington, New Zealand, e-mail [email protected]. Views expressed in this paper do not necessarily reflect those of the Reserve Bank. This work has benefited greatly from discussions and close work with Francisco Nadal De Simone. I am grateful to Peter Phillips, Nelson Mark, Robert Flood, Andrew Rose, Yin Wong Cheung, Menzie Chinn, John McDermott and participants of New Zealand Economic Meeting in Christchurch 1997 and the Far Eastern Meeting of the Econometric Society 1999 for their generous comments.

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Introduction Obstfeld and Rogoff (1996, p. 623) show a close graphical relationship between the trade-weighted dollar exchange rate and the real interest rate differential – uncovered interest rate parity, UIP – and ask why doesn’t the visual impression come through in empirical analysis? This voluminous literature’s conclusion is generally not in favor of the UIP condition.2 In their answer, Obstfeld and Rogoff (1996) appeal to Baxter’s (1994) argument that in order to find a link between the exchange rate and the interest rate differentials, one should filter the data properly. At very high frequencies (i.e., cycles 2-5quarters) one should expect to find nothing but noise related to the irregular component of the time series. Using the Baxter-King Band-Pass filter, she finds a correlation between the exchange rates and interest rate differentials at both trend and business cycle. There are two other separate papers that deal with long and short run movements in the exchange rate and interest rate differentials. Nadal De Simone and Razzak (1997) used the Phillips and Loretan (1991) nonlinear dynamic least squares and reported a very significant cointegrating relationship (trend) between nominal trade-weighted dollar exchange rate and the long-term nominal interest rate differential. Recently, Meredith and Chinn (1998) used regressions in differences and showed evidence in favor of UIP (short term) when long-term interest rates are used instead of short-term rates. These two papers indicate that there might be some evidence in favor of UIP at both the trend and the cycle. The objective of this paper is to examine nominal UIP condition at both the trend and the cycle. I use an asymptotically efficient and unbiased estimator (Phillips and Loretan, 1991) to estimate the coefficients that govern the longrun relationship between bilateral exchange rates and long-term nominal interest rates such as the 10-year government bond yield. At the cycle, I

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Examples are Frankel (1979), McNown and Wallace (1989), Baillie and Pecchenino (1991), Johansen and Juselius (1992), Edison and Pauls (1993), McNown and Wallace (1994), McCallum (1994), Kim and Mo (1995), Bartolini and Bodnar (1995), Meredith and Chinn (1998) and Edison and Melick (1999). The theoretical debate about how interest rates affect the exchange rates is found in Dornbusch (1976), Frankel (1979), Christiano et al (1998), Mussa (1976, 1979), Frenkel (1976) and Bilson (1978, 1979). For instance, see Messe and Rogoff (1983) and Messe and Rogoff (1988) and MacDonald and Nagayasu (1999) research on real exchange rates and real interest rate differentials.

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examine the relationship by assessing common serial correlation after removing common trends (Vahid and Engle, 1993). I find evidence of correlation between the exchange rate and long-term interest rate differentials at both the trend and the cycle. Section two consists of six sub-sections. Sub-section 2.1 defines the UIP condition; 2.2 describes the data; 2.3 discusses the Phillips-Loretan method; 2.4 examines the order of integration of the time series and tests for cointegration; 2.5 estimates the long-run coefficients using the Phillips-Loretan; and 2.6 examines the cyclical comovements using the Vahid-Engle (1993) method. Section three is a summary. 2. Empirical Evidence 2.1. The Definition and Empirical Form of UIP I briefly discuss the uncovered interest rate parity condition. UIP is expressed by: et = Et et + s + b(it ,t + s − it*,t + s ) ,

(1)

where et is the natural logarithm of the spot exchange rate, Et is the mathematical expectations operator conditional on the information set at time t , it is time t U.S. annualized nominal interest rate or yield to maturity on a pure discount bond that matures at time s . Similarly it* is the foreign magnitude. The size of the parameter b depends on the units of the interest rates. For example, if the interest rates are expressed as decimals of the 10year government bond yields, then b is 10 (e.g.,

s 40 ); b could also be 1 or = 4 4

0.10. When the error term in (1) is white noise, the interest rate differential is the only explanatory variables because it reflects all available information at time t . There are different ways to empirically estimate UIP and none of them escapes criticisms (see Edison, Melick, 1999).3 One standard empirical version that is typically tested in the literature is: 3

The expected exchange rate poses a problem for estimation because it unobservable. First differencing the spot rate is not the right way to deal with expectations. Making the expected exchange rate a function of a set of macroeconomic fundamentals is perhaps a reasonable way of dealing with problem.

