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On the linkage of real interest rates between the US and Canada: some additional empirical evidence
Journal of International Financial Markets, Institutions and Money 12 (2002) 279– 289 www.elsevier.com/locate/econbase
On the linkage of real interest rates between the US and Canada: some additional empirical evidence Hiroshi Yamada * Department of Economics, Hiroshima Uni6ersity, Kagamiyama, Higashi 1 -2 -1, Hiroshima 739 8525, Japan Received 16 June 2000; accepted 2 February 2002
Abstract This paper reexamines the linkage of real interest rates between the US and Canada. After examining the existence of a one-to-one long-run relationship between these two interest rates, we assess the degree of departure from the long-run relationship and the speed of adjustment to it following an exogenous shock in one of the markets. Our empirical results, based on data from approximately two decades, indicate that: (i) there exists a one-to-one long-run relationship and (ii) the extent of departure is small compared with the magnitude of a shock and the departure decays within a reasonably small number of periods. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Real interest rate linkage; Cointegrating relationship; Speed of adjustment JEL classification: F36
1. Introduction In this paper, we reexamine the linkage of real interest rates between the US and Canada using cointegration methods. We focus our attention on revealing the nature of the long-run equilibrium relationship between these two real interest rates and how important this relationship is in explaining the movement of individual * Tel.: +81-824-24-7214; fax: +81-824-24-7212. E-mail address: [email protected] (H. Yamada). 1042-4431/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved PII: S 1 0 4 2 - 4 4 3 1 ( 0 2 ) 0 0 0 0 7 - 0
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real interest rates. In order to accomplish our first task, we examine (i) whether or not there exists a long-run equilibrium relationship between the variables and (ii) whether the relationship, if it is found to exist, is a one-to-one correspondence. In addition, for our second task, we examine (iii) the degree of departure from the long-run relationship and the speed of adjustment to it following an exogenous shock in one of the markets. Evaluating the degree of integration between two different financial markets by testing for the equality of real interest rates across countries has attracted the attention of many researchers in the field of international finance. Examples in the 1980s include Mishkin (1984), Mark (1985), Cumby and Mishkin (1986) and Merrick and Saunders (1986).1 Examination of this hypothesis commenced in the 1990s using cointegration and stationarity tests. Examples include Goodwin and Grennes (1994), Chinn and Frankel (1995), Hutchison and Singh (1997) and Phylaktis (1999). Among these studies, a cointegration analysis of the linkage of real interest rates between the US and Canada was implemented by Goodwin and Grennes (1994) and they reported that (a) these two real interest rates are cointegrated and (b) the real interest rate differential is stationary. We firstly examine the validity of the results (a) and (b) stated above. Among their results, the first corresponds to our (i). Unlike previous works, we test this hypothesis by not allowing the existence of a linear time trend in the real interest rates. Because real interest rates apparently do not contain a linear time trend, our approach would thereby be expected to provide more reliable empirical results. This can be accomplished by the cointegration rank test shown in Johansen (1996) (Theorem 6.3).2 Among their results above, the second corresponds to our (ii). They examined the existence of a one-to-one long-run equilibrium relationship between real interest rates in the US and Canada by testing for stationarity of the real interest rate differential. Instead of their approach, we test this hypothesis more directly by testing the corresponding linear restriction on the cointegrating vector. Our approach is expected to provide alternative empirical evidence on this matter. We secondly try to assess the degree of departure from the long-run equilibrium relationship following an exogenous shock in one of the markets and to examine the speed of adjustment of these two real interest rates to the long-run equilibrium relationship. If the degree of departure is relatively small compared with the magnitude of an exogenous shock and the speed of convergence is sufficiently fast, the importance of the long-run relationship in explaining the movement of each real interest rate increases. Of the studies in the 1990s listed above, Hutchison and Singh (1997) and Phylaktis (1999) examined the speed of adjustment to the long-run relationship. The former deals with the real interest rate linkage between the US and Japan and the latter deals with market integration in the Pacific region. To the
1 Goodwin and Grennes (1994) (pp. 108–109) provide a brief survey of these studies. See also Phylaktis (1999) (p. 271). 2 Johansen (1996) (p. 99) also suggests employing this test statistic if it is clear there is no time trend in the variables.
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best of our knowledge, these linkages for the US and Canada have not yet been revealed. The remainder of the paper is organized as follows. Section 2 explains methodological issues. We show how the one-to-one long-run equilibrium relationship is characterized in a vector error-correction model and how the impulse response function of the cointegrating relationship is calculated. In Section 3 we discuss the data. In Section 4, we present the empirical results concerning the existence of the one-to-one long-run relationship. The results concerning the degree of departure and the speed of adjustment following an exogenous shock in one of the markets are also presented. A summary and conclusion are shown in Section 5. 2. Methodology
2.1. Econometric model cn us cn We suppose that Yt =(r us t , r t )%, where r t and r t denote the real interest rates in the US and Canada, respectively, is modeled as follows:
DYt = v +c1DYt − 1 +···+ ck − 1DYt − (k − 1) + hi%Yt − 1 + ut,
ut iidN(0, S),
t= 1,2,…,T
(1)
where both h and i are (2× r) full-column rank matrices and v is a bivariate constant vector. We further suppose that Yt is stationary (i.e. r= 2) or unit-root and r cn are both nonstationary (i.e. r =0,1). In the model, if r=1, then r us t t characterized as being cointegrated unit-root nonstationary series. The constant term v in Eq. (1) can be decomposed as v= h(h%h) − 1h%v +hÞ(h%ÞhÞ) − 1h%Þv,
(2)
where hÞ represents an orthogonal complement of h. As is well known, when h%Þv "0 and Yt is unit-root nonstationary (i.e. r=0, 1), Yt has a linear trend.3 Because including a linear trend in the real interest rates is inappropriate, we assume h%Þv= 0. Then, Eq. (1) can be rewritten as DYt = c1DYt − 1 +···+ ck − 1DYt − (k − 1) + hi*%Y* +ut, t−1
(3)
where i*%= (i%, l) and Y*t − 1 =(Y%t − 1, 1)% with l=(h%h) h%v. In this setting, r = 1 is a favorable result for the existence of a one-to-one long-run relationship. This indicates that these two real interest rates contain a common stochastic trend, h%Þts = 1 us. Consequently, as a first step, we test whether there exists only one cointegrating vector. To determine the cointegrating rank r in Eq. (3), we performed the trace test, which is shown in Johansen (1996) (Theorem 6.3). If this hypothesis is supported, we implement two subsequent analyses (i.e. (ii) and (iii) in Section 1). −1
3 4
See, for example, Johansen (1991) (pp. 1559 –1561). See, for example, Johansen (1992) (p. 387).
