An examination of real interest rates in the United States, Canada, France, and Germany during the recent floating exchange rate period

Review of Quantitative Finance and Accounting, 6 (1996): 277-292 �9 1996 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.

An Examination of Real Interest Rates in the United States, Canada, France, and Germany during the Recent Floating Exchange Rate Period

AJAY PATEL Babcock Graduate School of Management, Wake Forest University, 14qnston-Salem, NC 27109

SRINIVAS R. AKELLA Department of Economics and Finance, University of New Orleans, New Orleans, I.,4 70148

Abstract. This study reexamines the international linkage of e~-ante real interest rates using the theory of cointegrated processes. The univariate unit root tests suggest the existence of a nonstationary real interest rate in the United States, Canada, and (the former) West Germany. An ex-ante real interest rate is obtained by subtracting estimates of inflation from the nominal intew.st rate. The expected inflation rates are oblained by modeling changes in monthly CPI values as autoregressive moving average (ARMA) processes. A multivariate test for unit roots indicates that there are two cointegrating vectors, or one common stochastic trend, for the system of three nonstationary real interest rates. In addition, the log-likelihood ratio test fidls to reject the null hypothesis that, in the long run, real interest rates in the United States are equal to those in Canada and West Germany.

Key words: real interest rate, inflation, multivariate tests

1. Introduction

The international linkage of real interest rates is important because of its implications about the ability o f central banks to conduct independent monetary policy, and for the insight it provides about the integration of capital markets. For central bank monetary policy to have any impact on liquidity in the domestic economy, real interest rates across countries need to be independent of one another. I f capital markets are segmented, real interest rates across countries would be independent o f one another, since they would be determined by local demand and supply conditions. If, however, capital markets are integrated, capital flows should ensure that real interest rates across countries remain closely linked together even in the short run. The existence of this linkage of real interest rates may reduce the ability of central banks to isolate the impact of domestic monetary policy changes from that of the United States' major trading partners.

The concept of an international linkage of real interest rates across countries with unseg- mented capital markets is intuitively appealing given the size of the spot foreign exchange market, where current daily volume exceeds $I trillion and under 10% is related to the movement of goods and services. Since the foreign exchange market is driven by capital chasing risk-adjusted yields, real interest rates across countries should be related. In fact, for fully integrated capital markets, real interest rates across countries should be driven toward equality with the levels being determined by global demand and supply conditions.

278 A. PATEL AND S.R. AKELLA

Previous studies of the international linkage of real interest rates have focused on, and rejected, the null hypothesis that real interest rates are equal across countries. These studies rejecting the equality of real interest rates use methodologies that require the economic variables being examined to be stationary through time. Because the assumption of stationarity is important, other researchers have examined whatever economic and financial time series exhibit stationary or nonstationary trends. More recently, researchers have begun examining the time series properties of real interest rates. While the empirical findings on the nonstationarity of real interest rates are inconclusive, the issue has important statistical and modeling implications for tests of the equality of real interest rates. This is because, in general, nonstationary variables may not be related in the long run. They may, however, exhibit a long-run equilibrium if they are cointegrated. If nonstationary variables are found to be cointegrated, one can test the restriction that they are equal in the long run.

Rather than test for cointegration on a bilateral basis, recent theoretical developments permit multivariate estimation to determine the number of cointegrating vectors within a system of nonstationary variables. In addition, the number of common stochastic trends, or long-run components, can be determined. The common stochastic trends may be inter- preted as the sets of economic fundamentals driving the system of nonstationary variables. After estimating the cointegrating vectors, restrictions on the cointegrating vectors may be tested. In other words, one can test whether real interest rates are equal in the long run.

The purpose of this study is to reexamine the international linkage of real interest rates in the United States, Canada, France, and Germany by utilizing recent data and focusing on the theory of cointegrated processes. We first examine whether real interest rates in each of these countries are nonstationary. Our evidence, using univariate unit root tests on nominal interest rates and inflation rates for each country, suggests that during the sample period examined, real interest rates in the United States, Canada, and the former West Germany appear to be nonstationary. Next, we obtain estimates of the ex-ante real interest rate for these three countries by subtracting estimates of expected inflation from the nominal interest rate. Expected inflation rates are obtained by modeling changes in monthly CPI values as autoregressive moving average (ARMA) processes. A multvafiate test for unit roots suggests that there are two cointegrating vectors, or one common stochastic trend, for the system of real interest rates. In addition, the log-likelihood ratio test fails to reject the null hypothesis that, in the long run, real interest rates in the three countries are equal.

This article is organized into six sections, The next section contains a brief literature review. The third section presents the relevant theory. Section 4 motivates the multivariate tests by discussing the methodology and empirical findings of the univariate unit root tests. The fifth section presents the methodology and empirical results for the multivariate tests, and the final section contains the conclusions.

2. Literature review

In this section, we briefly survey the literature on international real rates of interest. The focus of this section is on prior research on the equality of real interest rates, and on recent studies examining the nonstationadty of the ex-ante real interest rate for the United States, Canada, and countries in the European Community.

