testing interest rate parity and rational expectations for canada and the united states

Testing Interest Rate Parity and Rational Expectations for Canada and the United StatesAuthor(s): Allan W. GregorySource: The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 20, No. 2(May, 1987), pp. 289-305Published by: Wiley on behalf of the Canadian Economics AssociationStable URL: http://www.jstor.org/stable/135362 .

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Testing interest rate parity and rational expectations for Canada and the United States ALLAN W. GREGORY University of Western Ontario and Queen's University

Abstract. The purpose of this paper is to examine the joint hypothesis of interest rate parity and rational expectations for Canada and the United States using monthly data. Two approaches to testing are considered: (1) the substitution method and (2) testing the cross-equation non-linear rational expectations restrictions from solving the interest rate parity equation. It is demonstrated that standard application of the substitution method requires the assumption that the interest rate parity equation holds deterministically. Tests for unit roots in the spot exchange rate and interest rate differentials are also investigated.

Un test de la parite des taux d'interet et des anticipations rationnelles pour le Canada et les Etats-Unis. L'objectif de cet article est d'examiner l'hypothese combinee de la parite des taux d'interet et des anticipations rationnelles pour le Canada et les Etats-Unis a I'aide de donnees mensuelles. Deux approches A la procedure de test sont examinees: (i) la methode de la substitution et (ii) le test des restrictions imposees par les anticipations rationnelles obtenues A partir de la solution de l'equation de parite des taux d'intere't. On montre que l'application standard de la methode de la substitution semble requerir une equation de la parite des taux d'interet valide d'une maniere deterministe. On examine aussi les tests des racines unitaires dans les differentiels de taux de change fermes et de taux d'interet.

INTRODUCTION

The joint assumption of uncovered interest rate parity and rational expectations has provided a convenient link between international financial assets in many theoretical open economy models. Although this assumption is not without appeal, there has been little research to establish its acceptability on empirical grounds. The tests that have been done are indirect in the sense that

The author would like to thank Jean-Marie Dufour, two referees, David Backus, Zvi Hercowitz, David Longworth, Michael McAleer, Tom McCurdy, Maurice Obstfeld, Michael Veall, and Stan Zin for helpful comments, suggestions, and criticisms.

Canadian Journal of Econornics Revue canadienne d'Economique, xx, No. 2 May mai 1987. Printed in Canada Imprime au Canada

0008-4085 / 87 / 289-305 $1.50 ( Canadian Economics Association



290 Allan W. Gregory

they have employed the substitution method (see Cumby and Obstfeld, 1981).1 For this approach, the expected value of the future spot rate is replaced by its actual future value, and under the null hypothesis of uncovered interest rate parity and rational expectations the ex post residuals from the parity equation should not be significantly different from white noise. A test of the null hypothesis then involves examining these residuals for serial correlation using various portmanteau tests.

Cumby and Obstfeld (1981) using Canadian/us weekly data for the period 1974-80 detected serial correlation in the ex post residuals and accordingly rejected the joint hypothesis of interest rate parity and rational expectations. One obvious advantage of this testing procedure is that it is 'model-free'; that is, no attempt is made to model the movements in exchange rates or the interest rate differential. However, the tests by themselves are not particularly informative since they do not indicate where specific weaknesses in the theory may lie. On the other hand, given that under rational expectations, the interest rate parity equation is itself a partial difference equation, a more direct and perhaps more revealing test is afforded by solving the equation using standard techniques and then testing the overidentifying rational expectations restrictions against the data. Of course, the usefulness of such tests must be evaluated in the context of the particular model presented.

Two further issues with respect to applying the substitution method of testing are important. The first concerns the stationarity of the ex post residuals. For examrrple, if the residuals are heteroskedastic then the Box-Pierce tests employed in Cumby and Obstfeld would not be valid and an alternative tests robust to heteroskedasticity should be calculated (for instance, the signed rank test for serial independence developed in Dufour, 1981). Following this line of reasoning, it is also worthwhile investigating time series properties such as the existence of unit roots in the individual variables (spot and interest rate differential) composing the residuals. Certainly any subsequent empirical modelling would require some joint stationarity assumption for these variables. Evidence recorded in Backus (1984) and others suggests that the spot rate is well approximated by a random walk.

The second important consideration in the substitution method of testing is the assumption that the interest parity relation itself holds identically and deterministically. However, Sims (1982) has argued that asset markets are never exactly in equilibrium, but instead, asset prices vibrate randomly around equilibrium. Yet if we relax this deterministic condition and permit serially independent mean zero departures from interest rate parity, then the substitution method does indeed produce ex post residuals that are correlated even under interest rate parity and rational expectations.2 This is despite the

I This method is discussed in Nelson (1975), McCallum (1976) and more recently in Pagan (1983). In contrast Hakkio (1981) has estimated the restricted structure in testing the term structure of the forward premium.

2 For that matter whenever a structural equation is stochastic and the substitution involves a



Testing interest rate parity 291

fact that the individual disturbances themselves are serially independent.3 It can be shown that in this case the residuals are only correlated at a lag length of one period, and that the usual test for serial correlation can be suitably adjusted.

The purpose of this paper is to examine the joint hypothesis of interest rate parity and rational expectations using monthly Canadian and American data over the period 1972(1)-1985(6). Throughout this study we consider the interest parity equation expressed both in levels and rates of change. As discussed below, the latter are not viewed as a transform of the level model but rather as an alternative formulation of the parity condition.

