Why are real interest rates not equalized internationally?

1. Introduction

International financial economists are intensely concerned with several parity conditions that relate goods prices and asset returns across countries. These include purchasing power parity (PPP), interest rate parity (IRP), and variations of these two, such as relative PPP, ex ante PPP, and real interest rate parity (RIP). Each of these parity conditions measures a degree of integration between the economies of the world. The greater the economic integration across countries, the greater the likelihood that the more restrictive of these parity conditions will have empirical support. For example, the strict PPP hypothesis implies that aggregate price levels will be equal in terms of a common currency. This condition is very stringent because it does not allow for even temporary deviations from the equilibrium condition. (1) A less restrictive version is long-run PPP, where price levels, measured in a common currency, across countries converge over some period of time, may be a very long period of time. Recognizing all of the impediments that keep strict PPP from holding, economists have put their efforts in studying the long-run version of this equilibrium condition.

There is a relatively large literature examining the equality of real interest rates internationally. (2) In all of these studies, the real interest rate is defined by the Fisher equation. These studies then go on to estimate and test for the simultaneous existence of uncovered interest parity (UIP), which is also called the open-economy Fisher relation, and ex ante PPP (EAPPP), the other conditions sufficient for RIP. The problem in this strategy is that the Fisher equation may not hold. Many studies have been devoted to examining the validity of the Fisher relationship for both the U.S. and other economies. (3) The results of this line of inquiry have been decidedly mixed. Even if the Fisher relation does hold, it is unlikely that the Fisher effect will be one-for-one, as implied by defining the real interest rate from the simple Fisher equation. Both Mundell--Tobin and tax effects can drive a wedge between inflation expectations and the effect on nominal interest rates. (4) It would seem more appropriate to treat the Fisher relation as another testable parity condition than as an assumption.

In this study, we examine the underlying parity conditions sufficient for RIP, treating them as hypotheses to be tested. The four parity conditions or equilibrium relationships upon which RIP is predicated, uncovered interest parity (UIP), ex ante PPP (EAPPP), the Fisher relation in each country (i.e., the Fisher relation in country A and the Fisher relation in country B), imply time series implications for the observable variables. Specifically, if any of the nominal interest rates or inflation rates can be characterized as integrated variables, evidence of which we will provide, then the four parity conditions imply one common stochastic trend between them. Therefore, an initial examination of the sufficient conditions for RIP should include a test for a unit root in the observable variables and then a test for the number of common stochastic trends in a given RIP system.

Because the power of univariate unit root and stationarity tests is notoriously low, many exploit the power gains available by using panel tests. However, there are serious drawbacks to some of these panel unit root tests, for example, O'Connell (1998), Taylor and Sarno (1998), and Breuer, McNown, and Wallace (2001). We overcome these drawbacks by taking advantage of recent innovations in univariate and multivariate unit root tests that have substantially greater power to reject a false null but are not subject to the size distortions associated with some popular panel-based tests. Our use of these methods leads us to conclude that each RIP system analyzed can be characterized, to some degree, as a system of integrated variables that share more than one common trend. This violates the sufficient conditions for RIP implied by the four parity conditions.

We extend the analysis to determine which of the parity conditions fail, resulting in the violation of RIP. Our results indicate that no single parity condition can explain the failure of RIP in all cases. But it does appear that the Fisher relation is the least likely to violate the RIP equilibrium. On the other hand, UIP appears to be the most commonly violated of the four conditions, with EAPPP somewhat more consistent with the data.

The rest of the article is organized as follows: Section 2 discusses the theoretical conditions sufficient for RIP and the time series implications of these conditions. Section 3 describes the econometric methodology and discusses the various hypothesis tests. Section 4 presents the empirical results and Section 5 concludes.

2. Real Interest Rate Parity

Four Parity Conditions

RIP is the condition where real rates of return on essentially identical assets are equalized across countries. There are many reasons why real interest rates will not always be equal across countries, for example, country-specific risk, transactions costs, information asymmetries, and/or differential tax treatment. For these reasons, our focus is on long-run RIP.

The nominal interest rate in country j can be related to the real interest rate by the Fisher relationship given in Equation 1:

(1) [i.sub.j,t] = [r.sub.j,t] + [[pi].sup.e.sub.j,t+k']

where [i.sub.j,t] is the nominal interest rate of country j in period t and [[pi].sup.e.sub.j,t+k] is the ex ante inflation in country j over the holding period k of the asset.

Nominal interest rates across countries are related via the UIP relationship. UIP relates the nominal interest rate differential to the expected exchange rate depreciation as in Equation 2:

(2) [i.sub.j,t] - [i.sub.i,t] + [DELTA][S.sup.e.sub.t+k']

where [s.sup.e.sub.t+k] is the log of the ex ante spot exchange rate in period t + k and [DELTA] is the first difference operator.

Inflation rates are related across economies via the EAPPP relationship. EAPPP relates the expected inflation differential to the expected exchange rate depreciation as in Equation 3:

(3) [[pi].sup.e.sub.j,t+k] - [[pi].sup.e.sub.i,t+k] = [DELTA][S.sup.e.sub.t+k']

Combining Equations 1, 2, and 3 yields the real interest parity relationship

(4) [r.sub.j,t] = [r.sub.j,t.]

If Equations 1, 2, and 3 are true, then real interest rates will be equalized internationally. RIP is itself based on the validity of four equilibrium relationships, a Fisher relation in both the foreign and domestic countries relating nominal rates to real rates, the UIP equation relating nominal rates across countries, and EAPPP relating expected inflation rates across countries.