Some of the studies examining the equality of real interest rates across countries are Mishkin (1984), Ctunby and Obstfdd (1984), Mark (1985), and Cumby and Mishkin (1986).

AN EXAMINATION OF REAL ESTATE INTEREST RATES 279

Mishkin (1984) analyzed quarterly real interest rates in the Euro deposit market from the second quarter of 1967 to the second quarter of 1979. The study first examined whether differences in quarterly ex-post real interest rates on Euro deposits were unforecastable at time t, given information available at t - 1. As stated by Mishkin, a rejection of the null hypothesis would be a rejection of the equality of real interest rates across countries. The results rejected the null hypothesis of equal real interest rates across countries. Mishkin also examined the equality of real interest rates by jointly estimating uncovered interest parity and ex-ante relative purchasing power parity. His results strongly rejected the null hypothesis.

Mark (1985) also examined real interest rates on Eurocurrency deposits, but used monthly observations between May 1973 and February 1982. His study differed from previous work by focusing on net-of-tax real interest rates across countries. Mark examined real interest rates on a monthly as well as a quarterly basis. His findings do not provide strong support for the hypothesis that net-of-tax real interest rates are equal across countries.

Rather than focus on real interest rate equality, Cumby and Mishkin (1986) examined the extent to which real interest rates in the major industrialized countries move together through time. The focus of their study was on (1) whether there has been a link between ex ante real interest rates in the United States and those in other countries, and (2) whether real interest rates within Europe were more closely linked together than they were with real interest rates in the United States. The study examined the real return on three-month Euro deposits and domestic money market instruments for eight industrialized countries from June 1973 until August 1983. Since the observations were drawn on a monthly basis, efficient parameter estimates were obtained using the two-step, two-stage least squares procedure of Cumby et al. (1983). The results indicated that real interest rates had climbed dramatically from the 1970s to the 1980s, and that, while real interest rates across countries were not equal, they were significantly positively related to each other. Moreover, real interest rates within Europe were not more closely related to each other than they were to the real interest rate in the United States.

While previous studies of the equality of real interest rates may have utilized different empirical methodologies, the data sets used in those studies have covered essentially the same time period; the data ends sometime in the first half of the 1980s. Moreover, there is at least one common assumption in each of these studies. Each of the studies assumes stationarity in the variables examined. Investigations of the stationarity assumption required to test the Fisher hypothesis, or the international linkage of real interest rates, have been fairly recent. In an important, yet controversial, paper Fama (1976) found that the T-bill market was efficient, that the Fisher hypothesis held in the short run, and that real interest rates in the United States were constant. Fama, however, did not explicitly test for stationarity or nonstationarity in nominal interest rates or inflation rates. Critics of Fama's article claimed that real interest rates were not constant. In fact, evidence suggested that the ex- post real interest rate was not distinguishable from a nonstationary (random walk) ex-ante real rate plus a stationary forecast error (Nelson and Schwert (1977), Garbade and Wachtel (1978), and Fama and Gibbons (1982)).

The methodology to test explicitly for a unit root (nonstationarity) was developed by Fuller (1976), and Dickey and Fuller (1979). Subsequently, theoretical advances by Dickey and Fuller (1981), Fuller, Hasza and Goebel (1981), Said and Dickey (1984), Phillips (1987)


and Phillips and Perron (1988) extended the original tests and are more robust to a wide variety of distributions. A review of the literature can be found in Dickey, Bell and Miller (1986) and Perron (1988). More recently, Granger (1986), and Engle and Granger (1987) developed tests for eointegration between nonstationary time series. The univariate cointegration tests were subsequently extended to their multivariate forms by Johansen (1988), Stock and Watson (1988) and Johansen and Juselius (1990). In addition, Johansen and Juselius (1992) developed a log-likelihood ratio test to investigate linear restrictions on the cointegrating vectors within systems of nonstationary variables.

In recent applications of the univariate methodologies, Rose (1988), and MacDonald and Murphy (1989) explicitly test for the presence of a unit root in nominal interest rate and inflation rate series. Then, they investigate the existence of cointegration between the two series. Both studies find evidence of a unit root in nominal interest rates. The finding is consistent for the United States and some countries in the European Community (EC), drawn over annual, quarterly, and monthly intervals. The two studies, however, yield conflicting results for inflation. MacDonald and Murphy find evidence of a unit root in quarterly rates of inflation, between 1955 and 1986, for the United States, United Kingdom, Canada, and Belgium. Rose, however, finds no evidence of a unit root in inflation drawn at annual (for the United States between 1892 and 1970) or quarterly intervals (for the United States and 17 other OECD countries between 1955 and 1985). The monthly inflation rates for the United States, however, in the post-1979 period indicate evidence of nonstationarity.

The evidence against cointegration between nominal interest rates and inflation rates is similar in the studies by Rose, and MacDonald and Murphy. In the investigation by Rose, the finding against cointegration is not surprising. Theoretically, a nonstationary interest rate series and a stationary inflation rate series cannot be cointegrated. In their investigation, MacDonald and Murphy found nonstationarity in both the nominal interest rate and inflation rate series. However, the data failed to reject the null hypothesis of no cointegration between the two series. Both studies do, however, imply a nonstationary real rate of interest.