4
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2.2. Cointegrating relationship Firstly, we estimate the long-run equilibrium relationship between these two real cn interests, r us t and r t . The long-run relationship can be written with i%= (b1, b2) as cn b1r us t = − l− b2r t +pt,
(4)
where pt denotes a univariate stationary variable with mean zero. Here we assume that the constant term in the cointegrating space, l= 0. This is because we found that this constant term makes the long-run relationship between these two real interest rates unstable.5 If we normalize the cointegrating vector i as i =(b1, b2)%= (1, 0)% + (0, 1)%,
(5)
then = − 1 corresponds to the one-to-one long-run relationship. This parameter is consistently estimable through the use of the maximum likelihood technique, as shown in Johansen (1996) (Theorem 6.1). Our concern here is to see how close the estimate for is to − 1. Next, we test the hypothesis corresponding to the one-to-one long-run relationship. This can be formulated in terms of a linear restriction on the cointegrating vector. H0: i = (1,− 1)%,
H1: i " (1,− 1)%.
(6)
In order to test the hypothesis above, we utilize the likelihood ratio test (LRT) statistic shown in Johansen (1996) (Eq. 7.2). This test statistic is asymptotically distributed as 2 with 1 dof (Johansen, 1996, Theorem 7.2).
2.3. Impulse response analysis In addition to examining the long-run equilibrium relationship between these two real interests, we assess the degree of departure from the long-run equilibrium relationship following a shock in one of the markets and examine the speed of adjustment of these two real interest rates back to the long-run equilibrium relationship. From Granger’s representation theorem (Johansen, 1996, Theorem 4.2), Eq. (3) can be transformed as t
Yt =C % us +D(L)ut +x.
(7)
s=1
The explicit form of matrices C, D(L) and x are given, for example, in Johansen (1996) (pp. 49– 52). In the model above, D(L)ut is a bivariate stationary series. It should be noted here that C and x can be written as iÞC0 and iÞx0 with iÞ being 5
Further discussion can be found in footnote 7.
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an orthogonal complement of i, which indicates that the cointegrating vector, i, can annihilate the matrices C and x. Then, defining the lower triangular matrix S such that SS%= , the cointegrating relationship, i%Yt, can be represented as,
j i%Yt = % d*L mt , j
(8)
j=0 j =i%D(L)S. Accordingly, mt is a bivariate standard where mt = S − 1ut and d*L j Gaussian white noise and d*j ( j =0, 1, 2,…) are (1×2) row vectors. From the equation above, the impulse response function of the cointegrating relationship, (r us +r cn), to a unit shock in r us is given by the sequence,
d*10, d*11, d*12, d*13,… where d*ij denotes the i-th element of the (1× 2) row vector d*. j In exactly the same way, the impulse response function of the cointegrating relationship, − (r us + r cn), to a unit shock in r cn is given by the sequence, − d*20, −d*21, − d*22, − d*23,…. Our first concern here is to assess the degree of departure from the long-run equilibrium relationship following a one-unit shock in one of the markets. In other words, the values of the first two or three responses are of interest. Our second concern is to see how many months it takes for the impulse responses to decrease sufficiently. We estimate these impulse response functions of the cointegrating relationship using the maximum likelihood estimates for .
3. The data We examine monthly data over approximately two decades, from 1980 to 1998. More precisely, the observation period of the real interest rate data for each country is from 1980:1 to 1998:12. In order to obtain robust empirical results, we examine different sample ends, namely, 1992:12, 1993:12, . . . , and 1998:12. Following the existing literature, such as Cumby and Mishkin (1986), Goodwin and Grennes (1994) and Phylaktis (1999), the real interest rate (rt ) on a j period bond held until maturity is defined from the Fisher condition as rt = it −yt,
(9)
where it is the nominal j period interest rate and yt is the rate of inflation from t to t+ j. Nominal interest rates (it ) for the money-market rates (3-month TB rates for both countries) and 3-month Eurocurrency rates for each country are used in the analysis. These interest rate data are transformed from an annual basis to a 3-month basis. The TB rates are obtained from the IMF International Financial Statistics and the Eurocurrency rates are obtained from the OECD Main Economic Indicators. The inflation rates (yt ) are calculated from the consumer price indices (1995 =100) that are obtained from the International Financial Statistics.
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4. Empirical results
4.1. Tests for the cointegration rank Table 1 tabulates the results for the cointegration rank test.6 The label ‘10% critical value’ in the table denotes the 10% critical values for the trace test, which are taken from Johansen (1996) (Table 15.2). Panel A of Table 1 tabulates the results for the case of money-market rates. From this table, we see that the hypothesis of a zero cointegrating vector (i.e. r= 0) is rejected in every sample period at the 10% level of significance. For the hypothesis of, at most, one cointegrating vector (i.e. r 5 1), the trace test statistic uniformly fails to reject the hypothesis at the 10% level of significance. Panel B of Table 1 shows the results for the case of Eurocurrency rates. We see qualitatively the same empirical results as for the case of money-market rates. From these results we conclude that Johansen’s cointegration tests support the hypothesis that there exists a long-run equilibrium relationship between these two real interest rates.