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et = a + b(it ,t + s − it*,t + s ) + ω t

(2)

In this regression, the expected exchange rate Et et + s is assumed to be a constant a , which is a very big assumption. The error term in (2) is assumed to be Niid and b is the slope coefficient. Least squares’ estimation of this equation is problematic because the exchange rate and probably interest rates have unit roots. Economists use differenced regressions to avoid spurious regressions. If the interest rates are 10-year government bond yields, the exchange rates have to be the 10-year changes. To avoid all problems associated with unit roots regressions and differencing, essentially, a more elaborate dynamic regression in levels of equation will be estimated in this paper using the Phillips-Loretan (1991) non-linear dynamic least squares method. The estimated coefficients aˆ and bˆ represent the cointegration coefficients. When ω t is white noise, and the parameter bˆ is not different from

s , the constant term aˆ is interpreted as the average long-run 4

exchange rate. 2.2 The Data Monthly data covering the period January 1980 to July 1997 from the International Monetary Fund’s International Financial Statistics (IFS) are used in this paper. The exchange rate is defined as the foreign currency price of the US dollar. The exchange rates are the DM-USD, GBP-USD, YENUSD, and the CAD-USD at the end of each month. The U.S. dollar-French Franc and the U.S. dollar-Italian Lire are excluded from the sample because France and Italy maintained some significant capital control measures until late 1980s which I think they may affect the tests of the UIP. The reason I start the sample in 1980 is to avoid periods of capital controls. Britain abolished exchange controls in 1979 (Artis and Taylor, 1989). Japan abolished its exchange control policies in 1980 (Fukao, 1990). The long-term interest rate is the 10-year government bond yield rate (IFS, line 61, where e.g., six percent is written 6.0 instead of 0.06). Figures 1 to 4 plot the exchange rates on the LHS axis and the interest rate differentials on the RHS axis. The interest rate differential is defined as the 10-year government bond yield of the United States minus that of the foreign

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country (i.e., Germany, Britain, Japan, and Canada respectively). Visually, there seems to be a strong relationship between the DM-USD and the GBPUSD exchange rates and the corresponding interest rate differentials. A weaker relationship appears between the CAD-USD exchange rates and the corresponding interest rate differentials. The YEN-USD interest rate differential relationship appears the weakest among all four pairs. If Obstfeld and Rogoff (1996) are right we should not uncover significant long and short-run relationships between the DM-USD and the GBP-USD and the corresponding interest rate differentials. Before estimating the UIP condition, I briefly discuss the theory of the Phillips-Loretan (1991) non-linear dynamic least squares (lags and leads regression). 2.3 The Phillips-Loretan (1991) method I briefly explain the methodology used in this paper to estimate the coefficients that govern the long-run relationship between the exchange rate and the interest rate differentials. This method is based on the Triangular Representation Theorem. Consider the following triangular system of equations: y1t = α + β ′y 2t + u1t ∆y 2t = u 2t ,

(3) (4)

where u t = [u1t u ′2t ]′ is a stationary vector. Phillips and Loretan (1991) consider estimation and inference in the above system. Equation (3) can be estimated using single equation methods, and provided that the equation is appropriately augmented the asymptotic properties of the estimator can be readily determined. The asymptotic (distributional) properties hinge on the interrelationships that exist between u1t and u 2t , which are assumed to be stationary. If the variance-covariance matrix of u t is block diagonal (so that the partitioned elements of the vector do not co-vary) and u t is also (i.i.d.) then equation (3) can be estimated using least squares. The estimates will be normally distributed, at least asymptotically, and will be equivalent to

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maximum likelihood estimates of the parameters of the system. If the variance covariance matrix of u t is block diagonal but the residuals are autocorrelated (but also block diagonal) the regression can be augmented with lagged error correction terms, counteracting the autocorrelation of u1t . This solution works because the regressors – y 2t – are super-exogenous. They are independent of the past and future history of u1t , i.e. they are orthogonal to {u1t }t+∞=−∞ . If u1t and u 2t are correlated one can alleviate the problems that this causes by augmenting the regression with leads and lags of ∆y 2t . One can think of this as projecting u1t against leads and lags of u 2t and then applying the Frisch – Waugh - Lovell theorem (see Davidson and MacKinnon, 1993). The error from the projection is uncorrelated with u 2t , and the simultaneity problem is thus dealt with. The inclusion of leads eliminates feedback from u1t back to u 2t and it is important for valid conditioning. The basic problem is that, a priori, we do not know whether u1t correlates with leads of u 2t or vice versa. The inclusion of lag differences eliminates the simultaneity problem, while the autocorrelation problem may need to be remedied by including lags of the error correction term too. Phillips and Loretan then estimate the following equation using least squares: K

y1t = α + β ′y 2t + ∑ d1k ( y1t −k − β ′y 2t −k ) + k =1

p

∑d

i =− p

2i

∆y 2t −i + ν 1t

(5)

Phillips and Loretan demonstrate that the parameter estimates from estimating this single equation are equivalent to maximum likelihood estimation of the system and hence are efficient. Furthermore, the parameters are asymptotically normally distributed. The distributions of the parameters of the cointegrating vector – and corresponding statistics – depend on the null hypotheses concerning the interrelationships. If the series are integrated but not cointegrated the regression is spurious, and the asymptotic theory is found in Phillips (1986). Barnhart, McNown and Wallace (1999) used this technique to test the forward rate unbiasedness hypothesis and demonstrated that the estimator is most