Perron (1990) develops the limiting distribution for the test of a unit root in a nonstationary series with a shift in the mean, against the stationary alternative that also assumes a shift in the mean value of the series. The test assumes that the point at which the shift occurs is known to the researcher. Based on the limiting distribution, Perron simulates critical values for the test statistic that are based on the sample size and the point at which the break occurs in the time series. He then applies the test to investigate whether ex-post real interest rates between 1961:1 and 1986:3 are nonstationary in the United States. Using the third quarter of 1980 as the break point, Perron is able to reject the null hypothesis of a unit root against the stationary alternative with a shift in the mean value of the series. His results suggest that the previous finding of nonstationary real interest rates by Rose (1988) is driven by the sudden shift in the mean of the series.

More recently, Evans and Lewis (1995) reexamine the long-run relationship between nominal interest rates and inflation rates in the United States. They find that over the January 1947 to February 1987 period, nominal interest rates move less than one-for-one with inflation rates, giving the appearance that the real interest rate may be nonstationary. They attribute this finding to the existence of a peso problem in inflation rates. Evans and Lewis explicitly test for structural shifts in the inflation rate series and find that the process appears


to have undergone shifts in 1961 and 1974. After accounting for the shifts, through use of a Markov switching model, they are unable to reject the hypothesis that real interest rates in the United States are stationary.

The evidence on the stationarity or nonstationarity of real interest rates in the United States is inconclusive. Recent studies indicate that structural shifts in the inflation rate process may explain the conflicting findings. Therefore, examining the post-1974 period would be a way of avoiding the problems associated with structural shifts in the time series being analyzed. Moreover, in order to examine whether real interest rates across countries are linked together, the time period chosen should be one during which capital is allowed to flow freely across countries, chasing risk-adjusted yields. The recent floating exchange rate period may be characterized as one during which there are fewer capital market im- pediments relative to the Bretton Woods period. Therefore, the empirical evidence to data seems to suggest that an examination of the linkage of real interest rates across countries, using recent time-series methodologies, should be conducted using data over the post-1974 period.

3. The Fisher hypothesis

The Fisherian framework decomposes the nominal interest rate on one-period bonds into an expected real interest rate on the one-period bond, and the expected rate of inflation during the holding period. Thus,

(1 + tnomt+l) = (1 + treal:+l) (1 + tinfl:+O, (1)

where tnomt+l = the nominal interest rate between t and t + 1 on a one-period bond; treal~+l = the expected one-period real interest rate between t and t + 1; and tinflt3+l = the expected inflation rate between t and t + 1.

A log-linear transformation of equation (1) yields the following additive expression.

tRt+l = tre+l + tl:+l, (2)

where

tRt+l = ln(1 + tnomt+O,

tr(+l = ln(1 + treal~+l),

ti7+1 = ln(1 + tinflT+O.

Since the ex-ante real interest rate is unobservable, researchers are forced to work with the ex-post real interest rate. If the expected one-period inflation rate at time t, differs from

1 the observed inflation rate, at time t + 1, tlt+l, by a stationary forecast error, #t+l,

tit+ 1 = tI[+ l "4- ~ t + 1" ( 3 )


Combining equations (2) and (3) yields the following expression for the ex-post real interest rate,

,R~+I = trT+l + t I ,+ l

or

tR t+ l - ,I~+1 = trT+l - l~+1. (4)

Equation (4) states that the ex-post real interest rate differs from the ex-ante rate by a stationary inflation forecast error. Therefore, the time series properties of the ex-ante real interest rate depend on those of the nominal interest rate and ex-post inflation rate.

In order to compare real interest rates on one-period bonds denominated in different currencies, default and maturity risk of the securities being examined must be equal. A popular security used in previous studies is the interest rate on Euro deposits. Euro deposits are offshore securities issued by the same bank, hence have similar default risk. The nominal returns on Euro-deposit securities in different currencies are related through uncovered interest parity (UIP), or the international Fisher effect (IFE). Uncovered interest parity is said to hold if the nominal return to a resident of country A for purchasing a Euro security denominated in the currency of country B, equals that obtained from investing in a similar maturity Euro security denominated in the currency of country A. For this resident, since the nominal returns from the two investments are equal, the real returns are also equal. If UIP holds for residents of country A and country B, the parity relation suggests that the expected change in the real exchange rate must equal the difference in real interest rates across the two countries. In other words, if UIP holds, real interest rates across countries are only equal if the expected change in the real exchange rate is zero. If UIP does not hold, as suggested by Mishkin (1984), the nominal interest rate differential equals the expected change in the nominal exchange rate plus the foreign exchange risk premium. Moreover, the real interest rate differential must equal the expected change in the real exchange rate plus the foreign exchange risk premium. In other words, any time-series variation in the real interest rate differential must (1) depend on the time-series properties of each individual real interest rate series; and (2) equal the time-series variation in the sum of the expected change in the real exchange rate and the foreign exchange risk premium.