4.2. Estimation and hypothesis testing for the cointegrating 6ector Next, we estimate the long-run equilibrium relationship between these two real interests and test the hypothesis corresponding to the one-to-one long-run relationTable 1 Trace test of the number of cointegrating vectors Sample period
1980/1–1992/12 1980/1–1993/12 1980/1–1994/12 1980/1–1995/12 1980/1–1996/12 1980/1–1997/12 1980/1–1998/12 10% Critical value
A. Money-market rates
B. Eurocurrency rates
r= 0
r51
r =0
r51
18.79* 19.61* 21.34* 23.80* 19.49* 18.01* 19.36* 17.85
2.87 3.02 3.87 4.12 4.37 4.28 4.98 7.52
19.34* 20.47* 22.76* 25.46* 21.31* 20.10* 21.57* 17.85
2.81 2.99 3.92 4.12 4.53 4.79 5.24 7.52
Critical values are taken from Johansen (1996) (Table 15.2). * Rejections of the null hypothesis at the 10% level of significance.
6
Examination of the cointegrating relationship among the variables requires the determination of an appropriate lag length k. We selected k using the Hannan – Quinn Criterion, setting the maximum lag at 12 and the minimum at 1. Note that, since we set the maximum age at 12, 1981:1 corresponds to t = 1 in Eq. (1). As a result, k = 10 is selected in every case apart from the sample periods 1 – 4 in the case of domestic money-market rates, for which k= 7 is selected. To treat all cases uniformly, we set k= 10. The detailed results concerning lag length selection are available from the author upon request.
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Table 2 Estimation and testing for the cointegrating vector Sample period
1980/1–1992/12 1980/1–1993/12 1980/1–1994/12 1980/1–1995/12 1980/1–1996/12 1980/1–1997/12 1980/1–1998/12
A. Money-market rates
B. Eurocurrency rates
.
LRT
P-value (%)
.
LRT
P-value (%)
−0.57 −0.54 −0.60 −0.57 −0.63 −0.66 −0.65
1.91 2.09 2.11 2.53 1.93 1.53 2.10
16.7 14.8 14.6 11.1 16.5 21.6 14.7
−0.79 −0.72 −0.79 −0.75 −0.81 −0.83 −0.82
0.78 1.14 1.00 1.40 0.89 0.68 0.94
37.7 28.5 31.7 23.7 34.5 40.8 33.1
The labels . and LRT denote, respectively, the maximum likelihood estimate for in Eq. (5) and the value of the likelihood ratio test statistic for testing the null hypothesis. H0: i =(1,−1)%. The test statistic is asymptotically distributed as 2 with 1 dof. The P value denotes a corresponding P-value to the LRT.
ship with one cointegrating vector.7 Panel A of Table 2 shows the empirical results in the case of money-market rates. The labels . and ‘LRT’ in the table denote, respectively, the maximum likelihood estimate for and the value of the likelihood ratio test statistic for testing the null hypothesis H0: i=(1, − 1)%. Let us recall that the estimator is consistent and that the test statistic is asymptotically distributed as 2 with 1 dof. The label ‘P-value’ in the table denotes the corresponding P-value to the LRT. From this table we see that the estimates of the parameter are uniformly negative and around −0.6. This suggests that, in equilibrium, a rise of 1% in the Canadian real interest rate is associated with an increase of 0.6% in the US real interest rate. From the table we can also find that P-values are uniformly above 10%. This means that the null hypothesis H0: i=(1, − 1)% cannot be rejected at the 10% level of significance. These two results reveal support for the one-to-one long-run relationship between these two real interest rates. The empirical results for the case of Eurocurrency rates are given in Panel B of Table 2. From this table, we see qualitatively the same empirical results as for the case of money-market rates. That is to say, empirical evidence for the one-to-one long-run relationship between these two real interest rates can be found. Here it is notable that the estimates of the parameter are uniformly around − 0.8, and this 7
At the beginning of this project, we estimated an error correction model without imposing the restriction on the constant term in the cointegrating space, such as l =0. Doing this, we found that estimates for become quite unstable. When we normalized i* as i*= (b1, b2, l)% =(1, 0, 0)%+ (0, I2)%(, l*)%, the estimates for were, for example, 66.1 and −10.2. On the other hand, we found in every case that the estimates for and l* are approximately equal in absolute value and their signs are opposite. These results seem to indicate that the existence of a constant term in the cointegrating space leads to unstable estimates. Consequently, we remove the constant term in the cointegrating space. Empirical results corresponding to not imposing the restriction on l are available from the author upon request.
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result is more favorable to the one-to-one long-run relationship than for the case of money-market rates. Although we can make no conclusion without statistical testing, these results indicate that the Eurocurrency market is more efficient than the domestic money market. In order to check the robustness of the empirical findings above in terms of sample periods, we conducted a supplementary analysis by splitting the sample period into the following three periods: (a) 1980– 1989, (b) 1985–1994 and (c) 1990–1998.8 We employed the same specification as the cases reported in Table 2. Table 3 shows the empirical results. As shown in the table, for cases (a) and (c), the qualitative results stated above do not change. On the other hand, for case (b), the estimates for depart slightly from − 1 compared with the other cases and more importantly, the null hypothesis H0: i = (1, − 1)%, which indicates the one-to-one long-run correspondence, was rejected at the conventional level of significance. The sample period (b) corresponds to the period during which two major international events (i.e. the Plaza Agreement and the Louvre Accord) took place, which may lead to this breakdown of the long-run relationship between these two real interest rates. It is noteworthy that the stable long-run relationship again appears in the 1990s, from which can be interpreted that there exist forces reestablishing the relationship.