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unbiased among many different methods used in testing. The assumptions of unit roots and cointegration need to be tested first, which is the next step. 2.4. Unit Roots and Cointegration Before estimation, tests for unit roots and cointegration must be carried out. There seems to be an agreement among economists that exchange rates, during the post Bretton-Woods era, may contain stochastic trends. The literature on unit roots and cointegration is vast and it will not be reviewed here. Suffice to say that there is a valid concern among economists about the appropriateness of the tests for unit roots and their power against stationary alternatives. The choice of a particular testing methodology is not straightforward. Ultimately, one may not be able to determine whether a particular time series contains a unit root or not. It seems inevitable, however, that one must make a choice. Because there is no consensus on the issues of unit root in long-term interest rate differentials and cointegration between the exchange rate and the interest rate differentials, it is perhaps useful to subject the data to various different tests. This strategy reduces the risk of being on the wrong side. A decision can then be based on whether the results of various tests converge or not. For example, when different tests for unit roots move in one direction, e.g., indicating a unit root, one can be a little more confident in the results than when the tests diverge.4 In this paper, four methods to test for unit roots are used. All of them have the same null hypothesis of a unit root and are available in most statistical packages. These are the DF test (Dickey and Fuller, 1979, 1981), the ADF test (Said and Dickey, 1984), the Z test (Phillips, 1987 and Perron, 1988), the DF-GLSJ test (Elliott, Rothenberg and Stock, 1996), and Perron (1997) test. Results for unit root tests are reported in table 1a and 1b.5 The tests that are considered in this paper seem to agree with conventional wisdom that the exchange rates have unit roots. 4

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For the comparison to be meaningful, it is important that the tests have the same null hypothesis.

The choice of the lag structure always has been an issue. The objective of the lags is to remove serial correlation. With this objective in mind, I look at different criteria to choose the lags. For example, the lag order is set as the highest significant lag order— using an approximate 95 percent confidence interval—from either the autocorrelation function or the partial autocorrelation function of the first-differenced series. The maximum lag order is the square root of the sample size. I also test backward using F tests, and look at

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However, there are two exceptions regarding the interest rate differentials. The ADF and the Z tests disagree twice regarding the interest rates differentials. The ADF test rejected the null hypothesis that the YEN-USD 10-year interest rate differential is I (1), while the Z test finds a unit root in it. Then the ADF test could not reject the null hypothesis of a unit root in the CAD-USD 10-year interest rate differential, while the Z test rejected it. The DF-GLS – a point-optimal invariant test, which has improved power when an unknown mean or trend is present in the data – indicates that the null of a unit root with a constant and a linear trend cannot be rejected for any variable. Finally, all the time series are tested using Perron test (1997).6 Table 1b shows the results for two models: the innovational outlier model (model 1) allows only a change in the intercept under both the null and the alternative hypothesis, and the additive outlier model (model 2) allows a change in both the intercept and the slope. According to the results of table 1b, the null hypothesis of unit root could not be rejected for the data except for the YENUSD 10-year interest rate differential. Therefore, one can conclude that the Perron test indicates that the YEN-USD 10-year interest rate differential does not have a unit root when allowance is made for a change in the intercept and the slope. The hypothesis of the unit in the YEN-USD interest rate differentials is rejected by two tests out of four. It is therefore assumed to be I (0). The DM-USD, GBP-USD and the CAD-USD interest rate differentials on the other hand seem to diverge for prolonged periods of time; they are likely to have stochastic trends. Schwarz IC and AIC. Unnecessary lags are eliminated. Every time, serial correlation is checked using the Bartlett - Kolmogrov - Smirnov test for white noise. 6

The test allows for a shift in the intercept of the trend function and a shift in the slope when the date of the possible change is not fixed a priori but is determined endogenously. Two methods are used to determine the break point (Tb). First, we select the breaking point that minimizes the t-statistic for testing the null of unit root ( tαˆ ). Second, we select the breaking point that minimizes the t-statistic on the parameter associated with the change in the intercept (model 1) ( tθˆ ), or the change in slope (model 2) ( t γˆ ). The lag parameter is chosen

following a general to specific recursive procedure so that the coefficient on the last lag in an autoregression of order k is significant and that the last coefficient in an autoregression of order k greater that is insignificant, up to a maximum order k max.

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To test the null hypothesis that the exchange rates are not cointegrated with the interest rate differentials, the same strategy used to test for unit root is followed; many different testing methods are used. And a search for a consensus is pursued. Note that if the interest rate differentials are I (0) then testing for “no cointegration” with the exchange rates is meaningless. For this reason, I will drop the YEN-USD from the cointegration analysis that follows. First, I use the Engle-Granger (1987) and the Engle and Yoo (1987) procedures. The Engle-Granger method is suitable for bi-variate systems. The null hypothesis that the residuals from the OLS regressions of the nominal exchange rate on a constant, or on a constant and a trend, and the interest rate differential is tested for unit root. Typically, the ADF test is used as recommended by Engle and Granger’s original paper. Second the PhillipsPerron-Phillips-Ouliaris test to test the same hypothesis is used.7 Given the sample, both tests seem to indicate that there is no statistical evidence of cointegration in all pairs of data.8 Third, I use the Johansen and Juselius (1990) maximum likelihood method. Although this method is more appropriate for multivariate cases, I use it to test for no-cointegration between nominal exchange rates and interest rate differentials because many papers report this test. Results are reported in table 2.9 The sample is adequate (in terms of the number of observations and the span), but I also report the corrected the critical values for small sample using the Cheung and Lai (1993) approach.10 Based on the λ max and the trace statistics 7

The same approach to selecting the lag structure in the tests for unit roots in individual series is used here.