Time Series Implications

We start the discussion of the time series implications of the previous section from the assumption that the observable variables [i.sub.j,t], [i.sub.i,t], [[pi].sub.j,t+k], [[pi].sub.i,t+k], and [s.sub.t+k] are integrated of order one or I(1). (5) This assumption implies that these variables have no tendency to revert toward a constant mean or deterministic trend. Such series may wander without bound.

Previous studies have used Equation 1 as the basis for testing the Fisher relation by noting that economic theory suggests that [r.sub.j,t] should be a stationary or I(0) variable. (6) In order for the Fisher relation to hold empirically when [i.sub.j,t] and [[pi].sub.j,t+k] are I(1) processes, they must be cointegrated. Cointegration implies that, while both individual variables will wander aimlessly through time, some linear combination of the two tends toward a constant value, that is, they share a long-run equilibrium. (7)

Similarly, if nominal interest rates and the spot exchange rate are first-order integrated or I(1), then the log difference of the spot exchange rate is I(0), and Equation 2 implies that the two nominal interest rates are also cointegrated. (8)

Finally, if [[pi].sub.j,t+k] and [[pi].sub.i,t+k] are both I(1) variables, then they must also be cointegrated if the EAPPP relation is to be empirically valid. Real interest rates will be equalized internationally if these four equilibrium relationships or parity conditions are satisfied in each two-country pairing, a Fisher relation in each country, UIP, and EAPPP. There are four observable variables in these four parity relationships: [i.sub.j,t], [i.sub.i,t], [[pi].sub.j,t+k] and [[pi].sub.i,t+k]. Because only three of the four parity conditions are independent, that is, if three hold, then the fourth must also be true, there should exist three stationary relations among the four variables in each RIP system. This means, as shown by Stock and Watson (1988), that there must be one common stochastic trend among the four variables.

In order to see this result more clearly, it is convenient to appeal to a vector autoregressive (VAR) model in the four observable variables as in Equation 5,

(5) [PHI](L)[X.sub.t] = [[mu].sub.t] + [[mu].sub.t],

where [X.sub.t] is the 4 x 1 vector of the observable variables: [i.sub.j,t], [i.sub.i,t], [[pi].sub.j,t+k], and [[pi].sub.i,t+k]. [PHI](L)is a pth-order matrix polynomial in the lag operator, [[mu].sub.t] is a vector of deterministic components, and [[mu].sub.t] is a vector of white-noise error terms. Equation 5 can be rewritten in error-correction model form as in Equation 6:

(6) [DELTA][X.sub.t] = [[mu].sub.t] + [PI][X.sub.t-1] = [p-1.summation over (i=1)][[GAMMA].sub.i][DELTA][X.sub.t-1] + [[mu].sub.t],

where the long-run impact matrix [PI] is of particular interest. There are three possibilities with respect to the rank of the [PI] matrix: (i) [PI] is of zero rank, implying that all of the variables in [X.sub.t] are I(1) and not cointegrated; (ii) [PI] is of full rank, implying that all of the variables in [X.sub.t] are stationary in levels; and (iii) [PI] is of reduced rank, implying that at least some of the variables in [X.sub.t] are integrated and some linear combination of the variables in [X.sub.t] are stationary. Under this last condition, the [PI] matrix can be decomposed into two 4 x r matrices [alpha] and [beta] such that [PI] = [apha][beta]', where r is the number of stationary or cointegrating relations in [X.sub.t.] [beta] represents the space spanned by the stationary relations and [alpha] is the matrix of error correction coefficients.

Equations 1, 2, and 3 imply the following representation for [beta]'[X.sub.t]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [eta], [gamma], and [delta] should equal one under a strict interpretation of the conditions given in Equations 1, 2, and 3, but may differ from unity for reasons alluded to earlier. (9)

From Equation 7, it can be seen that 1-i is of reduced rank equal to three when the conditions for RIP hold, implying that the variables in [[CHI].sub.t] are characterized by a single common stochastic trend. What if II has a rank of one or two? Then at least one of the parity conditions is being violated and equality of real rates internationally will not occur. (10) It is possible that all of the variables germane to the RIP parity conditions are stationary in levels form. Under these circumstances, the methodologies we suggest would be inappropriate. But if any of the four relevant variables have unit roots, Equations 1, 2, and 3 imply one common unit root among the four variables of each RIP system. By analyzing each of the parity relationships individually, it may be possible to shed light on the sources of RIP violations. In the next section, we present the empirical methodology used to test the four necessary conditions for RIP and those methods used to analyze the sources of the rejection of RIP.

3. Empirical Methodology

Unit Root Tests

Because the appropriateness of our suggested methodologies depends on the existence of unit roots in at least some of the RIP variables, the first task is to apply unit root tests to the individual variables. But it is well known that the power of such tests is notoriously weak when the autoregressive root is close to, but still less than, unity. Recently, several researchers have proposed using a panel unit root testing approach that significantly increases the power of the test against the null. (11)

However, O'Connell (1998) has shown that these tests suffer from extreme size distortion (rejects a true null too often) when the contemporaneous error terms are correlated across groups (referred to as spatial correlation in the literature). O'Connell further demonstrates that, once this spatial correlation is controlled for, the power of these tests drops significantly. Furthermore, Taylor and Sarno (1998) and Breuer, McNown, and Wallace (2001) have shown that rejection of the null hypothesis in the panel unit root tests cannot be interpreted as stationarity of all the series in the panel. These tests are uninformative about the number of series that are stationary versus the number that are nonstationary.