4.3. Impulse response analysis for the estimated cointegrating relationship In order to assess the degree of departure from the long-run equilibrium relationship following a shock in one of the markets and to examine the speed of adjustment of these two real interest rates to the long-run equilibrium relationship, we estimate the impulse response function of the cointegrating relationship, (r us + Table 3 Estimation and testing for the cointegrating vector (supplementary analysis) Sample period
1980/1–1989/12 1985/1–1994/12 1990/1–1998/12
A. Money-market rates
B. Eurocurrency rates
.
LRT
P-value (%)
.
LRT
P-value (%)
−0.63 −0.34 −0.61
1.66 10.15 0.38
19.8 0.1 53.8
−0.90 −0.49 −0.66
0.24 7.09 0.71
61.9 0.8 40.0
The labels . and LRT denote, respectively, the maximum likelihood estimate for in Eq. (5) and the value of the likelihood ratio test statistic for testing the null hypothesis. H0: i =(1,−1)%. The test statistic is asymptotically distributed as 2 with 1 dof. The P value denotes a corresponding P-value to the LRT. 8 Conducting a supplementary analysis by splitting the sample period was suggested by the referee. The referee especially recommended that we show the sensitivity of the empirical findings with respect to some major international events, such as the Plaza Agreement and the Louvre Accord. We are indebted to the referee for this suggestion.
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Fig. 1. The solid line shows the impulse response function of (r us +r cn) to a US real interest rate shock and the dashed line shows the impulse response function of − (r us +r cn) to a Canadian real interest rate shock.
r cn), using the maximum likelihood estimates for shown in Table 2. The smaller the degree of departure and the faster the speed of convergence, the higher is the degree of the linkage between these two real interest rates. Fig. 1 shows the impulse response function of the error correction term for the case of Eurocurrency rates. The sample period used in Fig. 1 is 1980–1998.9 In this figure, the solid line shows the impulse response function of (r us + r cn) to the 1% US real interest rate shock and the dashed line shows the impulse response function of − (r us +r cn) to the 1% Canadian real interest rate shock. From the figure we see that, regardless of the source of the shock, (i) a 1% shock in one of the real interest rate leads to approximately a 0.25% departure from the long-run equilibrium relationship and (ii) within about 4 months the departure decays under 0.1%. These results can be viewed as further empirical evidence indicating that these two markets are closely linked and that the long-run equilibrium relationship is quite important in explaining the movement of individual real interest rates.
5. Summary and conclusion In this paper, we have reexamined the linkage of real interest rates between the US and Canada using the cointegration technique, for the period 1980– 1998. We 9 We calculated impulse response functions for all cases shown in Table 2 and found that the qualitative results do not change. The other figures are available from the author upon request.
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examined whether or not there exists a long-run equilibrium relationship between them, and tested the hypothesis corresponding to the one-to-one long-run relationship. In addition, we assessed the degree of departure from the long-run equilibrium relationship following a shock in one of the markets and examined the speed of adjustment of these two real interest rates to the long-run equilibrium relationship. Our empirical findings are summarized as follows. (i) There exists a long-run equilibrium relationship between these two real interest rates. In other words, these two variables contain a common stochastic trend. Moreover, the relationship is consistent with a nearly one-to-one long-run correspondence. More precisely, the parameter estimates are around − 0.6 in the case of money-market rates and around − 0.8 in the case of Eurocurrency rates and, in addition, the null hypothesis H0: i=(1, − 1)% cannot be rejected at the 10% level of significance using Johansen’s likelihood ratio test statistic. These results indicate that the empirical findings stated in Goodwin and Grennes (1994) are confirmed by our alternative and more desirable approach. It should be noted here that our empirical results indicate this relationship broke down temporarily in the late 1980s. This is possibly due to two major international events, the Plaza Agreement and the Louvre Accord. (ii) Although an exogenous shock in one of the markets leads to departure from the long-run equilibrium relationship, the degree of departure relative to the magnitude of the exogenous shock is small enough. More precisely, a 1% shock in one of the real interest rates leads to approximately a 0.25% departure from the relationship. Moreover, the departure decays sufficiently (i.e. under 0.1%) within a reasonable time period of about 4 months. These results can be seen as further empirical evidence indicating that these two markets are closely linked. Our empirical results indicate that there exists a strong linkage between these two financial markets. In particular, the results (ii) stated above tell us that the nearly one-to-one long-run equilibrium relationship has quite an important role in explaining the movement of individual real interest rates in the US and Canada. Acknowledgements I wish to thank Professor Eiji Fujii for valuable comments on an earlier version of this paper. I also thank an anonymous referee for useful suggestions and comments. This paper was completed while I was vising the Institute of Economics, University of Copenhagen. I am grateful to Professor Hans C. Kongsted for his hospitality and support. This research was supported, in part, by the Zengin Foundation for Studies on Economics and Finance, the Danish Rectors’ Conference and the Japan Society for Promotion of Science. References Chinn, M.D., Frankel, J.A., 1995. Who drives real interest rates around the Pacific Rim: the USA or Japan? Journal of International Money and Finance 14, 801 – 821.