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I am not aware of any evidence of cointegration between the nominal exchange rate and interest rate differentials in the literature based on these tests. Results are not reported, but they are available upon request. The result is consistent with Meese and Rogoff (1988), Edison and Pauls (1993), and Kawai and Ohara (1997) for the real exchange rates. 9

Gonzalo (1994) compares five different residual-based tests for cointegration including the Engle-Granger test. Among them, he recommends using the Johansen-Juselius (1990) method. This test has been very popular in the literature, but just like any other unit root test, it is highly criticized for its lack of power in finite samples, and—among other problems—its sensitivity to the choice of the lag length. I start with a lag structure similar to that we adopted in the Engle-Granger test. The choice is based on the multivariate serial correlation tests recommended by Johansen and reported in CATS.

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The lag structure is determined using the same methodology explained earlier. The residuals of the models are carefully checked for whiteness each time using multivariate Ljung-Box, LM1 and LM4 tests.

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at the 95% level, the test indicates that the nominal exchange rate and the nominal interest rate differentials are cointegrated. There is a single cointegrating vector in each of the three pairs. The estimated long-run coefficients after imposing a rank of one and normalizing on the exchange rate are 0.18, 0.16 and 0.20 for the DM-USD the GBP-USD and the CADUSD pairs respectively. Having found a cointegrating relationship between each of the pairs of exchange rates and interest rate differentials, we move to using the PhillipsLoretan approach as a final check on the long-run comovement that we are interested in. 2.5 Estimation of the Long-run Coefficients As I explained earlier, this single-equation estimation technique of Phillips and Loretan is asymptotically equivalent to a maximum likelihood on a full system of equations under Gaussian conditions. This technique provides estimators that are statistically efficient, and whose t-ratios can be used for inference in the usual way. Most importantly, this method takes into account both the serial correlation of the errors and the endogeneity of the regressors that are present when there is a cointegration relationship. Recently, Barnhart, McNown and Wallace (1999, p 288-289) provided evidence that this procedure produces no bias in the estimates of the parameters in forward premium equation. I estimate two regressions, the restricted and the unrestricted given by equation (6) and (7) respectively: 11 et = a + b(i − i * ) t +

k

∑ δ ∆(i − i

i=− k k

et = a ′ + b1it + b2 it* +

i

*

) t −i + ρ[et −1 − a − b(i − i * ) t −1 ] + vt , k*

∑ ϕ i ∆it + ∑θ i ∆it* + ρ[et −1 − a ′ − b1it −1 − b2 it*−1 ] + η t ,

i=−k

(6) (7)

i =− k

I am mostly interested in the magnitudes a$ , b$ in the restricted regression and a ′ , bˆ1 and bˆ2 in the unrestricted regression, because they are the parameter 11

The Phillips Information Criteria (Phillips, 1996) is used to determine the optimal lag structure along with the cointegration vector. I find a three lag-lead structure to be sufficient to eliminate the serial correlation. However, because I am conscious about Phillips and Loretan warning of over-fitting, I start reducing the number of leads by one. Every time, I check the serial correlation, the parameter estimates, and their significance. I find that a structure of three lags and two leads gives the same results as a three lag-lead structure without compromising the whiteness of the residuals. A further reduction in the leads introduces some serial correlation.

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estimates that determine the long-run relationship between the nominal exchange rate and the nominal 10-year bond yield differential conditional on the dynamics. Unlike the regressions in differences reported in Meredith and Chinn (1998), the Phillips-Loretan is in levels and it is conditional on a more elaborate dynamic. Results of the nonlinear least square estimation for both pairs of currencies are reported in table 3. The parameters of the restricted regression are significant. I report the t-ratios and the p-values. These results are consistent with the Johansen-Juselius test results both in magnitudes and in quality. The various diagnostics of the residuals indicate that the residuals are white noise, serially uncorrelated, and homoskedastic. The magnitudes of b$ are almost identical for the DM-USD, the GBP-USD and the CAD-USD. These are 0.12, 0.10 and 0.10 respectively. In the long run, an increase in the nominal interest rate differential appreciates the home currency. When there is a significant slope relationship between the exchange rate and the interest rate differentials like the ones we obtained, the estimates of the constant terms a$ can be interpreted as average long-run values of the nominal exchange rate. For example, for the DM-USD the value is 0.34 and the antilog is 1.40. So the equilibrium DM-USD exchange rate over the sample is 1.40. It is 0.69 for the GBP-USD and 1.52 for the CAD-USD. The sample means are 1.95, 0.61 and 1.27 respectively. These numbers indicate that on average and during the period 1980:1-1997:7, the US dollar exceeded its long-run equilibrium value against the DM, but undershot its equilibrium rates against the Pound and the Canadian Dollar. The unrestricted regression produces similar results. However, the restrictions (7 restrictions on the levels and the dynamics) only hold in the DM-USD case as indicated by the F test. Also, I report a chi-square test of the restriction that the long-run coefficient on the United States interest rate is equal in magnitude (with opposite sign) to the foreign interest rate coefficient. We cannot reject the hypothesis that the long-run coefficients are equal in magnitude in the case of DM-USD. Also, I provide a graphical presentation of the long-run relationships. For each pair, the long run is computed from the RHS of equation (6), f t = aˆ + bˆ(i − i * ) t . Then I plot the exchange rates and f t . The plots are shown in