Given the recent evidence of nonstationarity in the real interest rate series, the international linkage of real interest rates needs reexamination. The question of importance, now, is not if the two nonstationary interest rate variables are equal, but whether these nonstationary processes move together through time. In the absence of capital market barriers, if real interest rates across countries are not equal, there exists an incentive for capital to flow to the country with the higher real interest rate. This flow of capital will cause a change in the real exchange rate and an increase in the real interest rate in the country with the lower rate. 2 This is expected to continue until equilibrium is restored, which is when exchange-adjusted real interest rates are equal across countries. These short-run dynamics (capital flows) should ensure that real interest rates across countries are related in the short run, and hence, linked together through time. For this linkage to exist, deviations in real interest rates must be stationary through time. For the deviations to be stationary, the sum of the expected change in the real exchange rate and the foreign exchange risk premium must also be stationary.


Recent research by Abuaf and Jorion (1990) and Kim (1990) indicate that PPP holds in the long run, or that the real exchange rate is stationary. The forward foreign exchange risk premium has also been documented to be a stationary process by Hakkio and Rush (1989). Since the sum of two stationary processes must be stationary, these findings suggest that differences in real interest rates must also be stationary. In other words, the two real interest rate series should not drift apart in the long run. When nonstationary variables do not drift apart from each other in the long run, they are said to be cointegrated, indi- caring a long-run equilibirum relationship between the nonstationary variables.

While the absence of capital market barriers and the resulting short-run dynamics may ensure that real interest rates across countries, have a long-run equilibrium relationship, the equality of real interest rates in the long run depends on whether the sum of the expected change in the real exchange rate and the foreign exchange risk premium equals zero in the long run. One way for the sum to be zero in the long run is for both the equilibruim expected change in the real exchange rate and the foreign exchange risk premium individually to equal zero in the long run. However, this is not a necessary condition for the equality of real interest rates in the long run. All that is needed is for the sum to equal zero in the long run.

Nonstationary real interest rates that are cointegrated may be driven by a common set of fundamentals, or, may be driven by independent country-specific factors. If real interest rates are cointegrated and equal in the long run, the equilibrium real interest rate is determined by global fundamentals, as opposed to country-specific factors. The system of real interest rates, then, is driven by a common stochastic trend, or a single set of economic fundamentals. On the other hand, if real interest rates are cointegrated but unequal in the long run, domestic monetary policy is able to affect real economic activity and there may be more than one common stochastic trend driving the system. If real interest rates are nonstationary but not cointegrated, the result would suggest that barriers to capital flows exist, since the real exchange rate and foreign exchange risk premium for most industrialized countries are expected to be stationary.

4. Are real interest rates nonstationary?

The focus of this article is on whether real interest rates are nonstationary and whether they are cointegrated and equal in the long run. The tests for nonstationarity are based on monthly inflation rates and one-month Euro-currency deposit rates for the United States, Canada, France, and West Germany. The time period covered is January 1975 through December 1989. The inflation rates are calculated from monthly Consumer Price Index (CPI) values. 3 The CPI values are obtained from the Citibase data tapes. The one-month Euro-currency deposit rates, obtained from Citicorp Database Service, are the LIBOR bid quotes from Barclay's Bank on the first trading day of each month. 4 Euro-currency deposit rates are used in this study, as opposed to domestic interest rates, to ensure that there are no risk, tax, or regulatory differences between the interest rate series. The data begins in 1975 for two specific reasons. Real interest rates across countries are more likely to exhibit a long-run equilibrium within a system of floating exchange rates, with little to no capital market barriers, than in a system of fixed exchange rates. By starting the data in


1975, we give the market sufficient time to adjust to the new floating exchange rate system. Furthermore, and more importantly, Evans and Lewis (1995) provide evidence of a structural shift in the inflation rate process in 1974. By beginning the data in 1975, we do not have the estimation and inference problems associated with structural shifts in the data that Perron (1990) and Evans and Lewis (1995) discuss. The univariate results of this study are based on 178 observations, except for those of Canada. The multivariate tests, however, are based on 157 observations:

In order to test whether the ex-ante real interest rate is nonstationary, we first test whether nominal interest rates and inflation rates are nonstationary in each of the countries. The unit root tests for nonstationarity are conducted using the adjusted t- and/Zstatistics described in Perron (1988). 6 The null hypothesis of a unit root in the data, with and without a drift term and allowing for a deterministic trend in the data, is tested against the stationary alternative. The tests are widely used at present and hence details about the statistics have been omitted. The critical values for the adjusted t-statistics are obtained from Fuller (1976), while those for the adjusted/Zstatistics are obtained from Dickey and Fuller (1981).

The test statistics are calculated for each of the four nominal interest rate and inflation rate series. Evaluation of the adjusted t- and F-statistics necessitates specifying a trunca- tion lag, l, corresponding to the maximum order of nonzero autocorrelations in the residuals. The statistics were computed for values of I ranging from 2 to 8, however, the results were found to be fairly similar. The results reported in the paper are for I = 3. A heteroscedasticity and autocorrelation consistent estimate of the variance-covariance matrix is obtained following Newey and West (1987).