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Cumby, R.E., Mishkin, F., 1986. The international linkage of real interest rates: the European – US connection. Journal of International Money and Finance 5, 5 – 23. Goodwin, B.K., Grennes, T.J., 1994. Real interest rate equalization and the integration of international financial markets. Journal of International Money and Finance 13, 107 – 124. Hutchison, M.M., Singh, N., 1997. Equilibrium real interest rate linkages: the United States and Japan. Journal of the Japanese and International Economies 11, 208 – 227. Johansen, S., 1991. Estimation and hypothesis testing of cointegration vectors in Gaussian autoregressive models. Econometrica 59 (6), 1551 – 1580. Johansen, S., 1992. Determination of cointegration rank in the presence of a linear trend. Oxford Bulletin of Economics and Statistics 54, 383 –397. Johansen, S., 1996. Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press, Oxford. Mark, N.N., 1985. Some evidence on international inequality of real interest rates. Journal of International Money and Finance 4, 189 –208. Merrick, J.J., Saunders, A., 1986. International expected real interest rates: new tests of the parity hypothesis and US fiscal-policy effects. Journal of Monetary Economics 18, 313 – 322. Mishkin, F., 1984. Are real interest rates equal across countries? An empirical investigation of international parity conditions. Journal of Finance 39, 1345 – 1357. Phylaktis, K., 1999. Capital market integration in the Pacific Basin region: an impulse response analysis. Journal of International Money and Finance 18, 267 – 287.
On the linkage of real interest rates between the US and Canada: some additional empirical evidence Hiroshi Yamada * Department of Economics, Hiroshima Uni6ersity, Kagamiyama, Higashi 1 -2 -1, Hiroshima 739 8525, Japan Received 16 June 2000; accepted 2 February 2002
Abstract This paper reexamines the linkage of real interest rates between the US and Canada. After examining the existence of a one-to-one long-run relationship between these two interest rates, we assess the degree of departure from the long-run relationship and the speed of adjustment to it following an exogenous shock in one of the markets. Our empirical results, based on data from approximately two decades, indicate that: (i) there exists a one-to-one long-run relationship and (ii) the extent of departure is small compared with the magnitude of a shock and the departure decays within a reasonably small number of periods. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Real interest rate linkage; Cointegrating relationship; Speed of adjustment JEL classification: F36
1. Introduction In this paper, we reexamine the linkage of real interest rates between the US and Canada using cointegration methods. We focus our attention on revealing the nature of the long-run equilibrium relationship between these two real interest rates and how important this relationship is in explaining the movement of individual * Tel.: +81-824-24-7214; fax: +81-824-24-7212. E-mail address: [email protected] (H. Yamada). 1042-4431/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved PII: S 1 0 4 2 - 4 4 3 1 ( 0 2 ) 0 0 0 0 7 - 0
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real interest rates. In order to accomplish our first task, we examine (i) whether or not there exists a long-run equilibrium relationship between the variables and (ii) whether the relationship, if it is found to exist, is a one-to-one correspondence. In addition, for our second task, we examine (iii) the degree of departure from the long-run relationship and the speed of adjustment to it following an exogenous shock in one of the markets. Evaluating the degree of integration between two different financial markets by testing for the equality of real interest rates across countries has attracted the attention of many researchers in the field of international finance. Examples in the 1980s include Mishkin (1984), Mark (1985), Cumby and Mishkin (1986) and Merrick and Saunders (1986).1 Examination of this hypothesis commenced in the 1990s using cointegration and stationarity tests. Examples include Goodwin and Grennes (1994), Chinn and Frankel (1995), Hutchison and Singh (1997) and Phylaktis (1999). Among these studies, a cointegration analysis of the linkage of real interest rates between the US and Canada was implemented by Goodwin and Grennes (1994) and they reported that (a) these two real interest rates are cointegrated and (b) the real interest rate differential is stationary. We firstly examine the validity of the results (a) and (b) stated above. Among their results, the first corresponds to our (i). Unlike previous works, we test this hypothesis by not allowing the existence of a linear time trend in the real interest rates. Because real interest rates apparently do not contain a linear time trend, our approach would thereby be expected to provide more reliable empirical results. This can be accomplished by the cointegration rank test shown in Johansen (1996) (Theorem 6.3).2 Among their results above, the second corresponds to our (ii). They examined the existence of a one-to-one long-run equilibrium relationship between real interest rates in the US and Canada by testing for stationarity of the real interest rate differential. Instead of their approach, we test this hypothesis more directly by testing the corresponding linear restriction on the cointegrating vector. Our approach is expected to provide alternative empirical evidence on this matter. We secondly try to assess the degree of departure from the long-run equilibrium relationship following an exogenous shock in one of the markets and to examine the speed of adjustment of these two real interest rates to the long-run equilibrium relationship. If the degree of departure is relatively small compared with the magnitude of an exogenous shock and the speed of convergence is sufficiently fast, the importance of the long-run relationship in explaining the movement of each real interest rate increases. Of the studies in the 1990s listed above, Hutchison and Singh (1997) and Phylaktis (1999) examined the speed of adjustment to the long-run relationship. The former deals with the real interest rate linkage between the US and Japan and the latter deals with market integration in the Pacific region. To the
1 Goodwin and Grennes (1994) (pp. 108–109) provide a brief survey of these studies. See also Phylaktis (1999) (p. 271). 2 Johansen (1996) (p. 99) also suggests employing this test statistic if it is clear there is no time trend in the variables.