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figures 5, 6 and 7. The correlation between the exchange rates and f t are visually clear in the DM-USD and the GBP-USD cases. The correlation values between the DM-USD exchange rates and the corresponding f t is 0.80 (equivalent R2 is 0.64), but smaller in the cases of the GBP-USD and the CAD-USD. Alternatively, one can also plot et − aˆ against bˆ(it − it* ) without any loss in information. 2.6 Examining Cyclical Comovements Finally, I use the Vahid-Engle (1993) method to test for common cycles for all pairs of currencies that are cointegrated with the interest rate differentials. The test is conditional on the presence of cointegration. This method exploits the serial correlation common between the first-difference of the two cointegrated variables to test for common-cycles. In essence it searches for comovements among the stationary components of the time series just like cointegration searches for comovements among the non-stationary or unit root components of the time series. Precisely, this method investigates common serial correlation among the first differences of a set of cointegrated variables. Thus, it implies that the remainders after removing their trends from their levels are common cycles. Results are reported in table 4. In the DM-USD, we cannot reject the null hypothesis that there is at least one common cycle. The null hypothesis of a common cycle in the pair GBP-USD is rejected although there is a common trend between that exchange rate and the GBP-USD interest rate differential. Also, there seems to be a significant relationship between the exchange rate CAD-USD and the interest rate differentials at the cycle. I conclude that the exchange rates contain unit roots. The interest rate differentials between the United States and Germany and Britain contain unit roots. The interest rate differentials between the United States and Canada may contain unit roots, but it is unclear whether the interest rate differentials between the United States and Japan is a unit root process. Consequently, I find the DM-USD, the GBP-USD and the CAD-USD exchange rates and the corresponding interest rate differentials to be cointegrated. The evidence of cointegration between the CAD-USD exchange rate and interest rate differentials is weaker than the other two. Further, the UIP holds very well in the cases of the DM-USD in the sense that the restrictions that the interest rate differentials in the levels, first difference, the lags and the leads have

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coefficients equal in size and opposite in sings could not be rejected. Also, there is evidence of common cycles between the DM-USD, and CAD-USD and the corresponding interest rate differentials. 3. Summary Different tests to test the null hypothesis that the nominal exchange rates and the long-term nominal interest rate differentials are unit root processes are used. Data are monthly for the DM-USD, the GBP-USD, the YEN-USD, and the CAD-USD from January 1980 to July 1997. Previous findings that the null hypothesis of a unit root in exchange rates cannot be rejected are confirmed. However, it is still unclear whether all of the long-term interest rate differentials are unit root processes. The 10-year bond rates differentials between the U.S. and Germany and the United States, the U.S. and Britain and between the U.S. and Canada contain a unit root, but it is more significant in the first two cases than in the case of the United States and Canada. It is unclear whether the interest rate differentials between the United States and Japan contain unit roots. It is highly probable that the relationship is an I(0). Consequently, I found that the DM-USD and the GBP-USD exchange rates and the corresponding long-term interest rate differentials are cointegrated. There is weaker evidence that the CAD-USD exchange rate is cointegrated with the corresponding interest rate differentials. Exactly like testing for unit roots, the finding of cointegration is sensitive to the power of the test used. The EngleGranger and Phillips-Perron-Phillips-Ouliaris type tests cannot reject the null hypothesis that the exchange rates and the interest rate differentials are not cointegrated for all pairs of countries. However, cointegration can be found using the Johansen-Juselius (1990) and confirmed by the Phillips-Loretan (1991) methods. Using long-term interest rate differentials suggests that there is strong statistical evidence in favor of UIP in the pairs DM-USD and GBP-USD, also between the CAD-USD and the long-run interest rate differentials, albeit weaker. Further. On the basis that nominal exchange rates and long-term nominal interest rate differentials are cointegrated, I investigate the presence of common cycles. Using the Vahid-Engle (1993) method, statistical evidence of common cycles between nominal exchange rates and interest rate differentials in the cases of DM-USD and the CAD-USD, but not in the case of GBP-USD.

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Johansen, S. and K. Juselius, “Testing Structural Hypotheses in a Multivariate Cointegration Analysis of the PPP and UIP for the UK,” Journal of Econometrics 53 (1992), 211-244. Johansen S. and K. Juselius, “The Full Information Maximum Likelihood Procedure for Inference on Cointegration--With Application to the Demand for Money,” Oxford Bulletin of Economics and Statistics 52 (1990), 169-210. Juselius, K. and R. MacDonald, “International Parity Relationship Between Germany and the United States: A joint Modeling Approach,” Working Paper, Institute of Economics, University of Copenhagen (2000). Kawai, M. and H. Ohara, “Non Stationarity of Real Exchange Rates in the G7 Countries: Are They Cointegrated with Real Variables,” Journal of the Japanese and International Economies 11 (1997), 523-547. Kim, B.J.C. and S. Mo, “Cointegration and the Long-Run Forecast of the Exchange Rate,” Economics Letters 48 (1995), 353-359. MacDonald, R. and Nagayasu, “The Long-Run Relationship between Real Exchange Rate and Real Interest Rate Differentials: A Panel Study,” IMF Working Paper No. 99/37.

- 15 -

Gonzalo, J., “Five Alternative Methods of Estimating Long-Run Equilibrium Relationships,” Journal of Econometrics 60 (1994), 203-233.

16

McCallum, B., “A Reconsideration of the uncovered interest parity relationship,” Journal of Monetary Economics Vol. 33 No. 1 (1994), 105-132. McNown, R. and M. Wallace, “Cointegration Tests of the Monetary Exchange Rate Model for Three High-Inflation Economies,” Journal of Money, Credit and Banking 26 (1994), 396-411. McNown, R. and M. Wallace, “National Price Levels, Purchasing Power Parity, and Cointegration: A Test of Four High Inflation Economies,” Journal of International Money and Finance 8 (1989), 533-545. Meese, R. and K. Rogoff, “Was it real? The exchange rate-interest rate differential relation over the modern floating-rate period,” Journal of Finance 43 (1988), 933948.