The unit root test results for the nominal interest rate series are presented in panel A of table 1. The adjusted t- and F-statistics fail to reject the null hypothesis of a unit root for one-month nominal interest rates during the post-1974 time period, in the United States, Canada, and West Germany. However, one-month nominal interest rates in France are stationary. 7 In addition, first differences of the nonstationary series were found to be stationary. s The results in panel A are consistent with the findings of Rose (1988) and MacDonald and Murphy (1989) and provide strong evidence that one-month nominal interest rates during the 1975 to 1989 period, in the United States, Canada, and West Germany are nonstationary and integrated of order 1, or I(1) in the terminology of Engle and Granger (1987).

The results for the four monthly inflation rate series are presented in panel B of table 1. The adusted t- and F-statistics reject the null hypothesis of a unit root in monthly inflation rates for each of the four series between 1975 and 1989. Our results cannot be com- pared to those of Rose (1988) or Evans and Lewis (1995) since our samples periods are different.

The results in tattle 1 indicate that nominal interest rates are nonstationary and I(1) for all the countries except France, while inflation rates are statioanry in each country. Given nonstationary interest rates and stationary inflation rates, the two series cannot be cointegrated, i.e., a linear combination of the two variables cannot be covariance stationary. This finding suggests that the ex-post real interest rate is nonstationary. The ex-post real interest rate differs from the ex-ante real interest rate by the inflation forecast error. If the inflation forecast errors are stationary, then the results in table 1 suggest that the ex-ante real interest rate in the United States, Canada, and West Germany are nonstationary.


Table 1. Tests for unit roots in the logarithms of nominal interest rates and inflation rates a.

Panel A: One-Month Euro-Currency Deposit Rates b

Country n c Z(t&) Z(ta, ) Z(~bl) Z(t&) Z(~b3) 7__,(~2)

United States 180 -0.864 -2.239 2.508 -2.238 2.503 1.672

Canada 158 -0.326 -2.110 2.257 -2.121 2.241 1.521

France 180 -1.604 --4.163"* 10,207"* -4.212"* 10.174"* 6.813"*

West Germany 180 -0.577 -1.551 1.202 -1.583 1.721 1.1470

Panel B: Inflation Rates d

Country n Z(t D Z(ta.) Z(~l) Z(t&) Z(63) Z(I~2)

United States 178 -2.360* --4.216"* 11.470"* -4.943** 14.981"* 10.006"*

Canada 178 -2.566* -7.425** 77.083** -4.293** 122.657"* 81.758"*

France 178 -1.898 -3.974** 12.126"* -7.207** 28.727** 19.169"*

West Germany 178 -4.007** -6.172"* 33.146"* -10.998"* 43.072** 28.721"*

aThe exact forms of the six test statistics are found in Perron (1988) and Phillips and Parron (1988). The critical values at the 1% and 5% levels for Z(t6) are -2.60 and - 1.95, respectively. The critical values of Z(t~) at the 1% and 5 % levels are -3.51 and -2.89, respectively, while those for Z(@I) are 6.63 and 4.68, respectively. The critical values ofZ( td) , Z(@3) and Z(@2) at the 1% and 5% levels are -4.04 and -3.45, 8.63 and 6.44, and 6.41 and 4.84, respectively.

bThe data are the natural logarithms of one plus the LIBOR bid quotes from Barclay's Bank on the first trading day of each month.

CRepresents the number of observations in the time series. aThe data are natural logarithms of monthly changes in the Consumer Price Index. **Rejection of the null hypothesis of a unit root at the 1% level. *Rejection of the null hypothesis of a unit root at the 5 % level.

Estimates of the ex-ante real interest rate are obtained by subtracting estimates of expected inflation from the nominal interest rate. In this study, estimates of expected inflation are obtained from the ex-post changes in monthly CPI values through estimation of an ARMA(1,1) model for each of the three countries. 9 The ARMA0,1) model is used in this study to provide reasonable, though not necessari ly optimal, estimates o f ex-ante inflation. The focus of this art icle is not on the optimal model for expectations of inflation, but on whether ex-ante real interest rates are equal in the long run. The estimates of expected inflation are used to calculate the ex-ante real interest rate. We then test whether these estimates of the ex-ante real interest rate are nonstationary as suggested by the previous results.

The unit root test results for nonstationarity in the three ex-ante real interest rate series

are presented in table 2. The results suggest that, at the 1% level, the ex-ante real interest rate in the United States, Canada, and West Germany is nonstationary. The test statistics for equations with and without a drift are significant at the 5 % level for the United States and West Germany. The adjusted t- and F-statistics for the regression model with a t ime trend are insignificant at the 5 % level for all three countries. The test statistic Z(@3) is marginally significant for the United States. The results in table 2 suggest that ex-ante real interest rates in the United States, Canada, and West Germany are nonstationary and I(I) during the 1975 to 1989 period.