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best of our knowledge, these linkages for the US and Canada have not yet been revealed. The remainder of the paper is organized as follows. Section 2 explains methodological issues. We show how the one-to-one long-run equilibrium relationship is characterized in a vector error-correction model and how the impulse response function of the cointegrating relationship is calculated. In Section 3 we discuss the data. In Section 4, we present the empirical results concerning the existence of the one-to-one long-run relationship. The results concerning the degree of departure and the speed of adjustment following an exogenous shock in one of the markets are also presented. A summary and conclusion are shown in Section 5. 2. Methodology
2.1. Econometric model cn us cn We suppose that Yt =(r us t , r t )%, where r t and r t denote the real interest rates in the US and Canada, respectively, is modeled as follows:
DYt = v +c1DYt − 1 +···+ ck − 1DYt − (k − 1) + hi%Yt − 1 + ut,
ut iidN(0, S),
t= 1,2,…,T
(1)
where both h and i are (2× r) full-column rank matrices and v is a bivariate constant vector. We further suppose that Yt is stationary (i.e. r= 2) or unit-root and r cn are both nonstationary (i.e. r =0,1). In the model, if r=1, then r us t t characterized as being cointegrated unit-root nonstationary series. The constant term v in Eq. (1) can be decomposed as v= h(h%h) − 1h%v +hÞ(h%ÞhÞ) − 1h%Þv,
(2)
where hÞ represents an orthogonal complement of h. As is well known, when h%Þv "0 and Yt is unit-root nonstationary (i.e. r=0, 1), Yt has a linear trend.3 Because including a linear trend in the real interest rates is inappropriate, we assume h%Þv= 0. Then, Eq. (1) can be rewritten as DYt = c1DYt − 1 +···+ ck − 1DYt − (k − 1) + hi*%Y* +ut, t−1
(3)
where i*%= (i%, l) and Y*t − 1 =(Y%t − 1, 1)% with l=(h%h) h%v. In this setting, r = 1 is a favorable result for the existence of a one-to-one long-run relationship. This indicates that these two real interest rates contain a common stochastic trend, h%Þts = 1 us. Consequently, as a first step, we test whether there exists only one cointegrating vector. To determine the cointegrating rank r in Eq. (3), we performed the trace test, which is shown in Johansen (1996) (Theorem 6.3). If this hypothesis is supported, we implement two subsequent analyses (i.e. (ii) and (iii) in Section 1). −1
3 4
See, for example, Johansen (1991) (pp. 1559 –1561). See, for example, Johansen (1992) (p. 387).
4
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2.2. Cointegrating relationship Firstly, we estimate the long-run equilibrium relationship between these two real cn interests, r us t and r t . The long-run relationship can be written with i%= (b1, b2) as cn b1r us t = − l− b2r t +pt,
(4)
where pt denotes a univariate stationary variable with mean zero. Here we assume that the constant term in the cointegrating space, l= 0. This is because we found that this constant term makes the long-run relationship between these two real interest rates unstable.5 If we normalize the cointegrating vector i as i =(b1, b2)%= (1, 0)% + (0, 1)%,
(5)
then = − 1 corresponds to the one-to-one long-run relationship. This parameter is consistently estimable through the use of the maximum likelihood technique, as shown in Johansen (1996) (Theorem 6.1). Our concern here is to see how close the estimate for is to − 1. Next, we test the hypothesis corresponding to the one-to-one long-run relationship. This can be formulated in terms of a linear restriction on the cointegrating vector. H0: i = (1,− 1)%,
H1: i " (1,− 1)%.
(6)
In order to test the hypothesis above, we utilize the likelihood ratio test (LRT) statistic shown in Johansen (1996) (Eq. 7.2). This test statistic is asymptotically distributed as 2 with 1 dof (Johansen, 1996, Theorem 7.2).
2.3. Impulse response analysis In addition to examining the long-run equilibrium relationship between these two real interests, we assess the degree of departure from the long-run equilibrium relationship following a shock in one of the markets and examine the speed of adjustment of these two real interest rates back to the long-run equilibrium relationship. From Granger’s representation theorem (Johansen, 1996, Theorem 4.2), Eq. (3) can be transformed as t
Yt =C % us +D(L)ut +x.
(7)
s=1
The explicit form of matrices C, D(L) and x are given, for example, in Johansen (1996) (pp. 49– 52). In the model above, D(L)ut is a bivariate stationary series. It should be noted here that C and x can be written as iÞC0 and iÞx0 with iÞ being 5
Further discussion can be found in footnote 7.
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an orthogonal complement of i, which indicates that the cointegrating vector, i, can annihilate the matrices C and x. Then, defining the lower triangular matrix S such that SS%= , the cointegrating relationship, i%Yt, can be represented as,
j i%Yt = % d*L mt , j
(8)
j=0 j =i%D(L)S. Accordingly, mt is a bivariate standard where mt = S − 1ut and d*L j Gaussian white noise and d*j ( j =0, 1, 2,…) are (1×2) row vectors. From the equation above, the impulse response function of the cointegrating relationship, (r us +r cn), to a unit shock in r us is given by the sequence,
d*10, d*11, d*12, d*13,… where d*ij denotes the i-th element of the (1× 2) row vector d*. j In exactly the same way, the impulse response function of the cointegrating relationship, − (r us + r cn), to a unit shock in r cn is given by the sequence, − d*20, −d*21, − d*22, − d*23,…. Our first concern here is to assess the degree of departure from the long-run equilibrium relationship following a one-unit shock in one of the markets. In other words, the values of the first two or three responses are of interest. Our second concern is to see how many months it takes for the impulse responses to decrease sufficiently. We estimate these impulse response functions of the cointegrating relationship using the maximum likelihood estimates for .
3. The data We examine monthly data over approximately two decades, from 1980 to 1998. More precisely, the observation period of the real interest rate data for each country is from 1980:1 to 1998:12. In order to obtain robust empirical results, we examine different sample ends, namely, 1992:12, 1993:12, . . . , and 1998:12. Following the existing literature, such as Cumby and Mishkin (1986), Goodwin and Grennes (1994) and Phylaktis (1999), the real interest rate (rt ) on a j period bond held until maturity is defined from the Fisher condition as rt = it −yt,
(9)
where it is the nominal j period interest rate and yt is the rate of inflation from t to t+ j. Nominal interest rates (it ) for the money-market rates (3-month TB rates for both countries) and 3-month Eurocurrency rates for each country are used in the analysis. These interest rate data are transformed from an annual basis to a 3-month basis. The TB rates are obtained from the IMF International Financial Statistics and the Eurocurrency rates are obtained from the OECD Main Economic Indicators. The inflation rates (yt ) are calculated from the consumer price indices (1995 =100) that are obtained from the International Financial Statistics.