Meredith, G. and M. D. Chinn, "Long-Horizon Uncovered Interest Rate Parity", working paper No 6797, National Bureau of Economic Research (1998). Mussa, M., “Exchange Rates in Theory and in Reality,” Essays in International Finance, No. 179 (1990), Princeton University. Mussa, M., “Empirical Regularities in the Behavior of the Exchange Rates and Theories of the Foreign exchange Market,” in Karl Brunner and Allan Meltzer, (eds.), Policies for Employment, Prices, and Exchange Rates. Carnegie-Rochester Conference, Vol. II Amsterdam, North Holland (1979), 0-58. Obstfeld, M. and K. Rogoff, Foundations of International Macroeconomics, MIT Press, Cambridge, Ma (1996). Perron, P., "Further Evidence on Breaking Trend Functions in Macroeconomic Variables," Journal of Econometrics 80 (1997), 355-385. Perron, P., “Trends and Random Walks in Macroeconomic Time Series,” Journal of Economic Dynamics and Control 12 (1988), 297-332. Phillips, P. C. B., “Time Series Regressions with a Unit Root,” Econometrica 55 (1987), 277-301.

- 16 -

Meese, R. and K. Rogoff, “Empirical Exchange Rate Models of the Seventies: Do They Fit Out-of-Sample?” Journal of International Economics (1983), 3-24.

17

Phillips, P. C. B., “Econometric Model Determination,” Econometrica 64 (1996), 763-812. Phillips, P. C. B. and M. Loretan, “Estimating Long-Run Equilibria,” The Review of Economic Studies 58(3) 1991, 407-436. Said, S. and D. A. Dickey, “Testing for Unit Roots in Autoregressive-Moving Average Models of Unknown Order,” Biometrika 71 (1984), 599-608. Vahid, F. and R. Engle, “Common Trends and Common Cycles,” Journal of Applied Econometrics Vol. 8 (1993), 341-360. - 17 -

Viner, A., “Inside Japanese Financial Markets", Dow Jones-Irwin, Homewood, Illinois, (1988)

18

Table 1a Tests for Unit Roots in Nominal Exchange Rates and Nominal Interest Rate Differentials k

∆ y t constant + trend + ρ y t −1 + ∑ δ i ∆ y t −i + ε t i =1

1980:1 to 1997:7

Lags

DM-USD GBP-USD YEN-USD CAN-USD i (US)- i (Germany) i (US)- i (Britain) i (US)- i (Japan) i (US)- i (Canada)

13 14 0 11 2 2 6 7

-2.55 -2.98 -1.41 -2.24 -1.82 -2.79 -3.73* -1.67

• •

•

Phillips – Perron z

DF-GLS

-10.30 -8.11 -5.98 -4.81 -10.14 -18.38 -11.93 -29.47*

-1.78 -1.10 -1.54 -2.16 -1.74 -1.65 -2.70 -1.89

The exchange rate is measured in natural logarithms. Interest rates are the 10-year bond rates. Lags are the same across all tests. ADF is the ADF statistic for H0: unit root. The 5% critical value is -3.41. Phillips-Perron is the Phillips-Perron statistic for H0: unit root. The 5% critical value is -21.7. We do not report the statistics for joint hypothesis tests for ρ and the constant, and ρ and trend. DF-GLS is the Elliott, Rothenberg and Stock statistic for Ho: unit root with a linear trend. The 5 percent critical value is –2.89. The test is the t-statistic for testing a o = 0 in the regression

∆y = α o y t −1 + a1 ∆y t −1 + ... + a p ∆ y t − p + ε t and y = y t − βˆ ot − βˆ1t .

- 18 -

y

Statistics for ρ =0 ADF τ

19

Table 1b Perron (1997) Unit Root Test for Nominal Exchange Rates and Nominal Interest Rate Differentials k

∑ c ∆y

Model 1 : yt = u + θDU t + β t + δD (Tb ) t + αyt −1 +

i

i =1

t −i

+ et ∧

∧

Tb

α

DM-USD GBP-USD YEN-USD CAN-USD i (US)- i (Germany) i (US)- i (Britain) i (US)- i (Japan) i (US)- i (Canada)

1986:4 1986:9 1985:7 1987:11 1988:11 1980:5 1994:1 1986:1

0.91 0.90 0.93 0.91 0.92 0.90 0.87 0.81

Model 2 : y t = u + θDU t + βt + δDTt + δD (Tb ) t + αy t −1 +

tα

-3.64 -4.08 -3.37 -3.92 -3.00 -4.14 -5.13* -3.93 k

∑ c ∆y i

t −i

+ et

i =1

•

∧

Y

Tb

α

DM-USD GBP-USD YEN-USD CAN-USD i (US)- i (Germany) i (US)- i (Britain) i (US)- i (Japan) i (US)- i (Canada)

1985:5 1992:7 1985:7 1987:11 1989:3 1985:2 1989:2 1995:10

0.89 0.92 0.92 0.91 0.89 0.85 0.85 0.79

∧

tα

-3.92 -3.65 -3.55 -3.74 -3.79 -4.67 -5.78* -4.32

Tb is the value that minimizes the t-statistic for testing α = 1 . Tests for Tb are chosen so as to minimize the t-statistic on the parameter associated with model 1 or model 2 gave the same results. The 5 percent critical values for model 1 and model 2 are –4.20 and –5.08, respectively.