In an effort to overcome the shortcomings of standard univariate and panel unit root testing procedures, we employ several recently developed methodologies. (12) The first of these is the univariate test proposed by Elliot, Rothenberg, and Stock (1996) based on generalized least squares (GLS) detrending. These tests have greater power than standard augmented Dickey-Fuller (ADF) tests and are denoted as Dickey-Fuller GLS (DF-GLS) tests. Ng and Perron (2001) extend these GLS detrending procedures to a class of univariate tests that are powerful and less size distorted, the socalled M-tests. (13)

We also examine the robustness of our unit root inference to possible structural breaks in the underlying series by applying the procedure advanced by Zivot and Andrews (1992). We use the model that allows for both changes in the level of the series and a change in the growth rate, which Zivot and Andrews call model (C).

We also apply three multivariate unit root tests that have been suggested in the literature. The first is the covariate ADF (CADF) test of Hansen (1995). This test is based on the standard ADF regression that is augmented with stationary covariates. This reduces the sample variance estimate, creating a more powerful test against the unit root null. Hansen demonstrates that the power of the CADF test is inversely related to the long-run (frequency zero) squared correlation between the series being tested and the covariates and is denoted as [[eta].sup.2]. He shows that, with a sample size of 100 observations and an autoregressive root of 0.95, the power envelope for the standard ADF test is 33%. This increases to 51% when [[eta].sup.2] = 0.7 and to 90% when [[eta].sup.2] = 0.3.

The second multivariate unit root test we employ was proposed by Breuer, McNown, and Wallace (2001). It treats the individual ADF regressions as a system of seemingly unrelated regressions (SUR). This ADF-SUR procedure yields substantial power gains over univariate unit root tests. The critical values for this procedure must, however, be simulated for each case. In our application, we used the estimated parameters on the lagged changes and the system covariance matrix from the ADF-SUR model to specify the simulation model. We then restricted the autoregression (AR) process to have a unit root and used 10,000 simulations to generate the appropriate critical values for each variable.

Finally, we use the procedure advocated by Taylor and Sarno (1998) in the context of testing purchasing power parity. They suggest using Johansen's (1988, 1991) test for the number of stationary relations (or cointegration vectors) in a vector of I(1) variables. If all of the series in the vector are stationary, then the number of stationary relations will equal the number of variables in the VAR and the II matrix from estimation of Equation 6 will be full rank. If at least one of the four variables is I(1), then this impact matrix will be of reduced rank. The likelihood ratio statistic testing the rank of [PI] is given by -T [[SIGMA].sup.p.sub.i=r+1] In(1 - [[??].sub.i]), where [[??.sub.i] is the estimated eigenvalues from the reduced rank estimation procedure suggested by Johansen (1991), which is commonly known as the trace statistic.

The distribution of this statistic is nonstandard and depends on, among other things, the specification of the deterministic components in the VAR. The appropriate deterministic specification can be tested using a likelihood ratio (LR) test procedure discussed in Johansen (1994).

Tests of Parity Conditions

We use two methodologies to examine the four individual parity conditions themselves for each bivariate pairing. The first technique employed in this phase of the analysis is the Johansen procedure. This methodology has the benefit of making no a priori restrictions or normalizations on the possible equilibrium relationships. If and when cointegration has been established, hypothesis tests on the cointegrating parameters can be conducted in order to determine if the particular parity condition, in fact, holds. For example, when testing for UIP, not only is it necessary that nominal interest rates are cointegrated across countries, but the cointegrating vector [beta] should span the space [1,- 1]'. (14) Hypothesis tests on [??], the estimated cointegrating vectors, can be conducted using LR tests with standard [chi square] inference.

The second technique used to examine the individual parity relationships is a cointegration test proposed by Horvath and Watson (1995) (hereafter HW), in which the cointegration parameters are prespecified. This technique is based on estimating Equation 6 while imposing a restricted [beta]. It has the advantage of increased power against the null of no cointegration, but does require knowledge of the cointegrating vector(s), usually derived from economic theory. Specifically, the HW test is a generalization of the Wald test for the significance of [alpha] in Equation 6 when [beta] is prespecified. (15)

Our strategy for the empirical analysis is straightforward, The first step is to test the univariate orders of integration for each of the variables in the RIP relationship. Once it has been established that we can reject the sufficient conditions for RIP, we proceed to analyze the individual equilibrium conditions of UIP, EAPPP, and the Fisher relation in each country, testing not only for an equilibrium between the germane variables but also testing the parameter restrictions implied by each of these parity conditions. The next section will discuss the data and the results from the empirical analysis.

4. Empirical Results

Data

The data used in our study consist of monthly observations on 12-month Eurocurrency deposit rates and annualized inflation rates for the United States (US), United Kingdom (UK), Canada (CN), Germany (GM), and Japan (JP) over the period February 1960 to April 1996. (16) The data are displayed in levels in Figure 1 and in first differences in Figure 2. This sample represents five of the largest and most open industrialized economies in the world, which we expect would be as likely as any to yield results favorable to the RIP hypothesis. (17)

[FIGURES 1-2 OMITTED]

Unit Root Tests

Table 1 displays the univariate unit root tests. These include the standard ADF test as well as the M[[Z.sub.[alpha] test of Ng and Perron and the DF-GLS test of Elliot, Rothenberg, and Stock (1996). The lag length in each augmented regression was determined by applying the modified Akaike information criterion (AIC) of Ng and Perron (2001). From these three tests, it appears that only the Japanese and German nominal interest rates are stationary. This result is robust to allowance for structural breaks in the data as evidenced in Figure 3, which displays the Zivot and Andrews tests lot each series. The plotted test statistics have been normalized by the appropriate 5% critical value of-5.08. Thus, values greater than one would imply statistical significance.