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4. Empirical results
4.1. Tests for the cointegration rank Table 1 tabulates the results for the cointegration rank test.6 The label ‘10% critical value’ in the table denotes the 10% critical values for the trace test, which are taken from Johansen (1996) (Table 15.2). Panel A of Table 1 tabulates the results for the case of money-market rates. From this table, we see that the hypothesis of a zero cointegrating vector (i.e. r= 0) is rejected in every sample period at the 10% level of significance. For the hypothesis of, at most, one cointegrating vector (i.e. r 5 1), the trace test statistic uniformly fails to reject the hypothesis at the 10% level of significance. Panel B of Table 1 shows the results for the case of Eurocurrency rates. We see qualitatively the same empirical results as for the case of money-market rates. From these results we conclude that Johansen’s cointegration tests support the hypothesis that there exists a long-run equilibrium relationship between these two real interest rates.
4.2. Estimation and hypothesis testing for the cointegrating 6ector Next, we estimate the long-run equilibrium relationship between these two real interests and test the hypothesis corresponding to the one-to-one long-run relationTable 1 Trace test of the number of cointegrating vectors Sample period
1980/1–1992/12 1980/1–1993/12 1980/1–1994/12 1980/1–1995/12 1980/1–1996/12 1980/1–1997/12 1980/1–1998/12 10% Critical value
A. Money-market rates
B. Eurocurrency rates
r= 0
r51
r =0
r51
18.79* 19.61* 21.34* 23.80* 19.49* 18.01* 19.36* 17.85
2.87 3.02 3.87 4.12 4.37 4.28 4.98 7.52
19.34* 20.47* 22.76* 25.46* 21.31* 20.10* 21.57* 17.85
2.81 2.99 3.92 4.12 4.53 4.79 5.24 7.52
Critical values are taken from Johansen (1996) (Table 15.2). * Rejections of the null hypothesis at the 10% level of significance.
6
Examination of the cointegrating relationship among the variables requires the determination of an appropriate lag length k. We selected k using the Hannan – Quinn Criterion, setting the maximum lag at 12 and the minimum at 1. Note that, since we set the maximum age at 12, 1981:1 corresponds to t = 1 in Eq. (1). As a result, k = 10 is selected in every case apart from the sample periods 1 – 4 in the case of domestic money-market rates, for which k= 7 is selected. To treat all cases uniformly, we set k= 10. The detailed results concerning lag length selection are available from the author upon request.
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Table 2 Estimation and testing for the cointegrating vector Sample period
1980/1–1992/12 1980/1–1993/12 1980/1–1994/12 1980/1–1995/12 1980/1–1996/12 1980/1–1997/12 1980/1–1998/12
A. Money-market rates
B. Eurocurrency rates
.
LRT
P-value (%)
.
LRT
P-value (%)
−0.57 −0.54 −0.60 −0.57 −0.63 −0.66 −0.65
1.91 2.09 2.11 2.53 1.93 1.53 2.10
16.7 14.8 14.6 11.1 16.5 21.6 14.7
−0.79 −0.72 −0.79 −0.75 −0.81 −0.83 −0.82
0.78 1.14 1.00 1.40 0.89 0.68 0.94
37.7 28.5 31.7 23.7 34.5 40.8 33.1
The labels . and LRT denote, respectively, the maximum likelihood estimate for in Eq. (5) and the value of the likelihood ratio test statistic for testing the null hypothesis. H0: i =(1,−1)%. The test statistic is asymptotically distributed as 2 with 1 dof. The P value denotes a corresponding P-value to the LRT.
ship with one cointegrating vector.7 Panel A of Table 2 shows the empirical results in the case of money-market rates. The labels . and ‘LRT’ in the table denote, respectively, the maximum likelihood estimate for and the value of the likelihood ratio test statistic for testing the null hypothesis H0: i=(1, − 1)%. Let us recall that the estimator is consistent and that the test statistic is asymptotically distributed as 2 with 1 dof. The label ‘P-value’ in the table denotes the corresponding P-value to the LRT. From this table we see that the estimates of the parameter are uniformly negative and around −0.6. This suggests that, in equilibrium, a rise of 1% in the Canadian real interest rate is associated with an increase of 0.6% in the US real interest rate. From the table we can also find that P-values are uniformly above 10%. This means that the null hypothesis H0: i=(1, − 1)% cannot be rejected at the 10% level of significance. These two results reveal support for the one-to-one long-run relationship between these two real interest rates. The empirical results for the case of Eurocurrency rates are given in Panel B of Table 2. From this table, we see qualitatively the same empirical results as for the case of money-market rates. That is to say, empirical evidence for the one-to-one long-run relationship between these two real interest rates can be found. Here it is notable that the estimates of the parameter are uniformly around − 0.8, and this 7
At the beginning of this project, we estimated an error correction model without imposing the restriction on the constant term in the cointegrating space, such as l =0. Doing this, we found that estimates for become quite unstable. When we normalized i* as i*= (b1, b2, l)% =(1, 0, 0)%+ (0, I2)%(, l*)%, the estimates for were, for example, 66.1 and −10.2. On the other hand, we found in every case that the estimates for and l* are approximately equal in absolute value and their signs are opposite. These results seem to indicate that the existence of a constant term in the cointegrating space leads to unstable estimates. Consequently, we remove the constant term in the cointegrating space. Empirical results corresponding to not imposing the restriction on l are available from the author upon request.