Table 2. Tests for unit roots in the ex-ante real interest rates, a

Country n c Z(t~,) Z(ta, ) Z(r Z(t~) Z(r Z(r )

United States 178 -1.99" -3.35* 6.36* -3.38 6.53* 4.37

Canada 157 -1.00 -2.44 3.04 -3.06 4.69 3.18

West Germany 178 -1.57 -3.22* 5.90* -3.28 4.03 5.99

aThe ex-post real interest rate between February 1975 and November 1989 is the difference between the natural logarithm of one plus the LIBOR bid quote from Barclay's Bank for one-month Eur~curreney deposit rates on the first Wading day of each month, and the natural logarithm of one plus monthly inflation. The forecasts of monthly inflation are obtained from the ex-post changes in the Consumer Price Index through estimation of an ARMA (1,1) model for each country. Details for the six test statistics are available in Perron (1988) and Phillips and Pert'on (1988). The critical values at the 1% and 5% levels for Z(t6) are -2.60 and -1.95, respectively. The critical values of Z(t~.) at the 1% and 5% levels are -3.51 and -2.89, respectively, while those for Z(~l) are 6.63 and 4.68, respectively. The critical values of Z(t6) , Z(cI,3) and Z(~2) at the 1% and 5% levels are -4.04 and -3.45, 8.63 and 6.44, and 6.41 and 4.84, respectively.

bRepresents the number of observations in the time series. eRejection of the null hypothesis of a unit root at the 5% level.

5. Are real interest rates cointegrated and equal in the long run?

Since the ex-ante real interest rate in each of the three countries exhibits I(1) behavior, we need to examine whether the three real interest rates are cointegrated and equal in the long run. The multivariate analysis would also indicate the number of common stochastic trends driving the system of three ex-ante real interest rates, l0

This issue is investigated through Johansen's (1988) multivariate procedure that estimates the number of common unit roots, or cointegrating vectors, within the system of nonstationary variables. The trace test for at most r cointegrating vectors, or at least ( p - r) common stochastic trends, in a system of p nonstationary variables is

-21nQr = - T ~ t = r + 1 ln(1 - ~i)]. (5)

The maximal eigenvalue statistic for testing the null hypothesis of (r - 1) cointegrating vectors against the alternative that there are r cointegrating vectors is given by

Xmax = - T [ l n ( 1 - ~r)]. (6)

Johansen and Juselius (1990) note, "One would, however, expect the power of this procedure [the trace test] to be low, since it does not use the information that the last three eigenvalues have been found not to differ significantly from zero. Thus one would expect the maximum eigenvalue test to produce more clear cut results." The Xi s are the squared canonical correlations between the two sets of residual vectors, Rot and Rlt, obtained from the following two regressions:

AYt = ~Z 1 ro,i AYt-i + Rot (7)

Y t - k = ~ii-~ 1 F l , i A Y t - i + Rlt (8)


where Yt is the p-vector of variables and l~jt are matrices of coefficient estimates. I f r = 0 cannot be rejected, there are no cointegrating relationships among the p variables in Yr. I f r = p cannot be rejected, then the hypothesis that Yt is a stationary process cannot be rejected. This may be thought of as a multivariate version of the augmented Dickey-Fuller test of the unit root hypothesis. Therefore, the test results provide evidence in favor of cointegration only in the case when p > r > 0. The critical values for the test statistics are available in Johansen and Juselius (1990).

Table 3 reports the results along with the 95 % quantiles. The order of the AR process (k) in Yt is taken as 3. The use of other values does not affect the results significantly. The results that have been reported in table 3 assume a linear deterministic trend in the variables in Yr. For the trace test, the null hypothesis that r < = 2 cannot be rejected, at the 5 % level of significance, while the hypothesis that r = 0 and r < = 1 can be rejected. Consequently, we conclude that there are two cointegrating vectors.

Turning to the maximal eigenvalue test, the hypothesis that r = 0 is rejected in favor of r > = 1. Similarly, the hypothesis that r < = 1 is also rejected in favor of r > = 2. However, the maximal eigenvalue test fails to reject the null hypothesis of r < = 2 in favor of r = 3. Thus, the two tests are consistent in that they both suggest that there are two cointegrating vectors, or long-run equilibrium relationships within the system of three real interest rates. In other words, we cannot reject the hypothesis that there is a single common stochastic trend driving the system of real interest rates.

Our results suggest that there is a single set of economic fundamentals driving real interest rates in the long run. This single set of economic fundamentals that drives the system of real interest rates in the long run must be globally determined. Otherwise, if country- specific fundamentals determined real interest rates in the long run, the results would have indicated more than one common stochastic trend within the system of three real interest rates. Given that there are two long-run equilibrium relationships in the system, the results suggest that real interest rates in the United States are cointegrated with those in Canada and West Germany.11

Table 3. Multivariate tests for unit roots in the logarithms of ex-ante real interest rates, a