[FIGURE 3 OMITTED]

Table 2 displays the results of the ADF-SUR procedure. Here we find evidence that only the German nominal interest rate is stationary. Based on simulations we conducted, with an AR root of 0.95 and 435 observations, these tests have size-adjusted power ranging from 41% for the Canadian inflation series up to 72% for the Japanese nominal interest rate.

Table 3 presents the CADF tests for unit roots. Because the asymptotic distribution of the test statistic depends on the value of [[eta].sup.2] the 5% critical values are also presented. (18) The results in Table 3 were estimated from a model that included a constant and time trend in the deterministic portion of the model and six lags of the covariates. These results suggest that all 10 series are I(1). To get an idea of the power increases associated with the CADF tests versus conventional unit root tests, we can examine the results from Table 5 of Hansen (1995). For example, the U.S. nominal interest rate CADF model yielded an [[eta].sup.2] estimate of 0.30. These values yielded power of 59-88%, depending on the empirical model specifications (included lags and so on). Even for the German inflation rate, which yielded the lowest power gains with estimates of [[eta].sup.2] on the order of 0.7, the power gain of the CADF test versus the ADF test is 40% (20% power for the ADF test vs. 28% power for the CADF test). (19)

The next unit root test we present is Johansen's test of the rank of II from Equation 6. Because the distribution of the test statistics, specifically the trace test, depends on the specification of the deterministic components in the vector error correction model (VECM), we must establish the proper specification for far in order to make valid inference using this test. (20) We consider the most general model relevant to our analysis to be one that includes a constant in the VAR of Equation 6, implying that the individual series have linear trends in their levels representation. This hypothesis is denoted as [H.sub.1] (r) to reflect that the null is dependent on the cointegration rank, r, being tested. (21) The alternative to this model is one that restricts the deterministic components of the level series to include a nonzero mean. We denote this hypothesis as [H.sup.*.sub.1](r). Table 4 presents the LR test statistics of [H.sub.i](r) versus [H.sup.*.sub.i](r). (22) There is no evidence against the restricted model for any choice of r, implying that the data do not contain significant linear trend components.

Table 5 presents the calculated LR tests of the hypothesis that there are no significant deterministic components in the VAR, that is, a test that each stationary relation has a zero mean. The evidence is somewhat mixed, but it generally suggests that it is inappropriate to eliminate the constant from the estimated stationary relationships. Therefore, we will conduct our tests of the rank of II based on a specification that does not include a constant in the VAR but does allow a constant term in the cointegration vector.

We use the Johansen trace test statistics to determine the rank of II as outlined in the previous section. We calculate a rolling estimate of the trace statistic in order to determine the sensitivity of our conclusions to the particular sample period. (23) We present the results graphically in Figure 4, where the test statistics have been normalized by the appropriate 10% critical value such that values greater than one imply statistical significance. In no case is the system being driven by one common trend. In all but the Japan--German and Japan-Canada systems, in which there is evidence of two cointegrating or stationary relationships, there is evidence of at most one cointegrating relation.

[FIGURE 4 OMITTED]

The evidence presented from examining the order of integration of the individual RIP component variables strongly implies that the data are nonstationary in levels, and we can conclude that the sufficient conditions for RIP to hold are not met. In the next section, we examine the various parity conditions that form the basis of RIP to see if we can determine a source of the RIP violations.

Testing Individual Parity Conditions

Table 6 displays the evidence from our analysis of the Fisher relation in each country. We present results from applying the Johansen tests and tests suggested by Horvath and Watson (1995). The first column of Table 6 references the country for which the Fisher relation is being tested. Columns 2 and 3 present the calculated trace test statistics for the null hypotheses of r = 0 and r [greater than or equal to] 1, respectively. The results from columns 2 and 3 imply strong evidence in favor of cointegration for all but the United Kingdom. (24) The tests of the restriction that the cointegrating relationship is one-for-one are displayed n the fourth column. Of the countries with evidence of cointegration between inflation and nominal interest rates, only for Japan can the Fisher restriction be rejected and then only at the 10% level. (25)

Columns 5 through 7 of Table 6 present results associated with the HW procedure. Column 5 displays HW test statistics when the cointegrating vector between the nominal interest rate and inflation is assumed to be known a priori. In this case, we assume that [beta] = [1, -1]'. The results from this test are in complete agreement with the Johansen test results. All but the United Kingdom reject the null of zero cointegrating vectors in favor of the alternative of one. Column 6 shows HW tests when the cointegrating relation is treated as unknown. Here, the evidence is not as strongly in favor of the Fisher relation, but the estimated Fisher effects in column 7 are consistent with the Fisher relationship in four out of the five countries examined.

Table 7 presents the results from testing the 10 UIP relationships possible. While there appears to be some strong evidence of cointegration in many of the relationships, there are only three bilateral relationships consistent with the UIP restrictions. These are the United States and the United Kingdom, Canada and the United Kingdom, and the United States and Canada. In the other seven bilateral relations, either the evidence in favor of cointegration is very weak or the UIP parameter restrictions are strongly rejected. The HW tests generally reinforce the Johansen results. The fact that Germany has a significant UIP cointegrating relationship with all four countries yet none of the UIP coefficient restrictions are supported can be explained by testing the null hypothesis that the German nominal interest rate is stationary by itself, or that it forms a trivial, [beta] = [1,0]', cointegrating relation. This restriction cannot be rejected in any of the German systems. The likelihood ratio tests of this restriction are 0.946 for the United States system, 0.773 for the Canada system, and 1.662 and 1.067 for the United Kingdom and Japanese systems, respectively. This statistic is distributed as a [[chi].sup.2](1) variate. (26) It would appear that UIP only holds in 3 out of the 10 bilateral relationships.