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result is more favorable to the one-to-one long-run relationship than for the case of money-market rates. Although we can make no conclusion without statistical testing, these results indicate that the Eurocurrency market is more efficient than the domestic money market. In order to check the robustness of the empirical findings above in terms of sample periods, we conducted a supplementary analysis by splitting the sample period into the following three periods: (a) 1980– 1989, (b) 1985–1994 and (c) 1990–1998.8 We employed the same specification as the cases reported in Table 2. Table 3 shows the empirical results. As shown in the table, for cases (a) and (c), the qualitative results stated above do not change. On the other hand, for case (b), the estimates for depart slightly from − 1 compared with the other cases and more importantly, the null hypothesis H0: i = (1, − 1)%, which indicates the one-to-one long-run correspondence, was rejected at the conventional level of significance. The sample period (b) corresponds to the period during which two major international events (i.e. the Plaza Agreement and the Louvre Accord) took place, which may lead to this breakdown of the long-run relationship between these two real interest rates. It is noteworthy that the stable long-run relationship again appears in the 1990s, from which can be interpreted that there exist forces reestablishing the relationship.
4.3. Impulse response analysis for the estimated cointegrating relationship In order to assess the degree of departure from the long-run equilibrium relationship following a shock in one of the markets and to examine the speed of adjustment of these two real interest rates to the long-run equilibrium relationship, we estimate the impulse response function of the cointegrating relationship, (r us + Table 3 Estimation and testing for the cointegrating vector (supplementary analysis) Sample period
1980/1–1989/12 1985/1–1994/12 1990/1–1998/12
A. Money-market rates
B. Eurocurrency rates
.
LRT
P-value (%)
.
LRT
P-value (%)
−0.63 −0.34 −0.61
1.66 10.15 0.38
19.8 0.1 53.8
−0.90 −0.49 −0.66
0.24 7.09 0.71
61.9 0.8 40.0
The labels . and LRT denote, respectively, the maximum likelihood estimate for in Eq. (5) and the value of the likelihood ratio test statistic for testing the null hypothesis. H0: i =(1,−1)%. The test statistic is asymptotically distributed as 2 with 1 dof. The P value denotes a corresponding P-value to the LRT. 8 Conducting a supplementary analysis by splitting the sample period was suggested by the referee. The referee especially recommended that we show the sensitivity of the empirical findings with respect to some major international events, such as the Plaza Agreement and the Louvre Accord. We are indebted to the referee for this suggestion.
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Fig. 1. The solid line shows the impulse response function of (r us +r cn) to a US real interest rate shock and the dashed line shows the impulse response function of − (r us +r cn) to a Canadian real interest rate shock.
r cn), using the maximum likelihood estimates for shown in Table 2. The smaller the degree of departure and the faster the speed of convergence, the higher is the degree of the linkage between these two real interest rates. Fig. 1 shows the impulse response function of the error correction term for the case of Eurocurrency rates. The sample period used in Fig. 1 is 1980–1998.9 In this figure, the solid line shows the impulse response function of (r us + r cn) to the 1% US real interest rate shock and the dashed line shows the impulse response function of − (r us +r cn) to the 1% Canadian real interest rate shock. From the figure we see that, regardless of the source of the shock, (i) a 1% shock in one of the real interest rate leads to approximately a 0.25% departure from the long-run equilibrium relationship and (ii) within about 4 months the departure decays under 0.1%. These results can be viewed as further empirical evidence indicating that these two markets are closely linked and that the long-run equilibrium relationship is quite important in explaining the movement of individual real interest rates.
5. Summary and conclusion In this paper, we have reexamined the linkage of real interest rates between the US and Canada using the cointegration technique, for the period 1980– 1998. We 9 We calculated impulse response functions for all cases shown in Table 2 and found that the qualitative results do not change. The other figures are available from the author upon request.
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examined whether or not there exists a long-run equilibrium relationship between them, and tested the hypothesis corresponding to the one-to-one long-run relationship. In addition, we assessed the degree of departure from the long-run equilibrium relationship following a shock in one of the markets and examined the speed of adjustment of these two real interest rates to the long-run equilibrium relationship. Our empirical findings are summarized as follows. (i) There exists a long-run equilibrium relationship between these two real interest rates. In other words, these two variables contain a common stochastic trend. Moreover, the relationship is consistent with a nearly one-to-one long-run correspondence. More precisely, the parameter estimates are around − 0.6 in the case of money-market rates and around − 0.8 in the case of Eurocurrency rates and, in addition, the null hypothesis H0: i=(1, − 1)% cannot be rejected at the 10% level of significance using Johansen’s likelihood ratio test statistic. These results indicate that the empirical findings stated in Goodwin and Grennes (1994) are confirmed by our alternative and more desirable approach. It should be noted here that our empirical results indicate this relationship broke down temporarily in the late 1980s. This is possibly due to two major international events, the Plaza Agreement and the Louvre Accord. (ii) Although an exogenous shock in one of the markets leads to departure from the long-run equilibrium relationship, the degree of departure relative to the magnitude of the exogenous shock is small enough. More precisely, a 1% shock in one of the real interest rates leads to approximately a 0.25% departure from the relationship. Moreover, the departure decays sufficiently (i.e. under 0.1%) within a reasonable time period of about 4 months. These results can be seen as further empirical evidence indicating that these two markets are closely linked. Our empirical results indicate that there exists a strong linkage between these two financial markets. In particular, the results (ii) stated above tell us that the nearly one-to-one long-run equilibrium relationship has quite an important role in explaining the movement of individual real interest rates in the US and Canada. Acknowledgements I wish to thank Professor Eiji Fujii for valuable comments on an earlier version of this paper. I also thank an anonymous referee for useful suggestions and comments. This paper was completed while I was vising the Institute of Economics, University of Copenhagen. I am grateful to Professor Hans C. Kongsted for his hospitality and support. This research was supported, in part, by the Zengin Foundation for Studies on Economics and Finance, the Danish Rectors’ Conference and the Japan Society for Promotion of Science. References Chinn, M.D., Frankel, J.A., 1995. Who drives real interest rates around the Pacific Rim: the USA or Japan? Journal of International Money and Finance 14, 801 – 821.
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