Maximal b

Null Alternative Trace b 95 % Null Alternative Eigenvalue 95 % Hypothesis Hypothesis Test Quantile Hypothesis Hypothesis Test Quantile

r = 0 r > = 1 57.63* 31.53 r = 0 r = 1 32.90* 21.07

r < = 1 r > = 2 24.74* 17.95 r < = 1 r = 2 20.02* 14.90

r < = 2 r = 3 4.72 8.18 r < = 2 r = 3 4.72 8.18

aThe ex-ante real interest rate between November 1976 and November 1989 is the difference between the natural logarithm of one plus the LIBOR bid quote on one-month Euro-currency deposit rates from Barclay's Bank on the first trading day of each month, and the natural logarithm of one plus monthly inflation. The estimates of expected monthly inflation are obtained from the changes in the Consumer Price Index through estimation of an ARMA(1,1) model for each country.

bDetafls for the trace test and the maximal eigenvalue test are available in Johansen (1988) and Johansen and Jusefius (1990). The statistics test for the number of linearly independent cointegrating vectors r, or equivalently the number of common stochastic trends (3 - r), in a VAR(3) for the set of three real interest rates. The estimation process assumes trended variables, but no trend in the data generation process.

*Significant at the 5 % level.


The results in table 4 report estimates of the cointegrating vectors for the system of real interest rates. The coefficients of the normalized vectors (in brackets), suggest that cointegrating vector 1 describes the long-run equilibrium between the United States and West Germany. Similarly, cointegrating vector 2 appears to describe the long-run relationship between real interest rates in the United States and Canada. To examine whether real interest rates are equal in the long run, we need to test the restrictions imposed on the cointegrating vectors. The restrictions imposed on vector 1 are that the normalized coefficients for the United States, Canada, and Germany equal minus one, zero, and one, respectively. The restrictions imposed on cointegrating vector 2 are that the normalized coefficients for the United States, Canada, and Germany equal minus one, one, and zero, respectively. The test follows hypothesis H 5 outlined in Johansen and Juselins (1992). The log- likelihood ratio test statistic is asymptotically distributed as a chi-squared variate with two degrees of freedom.

The results for the joint test are striking. The test statistic cannot reject the restrictions imposed on the cointegrating vectors at the 5 % level of significance. The results suggest that real interest rates in the United States are cointegrated with those in Canada and West Germany. In addition, in the long run, equilibrium real interest rates during the floating exchange rate period are equal across the three countries.

The multivariate results, while striking, are internally consistent. If real interest rates are equal in the long run across the United States, Canada, and West Germany, the equilibrium real interest rate must be determined by aggregate real demand and supply at the global, or multinational, level as opposed to the country-specific level. This suggests that real interest rates in the long run, across the three countries studied, are driven by the same set of economic fundamentals, i.e., there is a single common stochastic trend in the system of real interest rates. The multivariate results in this paper suggest such a scenario.

Table 4. Test for the equality of real interest rates)

Cointegrating Vector 1 Cointegrating Vector 2 Country (Normalized Vector) (Normalized Vector) X2(2) b

United States 43.1202 28.5270 4.849 (-1.000) (-1.000)

Canada 1.8608 -50.9068 (-0.043) (I .785)

West Germany -66.1513 24.9961 (1.534) (-0.876)

aThe cointegmting vectors are obtained through Johansen's (1988) multivariate test for unit roots on the system of monthly ex-ante real interest rates between November 1976 and November 1989. The eointegrating vectors have been normalized on the United States. Each vector represents a long-run equilibrium relationship.

bThe test for the equality of real interest rates imposes restrictions on the values of the cointegrating vectors. Based on the values of the normalized cointegrating vectors, it appears that vector 1 represents equilibrium between real interest rates in the United States and West Germany, hence the restriction is [ - 1,0,1] '. Vector 2, then, must represent equilibrium between the United States and Canada, hence the resUiction is [-1,1,0] '. Johansen and Iuselius' (1992) log-likelihood ratio statistic, that jointly tests the restriction, is asymptotically distributed as a chi-squared variate with two-degrees of freedom. The critical value at the 5 % level is 5.99.


Our finding that real interest rates are equal in the long run have extremely different implications from a finding that real interest rates are equal at all times, or move one for one through time. If real interest rates are equal at all times, then monetary authorities have no control over the domestic real interest rate relative to the global real interest rate (Mishkin 1984). If real interest rates across countries move one for one, "an important avenue for monetary policy to influence the domestic economy emphasized in many open economy macroeconomic models is removed" (Cumby and Mishkin 1986). As opposed to these implications, our results in no way preclude domestic monetary policy from affecting real economic activity in the short run. Differences in monetary policy should result in different rates of real economic activity which should show up as differences in real interest rates. Real interest rate differences across countries in the short run should result in capital flows across countries. Recent estimates suggest that over 90% of the $1 trillion daily volume in the foreign exchange market is due to capital flows seeking higher risk-adjusted real interest rates. These flows provide empirical verification that short run differences do exist across countries. However, it is these short-run dynamics that ensure that real interest rates across countries are linked together even in the short run. Our results on the equality of real interest rates in the long run suggest that for industrialized countries with integrated capital markets, real interest rates in the long run are determined by global factors.