- 19 -

Y

20 Table 2 The Johansen-Juselius Maximum Likelihood Test for Cointegration 1980:1 to 1997:9

DM-USD Residuals 2 L-B ~ χ 126 LM1 ~ χ 4

2

LM4 ~ χ 4

2

GBP-USD Residuals 2 L-B ~ χ 122 2

LM4 ~ χ 4

2

CAD-USD Residuals 2 L-B ~ χ 202 LM1 ~ χ 4

2

LM4 ~ χ 4

2

• • • • •

λ max

Trace

H0: r

p-r

λ max 95%

Trace 95%

lag

0.1438 0.0012

20.33* 0.15

20.49* 0.15

0 1

2 1

12.33 4.18

16.51 4.18

1

115.656 (0.74) 2.209 (0.70) 0.343 (0.99) 0.1542 0.0236

21.77* 3.10

24.87* 3.10

0 1

2 1

12.45 4.22

16.67 4.22

2

15.17* 1.25

16.42 1.25

0 1

2 1

12.45 4.22

16.67 4.22

2

102.446 (0.90) 7.106 (0.13) 3.672 (0.45) 0.0700 0.0060

- 20 -

LM1 ~ χ 4

Eigen Values

214.305 (0.26) 10.395 (0.03) 6.869 (0.14)

r is the number of cointegrated vectors. p is the number of variables. The 95% critical values corrected for small samples using Cheung and Lai (1993) are used to evaluate the results. The models include drift terms. L-B is a multivariate Ljung-Box test based on estimated auto crosscorrelation of the first T/4 lags. The LM1 LM4 are the Lagrange multiplier tests.

21 Table 3 The Phillips-Loretan Non -Linear Dynamic Least Square Estimator 1980:1 to 1997:7 Restricted Model et = a + b(i − i )t + *

k

∑ δ ∆(i − i ) *

i

i=−k

t −i

Unrestricted Model

+ ρ[et −1 − a − b(i − i )t −1 ] + vt *

GBP-USD -0.36 (-7.07) [0.0001] 0.10 (3.04) 0.0023 0.93 (50.1) [0.0001] -

CAD-USD 0.42 (3.65) [0.0002] 0.10 (1.50) [0.1350] 0.97 (79.8) [0.0001] -

b2

-

-

-

Se DW Breusch-Pagan

0.03 1.85 6.10 [0.6349] 10.35 [0.5846] 0.0611

0.03 1.77 9.53 [0.2993] 15.52 [0.2142] 0.0982

0.01 2.01 7.16 [0.5190] 6.86 [0.8662] 0.0659

2.048 3.77 [.0518]

4.68* 5.611 [.0178]

10.25* 0.77 [.3772]

b ρ

b1

ARCH (12) Bartlett’s-KolmogrovSmirnov F

χ • • • •

2 1

k

i =−k

i =−k

DM-USD 0.09 (0.49) [0.6174] -

GBP-USD -0.12 (-1.10) [0.2690] -

CAD-USD 0.55 (2.75) [0.0058] -

0.93 (51.8) [0.0001] 0.11 (5.80) [0.0001] -0.06 (1.83) [0.0665] 0.03 1.90 17.88 [0.2689] 12.00 [0.4456] 0.0583

0.90 (33.6) [0.0001] 0.09 (3.38) [0.0001] -0.11 (-4.88) [0.0001]

0.97 (68.8) [0.0001] 0.09 (1.50) [0.1389] -0.10 (-1.62) [0.1052]

0.03 1.80 13.41 [0.5700] 21.86 [0.0390] 0.0810

0.01 2.11 11.79 [0.6947] 13.31 [0.3469 0.0714

Se: Standard error of the estimates. The t-ratios are in parentheses and P-values are in square brackets. Bartlett’s-Kolmogrov-Smirnov 5% critical value is 0.1587. F tests the restriction on the long-run parameters and the dynamics. χ 12 is to test the restrictions on the long-run parameters only.

- 21 -

DM-USD 0.34 (8.35) [0.0001] 0.12 (5.60) [0.0001] 0.94 (51.3) [0.0001] -

a

k

et = a + b1it + b i + ∑δ i ∆it −i + ∑θ t −i ∆it*−i + ρ[et −1 − a − b1it −1 − b2 it*−1 ] + ηt * 2 t

Table 4 The Vahid-Engle Test for Common Cycles 1980:1 to 1997:9

DM-USD

Squared Canonical Correlation 0.012417 0.078758

C ( k , s) ~ χ 2

H0: s

Degrees of Freedom

P-value

1

2.63 19.93*

s>0 s>1

2 6

0.2685 0.0003

k

GBP-USD

0.037011 0.088987

2

7.95* 27.61*

s>0 s>1

2 6

0.0188 0.0001

CAD-USD

0.021533 0.083491

1

4.59 22.98*

s>0 s>1

2 6

0.1008 0.0001

s

•

C ( k , s) = − ( T − k − 1) ∑ ln (1 − λ2i ) , where T is the sample size, k is the lag length, and λ is the canonical correlation.

•

s

i =1

is the smallest squared canonical correlation between the exchange rate and the interest rate differential.