Because the UIP relationship is the analog to the forward premium in the foreign exchange market, these results can be compared with those in the forward premium literature. Specifically, the forward premium can be decomposed into spot rate changes, expectation errors, and a risk premium. Failure of UIP can be interpreted as failure of a stationary forward premium. If we assume that expectations are rational, then this failure implies a nonstationary risk premium in the foreign exchange market.

Table 8 presents the evidence for bilateral EAPPP relationships. The Johansen test evidence here is quite mixed. Half of the bilateral relations exhibit evidence of cointegration at the 10% level, but only three at the 5% level. Of the five, only two cannot reject the EAPPP parameter restrictions at the 10% level. The EAPPP relation seems to be strongest for the Japan and United Kingdom and the United States and Canada systems. Germany seems the least likely to share an EAPPP equilibrium relationship with any of the other countries. Again, as in Tables 6 and 7, the HW tests tend to confirm the Johansen results.

These results are difficult to reconcile given that EAPPP implies that changes in the real exchange rate should be stationary, of which there is much evidence in the literature. (27) In fact, most studies have concluded that nominal exchange rate changes are stationary. When both nominal and real exchange rate changes are stationary, then inflation differentials should also be stationary. The HW [W.sub.0.1](0, [[alpha].sub.k]) tests in Table 8, where the imposed cointegrating relation implies that the inflation differential is stationary, should generate a more powerful test than standard unit root tests because it does not impose the common-factor restriction. (28) One possible explanation of this seemingly contradictory evidence has been proposed by Newbold et al. (1998) within the context of the forward premium anomaly. The real exchange rate depreciation is the sum of the spot depreciation and the inflation differential. Nominal exchange rate depreciation is a stationary series with a very large variance, while inflation differentials appear (in some instances) to be nonstationary with a small permanent component, in the sense that the variance of the unit root component is small relative to that of the stationary component. Therefore, adding the stationary nominal exchange rate depreciation, with a large variance, to the inflation differential, which already has a small permanent component, makes the unit root extremely difficult to detect in finite samples. Under these circumstances, the more informative test of EAPPP is a test of cointegration between inflation rates, which is exactly the test we have conducted.

5. Conclusions

In this study, we set out to achieve a couple of objectives. First, we wanted to demonstrate that real interest rate parity and the underlying parity conditions, uncovered interest parity, ex ante PPP, and the Fisher relation in each country, imply that, if any one of the variables is integrated of order one or I(1), then the four must share a single common trend if RIP is to be a valid long-run characterization of the data. Using several testing procedures, we find little evidence that these sufficient conditions for RIP hold empirically for a group of five industrialized nations.

Our second objective was to identify which of the four underlying parity conditions may be responsible for the RIP rejections. We estimated and tested 25 bilateral equilibrium relationships. From these, we were able to conclude that the Fisher relation has wide support empirically but that UIP and EAPPP have much more limited support. The rejections of UIP are consistent with the literature that finds a significant risk premium in foreign exchange markets. The results in the present study go further in that the evidence also suggests that this risk premium may be nonstationary. The rejections of the EAPPP condition are difficult to reconcile with the implications for the time series behavior of real exchange rate changes and suggest further inquiry into the dichotomy between the results from examining unit root tests on real exchange rate changes and cointegration tests between inflation rates.

Table 1. Univariate Unit Root Tests

Variable            ADF   DF-GLS   M[Z.sub.a]  k

[[pi].sub.US]      -2.14   -1.77      -5.19     8
[i.sub.US]         -2.62   -2.51     -16.52    13
[[pi].sub.UK]      -2.00   -1.72      -3.85    12
[i.sub.UK]         -2.54   -2.49     -13.43     2
[[pi].sub.CN]      -1.84   -1.30      -2.61    11
[i.sub.CN]         -2.23   -2.28     -12.02     4
[[pi].sub.GM]      -2.31   -1.35      -0.91    13
[i.sub.GM]         -3.86   -3.71     -64.39    13
[[pi].sub.JP]      -2.97   -2.51      -2.28    11
[i.sub.JP]         -3.71   -3.63     -31.23     3
5% critical value  -3.41   -2.91     -17.30

Results calculated from model with a constant and trend included. The
number of augmentation lags k determined using the modified AIC
criteria of Ng and Perron (2001). US, United States; UK, United
Kingdom; CN, Canada; GM, Germany; JP, Japan.

Table 2. ADF-SUR Unit Root Tests

Variable       ADF-SUR  5% Critical  k

[[pi].sub.US]   -3.95      -3.78      8
[i.sub.US]      -3.69      -3.71     13
[[pi].sub.UK]   -2.62      -4.03     12
[i.sub.UK]      -2.49      -3.65      2
[[pi].sub.CN]   -1.87      -3.29     11
[i.sub.CN]      -2.67      -3.86      4
[[pi].sub.GM]   -2.62      -3.66     13
[i.sub.GM]      -4.03      -3.99     13
[[pi].sub.JP]   -3.31      -3.46     11
[i.sub.JP]      -3.49      -3.62      3

Results calculated from model with a constant and trend included. The
number of augmentation lags k determined using the modified AIC
criteria of Ng and Perron (2001). Critical values were determined from
10,000 iterations of a Monte Carlo simulation as suggested by Breuer,
McNown, and Wallace (2001). US, United States; UK, United Kingdom; CN,
Canada; GM, Germany; JP, Japan.