6. Conclusions

The use of the Perron (1988) and Phillips and Perron (1988) univariate tests, which are robust to the presence of serial correlation and time-dependent heteroscedasticity, provide strong evidence that one-month Euro-currency (deposit) interest rates in the United States, Canada, and West Germany are nonstationary and integrated of order 1, or I(1), while that in France is stationary. Similarly, the tests provide evidence that monthly inflation rates are stationary. The results suggest that the real interest rate in the United States, Canada, and West Germany is nonstationary. An implication of this result is that standard OLS inference of the Fisher effect is not valid, since the residuals are nonstationary.

The ex-ante real interest rate in the United States, Canada, and West Germany is obtained by subtracting estimates of expected inflation from the nominnal one-month Euro- currency interest rate. The estimates of expected inflation are obtained by modeling ex- post inflation rates as ARMA(1,1) processes. The ex-ante real interest rates, in each of the three countries, were found to be nonstationary through use of univariate unit root tests.

Application of Johansen's (1988) and Johansen and Juselius' (1990) multivariate test for unit roots indicates two cointegrating vectors, or long-run equilibrium relationships, in the system of nonstationary real interest rates. The results suggest that the three real interest rates are driven by a single common stochastic trend. Further, application of Johansen and Juselius' (1992) log-likelihood ratio test indicates that real interest rates in the three countries are equal in the long run. These findings of a single set of economic fundamentals driving real interest rates that are equal in the long run, while striking, contradict the findings in previous research. Thus, while many models assume that real interest rates across countries are driven by separate sets of fundamentals, our results suggest that there is a strong trend component binding the rates together in the long run.


A c k n o w l e d g m e n t s

The second author was at the Univers i ty o f New Orleans whi le the paper was being com-

pleted. Unfor tunate ly he passed away in 1994. This paper has benef i t ted f rom the com-

ments o f par t ic ipants at the annual meet ings of the F M A and SFA in 1992. In addit ion,

we especia l ly acknowledge the c o m m e n t s of RQFA edi tor C.F. Lee and two anonymous

referees.

Notes

1. Evans and Lewis (1995) find that inflation forecast errors may be biased ex-post with the bias due to the peso problem or learning effects; however, Aggarv, al et al. (1995) find that survey forecasts of consumer prices are stationary and rational. Even if the forecast crrors are biased ex-post, they should represent a stationary process.

2. The change in the level of the real exchange rate and the real interest rate will depend on the sensitivities of the variables to real shocks. For example, if an increase in foreign demand caused an increase in foreign real interest rates, the sensitivity of U.S. demand to the real exchange rate and the U.S. real interest rate would determine the magnitude of the change in the levels of the variables. If U.S. demand is more sensitive to the real interest rate, then smaller chan~es in the real interest rate would be necessary to restore equilibrium.

3. Data on Consumer Price Indexes were obtained from January 1975 through November 1989 for the United States, Canada, France, and West Germany. This resulted in inflation rates being available from February 1975 through November 1989, yielding 178 observations for the countries.

4. The one-month Euro-currency deposit rates obtained from Citicorp Database Service for the United States, France, and West Germany cover the period from January 1975 through December 1989. The data for Canada, however, begins in November 1976 and ends in December 1989. Therefore, there are 180 observations on nominal interest rates for the United States, France, and West Germany, and only 158 observations for Canada.

5. The multivariate tests require an equal number of observations for each of the countries. The data for Canada covers the smallest time period, hence, the multivariate tests are based on data over the period November 1976 through November 1989 resulting in 157 observations.

6. This study does not test for nonstationarity using the methdology of Perron (1990) because the test requires knowledge of the point in time at which the shift in the mean value of the series occurs. Since this study investigates the real interest rate in four countries, the use of a single break point for all four countries appears as inappropriate as assuming no break at all. Since the focus of this article is on whether real interest rates across countries move together through time and are equal in the long run, as opposed to whether they are stationary or nonstationary, the standard univariate and multivariate tests for unit roots are conducted.

7. The unique behavior of French data has been documented previously by Mishkin (1984). Given Mishkin's previous documentation, it is not surprising that French data behaves differently from that for the other three countries.

8. Unit root tests on the first differences of the nominal interest rote series tests whether the null hypothesis of two unit roots can be rejeced. The tests strongly rejected the presence of two unit roots in the three series. The results for the six test statistics have not been reported for space considerations, but are available from the authors. The ability to reject the null hypothesis of two unit roots is similar to the findings of Rose.

9. Since monthly changes in the Consumer Price Index in each of the countries was found to be stationary, or 1(0), the use of autoregressive integrated moving average models (ARIMA) was not necessary.

10. Univariate tests for cointegration were also conducted. Real interest rates in the United States were found to be cointegrated with those in Canada and West Germany. In addition, real interest rates in Canada were also found to be cointegrated to those in West Germany. Therefore, on a bilateral basis, all possible pairs of the three real interest rate series are cointegrated.

11. The multivariate results confirm the univariate results indicating cointegration between the three rates on a bilateral basis.


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Documents

An examination of real interest rates in the United States, Canada, France, and Germany during the recent floating exchange rate period