• The degrees of freedom are s − spk + sr − sp , where (i.e., 2). An asterisk means significant at the 5% level. 2

r is the number of cointegrated vectors (i.e., 1), and p is the dimension of the system

Ja n8 N 0 ov -8 Se 0 p81 Ju l-8 M 2 ay -8 M 3 ar -8 Ja 4 n8 N 5 ov -8 Se 5 p86 Ju l-8 M 7 ay -8 M 8 ar -8 Ja 9 n9 N 0 ov -9 Se 0 p91 Ju l-9 M 2 ay -9 M 3 ar -9 4 Ja n9 N 5 ov -9 Se 5 p96 Ju l-9 7

Ja n8 N 0 ov -8 Se 0 p81 Ju l-8 M 2 ay -8 M 3 ar -8 Ja 4 n8 N 5 ov -8 Se 5 p86 Ju l-8 M 7 ay -8 M 8 ar -8 Ja 9 n9 N 0 ov -9 Se 0 p91 Ju l-9 M 2 ay -9 M 3 ar -9 Ja 4 n9 N 5 ov -9 Se 5 p96 Ju l-9 7

DM-USD

0 DM-USD

GBP-USD

i-i*

1.2

1

0.8 2

0.6 1

0.4

0.2

0

i-i*

-0.1

-0.4

-0.5

-0.6

-0.9

-1 -1

-0.7 -2

-0.8 -3

-4

-5

i-i*

1.4

i-i*

- 23 -

GBP-USD

23

Figure 1: Exchange Rate & Interest Rate Differential USD-DM (80:1-97:7) 6

5

4

3

0

-1

-2

-3

Figure 2: Exchange Rate & Interest Rate Differential USD-GBP (80:1-97:7) 3

-0.2 2

-0.3 1

0

Ja n8 N 0 ov -8 Se 0 p81 Ju l-8 M 2 ay -8 M 3 ar -8 Ja 4 n8 N 5 ov -8 Se 5 p86 Ju l-8 M 7 ay -8 M 8 ar -8 Ja 9 n9 N 0 ov -9 Se 0 p91 Ju l-9 M 2 ay -9 M 3 ar -9 Ja 4 n9 N 5 ov -9 Se 5 p96 Ju l-9 7

0.4 CAD-USD

i-i*

5.6

5.4

4

4.8

4.6 1

4.4 0

i-i*

0.35

0.3

0.25

0.2

0.15 -1

0.1

0.05

0 -1.5

-2

-2.5

-3

i-i*

YEN-USD

i-i*

Ja n8 N 0 ov -8 Se 0 p81 Ju l-8 M 2 ay -8 M 3 ar -8 Ja 4 n8 N 5 ov -8 Se 5 p86 Ju l-8 M 7 ay -8 M 8 ar -8 Ja 9 n9 N 0 ov -9 Se 0 p91 Ju l-9 M 2 ay -9 M 3 ar -9 Ja 4 n9 N 5 ov -9 Se 5 p96 Ju l-9 7

YEN-USD 5.8

- 24 -

CAD-USD

24

Figure 3: Exchange Rate & Interest Rate Differential US-YEN (80:1-97:7) 8

7

6

5.2 5

5 3

2

Figure 4: Exchange Rate & Interest Rate Differential USD-CAD (80:1-97:7) 1.5

1

0.5

0

-0.5

Ja nN 80 ov -8 Se 0 p8 Ju 1 l-8 M 2 ay -8 M 3 ar -8 Ja 4 nN 85 ov Se 85 p8 Ju 6 l-8 M 7 ay -8 M 8 ar -8 Ja 9 nN 90 ov Se 90 p9 Ju 1 lM 92 ay -9 M 3 ar -9 Ja 4 nN 95 ov Se 95 p9 Ju 6 l-9 7

-0.2

-0.4

-0.6

- 25 -

Ja nN 80 ov Se 80 p8 Ju 1 l-8 M 2 ay -8 M 3 ar -8 Ja 4 nN 85 ov Se 85 p8 Ju 6 l-8 M 7 ay M 88 ar -8 Ja 9 nN 90 ov Se 90 p9 Ju 1 l-9 M 2 ay -9 M 3 ar Ja 94 nN 95 ov Se 95 p9 Ju 6 l-9 7

Ja n8 D 0 ec -8 N 0 ov -8 O 1 ct -8 Se 2 p8 Au 3 g84 Ju l-8 Ju 5 nM 86 ay -8 Ap 7 r-8 M 8 ar -8 Fe 9 b9 Ja 0 n9 D 1 ec -9 N 1 ov -9 O 2 ct -9 Se 3 p9 Au 4 g95 Ju l-9 Ju 6 n97

25

Figure 5: Nominal Exchange Rate & a+b(i-i*) DM-USD (80:1-97:7)

1.4 1.2 1 0.8 0.6 0.4 0.2 0 DM-USD

GBP-USD

CAD-USD

a+b(i-i*)

Figure 6: Nom inal Exchange Rate & a+b(i-i*) GBP-USD (80:1-97:7) a+b(i-i*)

0

-0.8

-1

Figure 7: Nom inal Exchange Rate & a+b(i-i*) CAD-USD (80:1-97:7)

0.6 0.5 a+b(i-i*)

0.4 0.3

0.2 0.1

0

26

- 26 -

27

- 27 -

28

- 28 -

29

- 29 -