Table 3. Covariate Unit Root Tests

Variable       CADF   [[eta].sup.2]  5% CV

[[pi].sub.US]  -0.07      0.28       -2.69
[i.sub.US]      0.02      0.07       -2.31
[[pi].sub.UK]  -1.92      0.46       -2.94
[i.sub.UK]     -1.33      0.57       -3.06
[[pi].sub.CN]  -1.52      0.58       -3.07
[i.sub.CN]      0.18      0.27       -2.68
[[pi].sub.GM]  -2.34      0.78       -3.25
[i.sub.GM]     -1.66      0.44       -2.92
[[pi].sub.JP]  -2.38      0.46       -2.93
[i.sub.JP]     -1.36      0.40       -2.87

Results calculated from CADF model with a constant and trend included.
The number of augmentation lags k determined using the modified AIC
criteria of Ng and Perron (2001). Six lags and zero leads of the
covariates were included. [[eta].sup.2] estimated using the quadratic
spectral kernel with 24 included autocorrelations. US, United States;
UK, United Kingdom; CN, Canada; GM, Germany; JP, Japan.

Table 4. Test for Linear Trends in Data

                             r [less      r [less      r [less
                             than or      than or      than or
Country Pair        r = 0  equal to] 1  equal to] 2  equal to] 3

JP-UK               0.747     0.500        0.500        0.306
CN-UK               0.034     0.034        0.034        0.028
CN-JP               0.549     0.452        0.311        0.290
CN-US               0.190     0.188        0.153        0.034
JP-US               0.790     0.669        0.551        0.496
UK-US               0.128     0.116        0.105        0.102
GM-US               0.315     0.290        0.172        0.169
GM-CN               0.095     0.048        0.015        0.006
GM-JP               0.466     0.456        0.020        0.000
GM-UK               0.194     0.109        0.066        0.062
5% critical value   9.49      7.81         5.99         3.84

Entries represent the calculated LR test statistics where the null
hypothesis is no linear trends in the data, only a nonzero mean in the
stationary relationships. r represents the hypothesized number of
cointegrating vectors under the null hypothesis. JP, Japan, UK, United
Kingdom; CN, Canada; GM, Germany; US, United States.

Table 5. Test for Nonzero Means in Data
                              r [less      r [less      r [less
                              than or      than or      than or
Country Pair        r = 0   equal to] 1  equal to] 2  equal to] 3

JP-UK              0.880      4.904        8.112 *     11.746 *
CN-UK              4.308 *    5.159        9.599 *     12.575 *
CN-JP              5.432 *   13.379 *     16.293 *     20.186 *
CN-US              0.038      4.488        6.218       10.847 *
JP-US              1.600      4.612       11.315 *     14.596 *
UK-US              5.770 *    6.028 *      8.712 *     12.926 *
GM-US              0.514      8.568 *     11.017 *     16.113 *
GM-CN              0.180     12.235 *     12.424 *     17.476 *
GM-JP              0.057      3.437        7.205       11.247 *
GM-UK              0.884      2.731        5.635       10.887 *
5% critical value  3.84       5.99         7.81         9.49

Entries represent the calculated LR test statistics where the null
hypothesis is a zero mean in the stationary relationships. r represents
the hypothesized number of cointegrating vectors under the null
hypothesis. JP, Japan; UK, United Kingdom; CN, Canada; GM, Germany; US,
United States.

* Significance at the 5% level.

Table 6. Test for the Fisher Relation

                                   r [less    [H.sub.O]:
                                   than or     [beta] =
Country               r = 0      equal to] 1   [1,-1]'

US                   17.11 **      7.37 **     0.01
UK                   12.84         4.70        3.20 **
CN                   16.77 **      5.08        0.71
GM                   33.82 *       7.61        1.08
JP                   24.69 *       7.09        3.66 **
5% critical value    17.98         7.56        3.84

                   [W.sub.0,1]
                   (0, [[alpha]  [W.sub.0,1]
Country              .sub.k])       (0,0)       [beta]

US                   21.25 *      12.21         -0.95
UK                   12.70         8.06          2.24
CN                   16.56 **     11.51         -1.32
GM                   23.47 *      16.62 *       -1.46
JP                   17.62 **     12.62 **      -1.73
5% critical value    18.17        14.18          NA

Entries in columns 2 and 3 represent the calculated trace test
statistics where r is the hypothesized number of cointegrating vectors.
These statistics were estimated from a VECM that included a constant
and lags determined by minimizing the Schwarz Bayesian criterion (SBC).
Column 4 is the LR test statistic that the cointegrating vector is
equal to [1, -1]' of which the null hypothesis [H.sub.0] is distributed
as a [[chi square].sub.1] variate. Column 5 is the HW statistic
calculated under the assumed null of one known cointegrating vector
equal to [1, -1]' and one unknown vector under the alternative. Column
6 is the HW statistic calculated under the null of zero known
cointegrating vectors and one unknown vector under the alternative.
Column 7 is the estimated cointegrating parameter from the estimated
model of column 6. HW estimated model included a constant and lags
determined by minimizing the SBC criterion. NA, not applicable; JP,
Japan; UK, United Kingdom; CN, Canada; GM, Germany; US, United States.

* Significance at the 5% level.

** Significance at the 10% level.

Table 7. Test of UIP Relation

                                    r [less    [H.sub.O]:
                                    than or     [beta] =
Country Pair           r = 0      equal to] 1   [1,-1]',

JP-UK                 12.21          4.32        3.56 **
CN-UK                 22.81 *        6.24        0.03
CN-JP                 13.58          5.85        1.82
CN-US                 17.97 **       7.05        0.23
JP-US                 14.38          4.84        4.41 *
UK-US                 18.76 *        6.63        0.18
GM-US                 30.27 *        4.23       18.91 *
GM-CN                 30.78 *        5.65       16.44 *
GM-JP                 34.30 *        7.41       16.56 *
GM-UK                 30.55 *        6.19       13.71 *
5% critical value     17.98          7.56        3.84

                    [W.sub.0,1]
                    (0, [[alpha]  [W.sub.0,1]
Country Pair          .sub.k])       (0,0)       [beta]

JP-UK                 14.40          8.22          1.80
CN-UK                 34.48 *       29.76 *       -1.00
CN-JP                 14.01          8.69         -1.54
CN-US                 21.49 *       14.89 *       -1.00
JP-US                 14.01          8.51         -1.68
UK-US                 23.42 *       18.59 *       -1.34
GM-US                 23.52 *       17.90 *      -25.54
GM-CN                 21.22 *       16.40 *       -6.55
GM-JP                 22.00 *       15.88 *        0.21
GM-UK                 22.46 *       15.89 *       -8.35
5% critical value     18.17         14.18          NA

Entries in columns 2 and 3 represent the calculated trace test
statistics where r is the hypothesized number of cointegrating vectors.
These statistics were estimated from a VECM that included a constant
and lags determined by minimizing the Schwarz Bayesian criterion (SBC).
Column 4 is the LR test statistic that the cointegrating vector is
equal to [1, -1]' of which the null hypothesis [H.sub.0] is distributed
as a [[chi square].sub.1] variate. Column 5 is the HW statistic
calculated under the assumed null of one known cointegrating vector
equal to [1, -1]' and one unknown vector under the alternative. Column
6 is the HW statistic calculated under the null of zero known
cointegrating vectors and one unknown vector under the alternative.
Column 7 is the estimated cointegrating parameter from the estimated
model of column 6. HW estimated model included a constant and lags
determined by minimizing the SBC criterion. NA, not applicable; JP,
Japan; UK, United Kingdom; CN, Canada; GM, Germany; US, United States.

* Significance at the 5% level.

** Significance at the 10% level.

Table 8. Test of EAPPP Relation

                                   r [less    [H.sub.O]:
                                   than or     [beta] =
Country Pair          r = 0      equal to] 1   [1,-1]',

JP-UK                19.62 *        3.61       1.03
CN-UK                17.87 **       2.45       3.36 **
CN-JP                12.32          3.79       0.01
CN-US                24.30 *        4.38       0.87
JP-US                12.22          5.68       0.12
UK-US                20.33 *        6.41       3.11 **
GM-US                16.12          5.67       2.92 **
GM-CN                14.39          3.65       4.01 *
GM-JP                17.78 **       4.55       5.32 *
GM-UK                14.69          4.45       4.68 *
5% critical value    17.98          7.56       3.84

                   [W.sub.0,1]
                   (0, [[alpha]  [W.sub.0,1]
Country Pair         .sub.k])       (0,0)       [beta]

JP-UK                19.30 *      15.72 *       -1.34
CN-UK                17.58 **     15.14 *       -0.65
CN-JP                12.19         8.42         -1.07
CN-US                23.78 *      19.46 *       -1.14
1P-US                12.07         6.43         -1.63
UK-US                20.03 *      13.69 **      -1.72
GM-US                15.91 **     10.30         -0.36
GM-CN                14.22        10.59         -2.77
GM-JP                17.49 **     12.97 **      -2.92
GM-UK                14.51        10.09         -4.09
5% critical value    18.17        14.18          NA

Entries in columns 2 and 3 represent the calculated trace test
statistics where r is the hypothesized number of cointegrating vectors.
These statistics were estimated from a VECM that included a constant
and lags determined by minimizing the Schwarz Bayesian criterion (SBC).
Column 4 is the LR test statistic that the cointegrating vector is
equal to [1, -1]' of which the null hypothesis [H.sub.0] is distributed
as a [[chi square].sub.1] variate. Column 5 is the HW statistic
calculated under the assumed null of one known cointegrating vector
equal to [1, -1]' and one unknown vector under the alternative. Column
6 is the HW statistic calculated under the null of zero known
cointegrating vectors and one unknown vector under the alternative.
Column 7 is the estimated cointegrating parameter from the estimated
model of column 6. HW estimated model included a constant and lags
determined by minimizing the SBC criterion. NA, not applicable; JP,
Japan; UK, United Kingdom; CN, Canada; GM, Germany; US, United States.

* Significance at the 5% level.

** Significance at the 10% level.

We would like to thank, without implicating, two anonymous referees for suggestions that improved the article.

(1) Presumably, goods arbitrage will be instantaneous in such a world, preventing price levels from diverging.

(2) Recent contributions include Mark (1985). Dutton (1993). Kugler and Neusser (1993), Jorion (1996), Marston (1997), and Wu and Chen (1998).

(3) Recent contributions in this area are Mishkin (1992), Crowder and Hoffman (1996). Crowder (1997). Koustas and Serletis (1999), and Malliaropulos (2000)

(4) For an examination of the importance of tax effects in the Fisher relation, see Crowder and Wohar (1999).

(5) It is not necessary that all of the series under consideration be integrated processes to make our methodology valid.

(6) This result has been exploited by Rose (1988), Mishkin (1992), Crowder and Hoffman (1996), and Crowder (1997), among others.

(7) While cointegration is a necessary condition for the Fisher relation to be true when nominal interest rates and inflation rates are I(1), it is not sufficient. A generalized version of Equation 1 will include a risk premium term that reflects the conditional covariance between inflation and the market portfolio return or consumption growth and Siegel's paradox.

(8) Again, this cointegration restriction represents a necessary but not sufficient condition for UIP to hold. It is possible that a time-varying but stationary risk premium exists in the foreign exchange market. Such a risk premium would violate the UIP relationship but still allow nominal interest rates to be cointegrated across countries.

(9) [PI] is only identified up to a matrix normalization. The normalization depicted in Equation 7 is convenient, as it implies a triangular representation of the system. Other normalizations are, however, possible and are often used in the cointegration literature. Although the parity conditions yield no direct relationship between [i.sub.i,t] and [[pi].sub.j,t+k], an indirect relationship can be easily derived.

(10) As a referee has pointed out. RIP may still hold if any two of the parity conditions hold while the other two parity conditions have disequilibria that just cancel each other, that is, the equilibrium errors would be I(1) and cointegrated. This would still imply a single common trend among the four variables. What if none of the four individual parity conditions are true'? For example, is it possible that Fisher deviations across countries just cancel each other out while EAPPP deviations just cancel out UIP deviations such that RIP holds? The answer is no. While this would imply two stationary linear combinations among the same four variables, the two relations are not uniquely identified. In this example, the following cointegrating relationships exist:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

These can always be normalized to reflect a just identified system as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This set of equilibria is inconsistent with the premise that Fisher deviations just cancel and UIP deviations just cancel EAPPP deviations. The theory does not yield two separately identified relationships in this case. If we can restrict [[beta].sup.*.sub.12] = [[beta].sup.*.sub.21] = 0, then RIP may still hold as a cointegrating relationship between nonstationary real interest rates. But these two restrictions further imply that the conditions for both EAPPP and UIP are valid. This is precisely what we test when we examine the individual parity conditions.

(11) See Papell (1998) and the references therein for a discussion of the panel unit root tests as applied to the PPP hypothesis.

(12) Due to space constraints, we only briefly describe each procedure and refer the reader to the relevant citations for a thorough development of each procedure.

(13) The size distortion in unit root tests associated with large negative moving average errors is particularly relevant for inflation rates as demonstrated in Crowder (1996).

(14) A finding of cointegration between nominal interest rates that is statistically different from the theoretical vector may be explained by such things as differential tax treatments or capital controls. In such a case, it may be more appropriate to conclude that a version of UIP holds. A similar conclusion applies to EAPPP and the Fisher relations. The sign and magnitude of the estimated cointegration vector parameters will facilitate such judgments.

(15) It may seem that a natural test of the parity conditions is to impose the equilibrium relationship and then test this for a unit root. This strategy, however, imposes common factor restrictions that the HW test does not. It is in this way that the HW test is a more powerful test of cointegration.

(16) The inflation rates are calculated as 12 times the log difference of the respective consumer price indices (CPIs). All CPI data are taken from the international finance statistics (IFS) database of the International Monetary Fund (IMF). The Eurocurrency deposit rates are from the Standard & Poors DRI database.

(17) The classic errors-in-variables bias created by using ex post measures of ex ante variables, as we do here, is dominated by the superconsistency result of the estimators and thus does not present a problem for our analysis.

(18) The covariates used in the analysis were the log differences of industrial production for Canada, the United States, the United Kingdom, Japan, and France and West Texas Intermediate oil price and unemployment from Canada, the United States and the United Kingdom. These variables were chosen on the basis of data availability over the entire sample period.

(19) Hansen's Table 4 shows that the CADF tests are correctly sized as well.

(20) Crowder (2001) presents simulation evidence of serious size distortion in the trace tests when the empirical model allows for linear trends in the data but the true data generating process has only nonzero means.

(21) Our notation follows Johansen (1994).

(22) The lag length in the VAR was chosen such that it was the minimum necessary to eliminate all statistically significant residual serial correlation. This resulted in a lag length of 12 in all cases except for Germany--Canada, which required 13 lags.

(23) These rolling estimates of the trace statistics are calculated using the model in Equation 6. with the short-run dynamics held constant and equal to their full sample estimates as suggested by Hansen and Johansen (1993).

(24) It turns out that the data for the Fisher relationships generally favor the [H.sub.2](r) specification, that is. no deterministic components in the equilibrium relationship. When this specification is used. the cointegration results are stronger and parameter restrictions are more consistent with the theoretical model.

(25) Even in this case. however, the estimated cointegrating relation is consistent with a Fisher effect that includes the effects of taxes on interest income, that is, the so-called Darby effect. See Crowder and Wohar (1999) for a more detailed discussion of this effect.

(26) Interestingly, this result is at odds with the German Fisher relation results, where the stationarity restriction on the German nominal interest rate can be rejected with a calculated LR statistic of 7.238. This discrepancy bears further examination but is outside the scope of the present analysis and so is left for future study.

(27) The real exchange rate changes between the countries in the present study all appear to be stationary when subjected to the same battery of unit root tests used in the rest of the analysis.

(28) See Kremers, Ericsson, and Dolado (1992) for a detailed discussion on this point.

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S. Young Chung * and William J. Crowder ([dagger])

* Division of Business Administration, Wonkwang University, 344-2 Shinyong-Dong, Iksan-Shi, Chollabuk-Do, 570-749, Korea; E-mail [e-mail].

([dagger]) Department of Economics, Box 19479, The University of Texas at Arlington, Arlington, TX 76019, USA; E-mail [e-mail]; corresponding author.

Received August 2003; accepted February 2004.