The investigation follows two paths. The first is to test the relationship via the model free approach; that is, we conduct tests for serial correlation of the ex post residuals paying close attention to the memory of the process. Special concern will be given to the sensitivity of the results to subsampling over the period.4 Hence we will determine whether there have been periods of time for which the joint hypothesis of rational expectations and interest rate parity is a reasonable characterization. Also, we shall be interested in testing whether the spot rate and interest rate differentials have time-series representations with unit roots and hence provide some motivation to the parity equation expressed as rates of change.

The other strand of the inquiry is: (1) to specify and to estimate a simple model of exchange rate determination based upon interest rate parity and rational expectations over the same time period; and (2) to test the corresponding cross-equation rational expectations restrictions. A key question of this testing is to determine whether the time-series representation of the interest rate differential process appears in the exchange rate equation in the form prescribed by interest rate parity and rational expectations. As in the model-free approach, we examine whether conclusions based upon full-sample estimation continue to hold for various subperiods. In this way we may appreciate when the postulated model of exchange rate determination accurately describes the facts and when it does not.

future variable (as in the interest parity relation) the subsequent residual in the equation is serially correlated (see Nelson, 1975 and McCallum, 1976). Recently Hayashi and Sims (1983) have proposed a potentially more efficient procedure for estimating these kinds of rational expectations models.

3 These remarks should not be interpreted as a general condemnation of the substitution model. Indeed quite the contrary is intended. The substitution method often provides the sim- plest way possible in which to estimate many rational expectations models. The principal advantage lies in the fact that no particular model need be specified to generate the required expectations. Hence potential misspecification from using an 'incorrect' model can be avoided and the estimation problem becomes one of errors in variables.

4 In a related study investigating the unbiasedness hypothesis in the forward foreign exchange market, Gregory and McCurdy (1984 and 1986) have found that conclusions based upon full sample estimation could be very misleading. The estimates of the test equation could be very unstable and subsampling can give some indication of the robustness of the results.




On balance the evidence from the unit root tests suggests that both the spot rate and interest rate differentials be cast in first differences. With the variables expressed as rates of change, both approaches show support for the joint hypothesis of interest rate parity and rational expectations over the period 1972 to 1976. Afterwards, the results diverge somewhat. However, a common conclusion is that observations from the 1980s should not be grouped together with those from the 1970s. Evidence is presented indicating a pronounced change in econometric structure at about the close of the decade.

The organization of the paper is as follows. In the following section the model is developed and related to conventional tests for serial correlation of the ex post residuals in the parity relation. Results from these tests as well as tests of the overidentifying rational expectations restrictions are presented and interpreted in the third section. Concluding remarks are given in the final section.

THE MODEL AND TESTING INTEREST RATE PARITY AND RATIONAL

EXPECTATIONS

In this section we develop a simple model of exchange rate determination based upon interest rate parity and rational expectations. Two distinct forms of the parity condition are specified and tested. We consider the relation in level form and an equation with variables cast in first differences. The version in first differences is not viewed as a simple transformation of the levels equation but rather as an alternative (though related) formulation of the parity condition and worthy of study in its own right. The strategy here is to determine whether either version of the interest parity condition is supported by the data. In this we follow the approach adopted in testing purchasing power parity in first differences (called relative purchasing power parity). We attempt to link the two alternative models by applying unit root tests to the variables comprising the interest rate parity condition. However, it should be kept in mind that the solution for the rational expectations problem requires that the variables of the model be stationary. Thus in solving either formulation of the parity condition we maintain the stationarity assumption for the arguments for that particular equation. For reasons of tractability, internal consistency across the two formulations is not imposed.

Uncovered interest rate parity relates the expected change in the level of spot rate to the interest rate differential on similar assets in the different currencies. This relationship may be written as

St -EtSt+I = it* - it + ut t = 1 ... N, (1)

where st is the logarithm of the spot exchange rate at time t, Et is the expectation operator conditional upon the inform4tion available at time t, it and i * are the domestic and foreign nominal interest rates respectively and u is




a mean zero serially independent constant variance disturbance term.5 The fact that ut appears in equation (1) implies that we are assuming that interest rate parity holds up to a stochastic serially independent error term. Thus we permit non-systematic departures from interest rate parity and assume Etut+. = 0 for all] = 1 .I..

The other formulation of the interest parity condition is to cast the variables in first differences:

-Et = - -4t) + -r (1t)

where it = xt xt-1 and nt is an independently and identically distributed error term such that Etrt+j= 0 for allj = 1 ... . As specified, equation (1') is not simply a first difference of equation (1).6 This formulation states that the expected rate of change of the spot rate is equal (up to a serially independent error term) to the change in the nominal interest rate differentials. In some sense equation (1') is a somewhat weaker condition than equation (1).

As mentioned earlier, the rational expectations solution requires stationarity of the variables, and so we treat for analytical purposes equation (1) and equation (1') separately as alternative statements for interest parity. For expositional reasons, we deal below with the solution of (1); however, the analysis of (1') is the same save for variable definition. In order to solve the partial difference equation (1), the data generating process for the interest rate differential, zt = it * - it must be modelled. If we assume that the interest rate differential is stationary and ergodic, they by Wold's decomposition theorem, the process has an infinite moving average representation.7 Following Baillie, Lippens, and McMahon (1983), we approximate the infinite moving average process by a finite autoregressive process of order p:

p 2 Zt= iZt-i + C (2)

where the roots of the polynominal in the lag operator, 1 - 1L - ...-PLP lie outside the unit circle, and E, is white noise.

5 In accordance with rational expectations, we have assumed that it is the conditional mathe- matical expectation of the future spot rate that appears in equation (1). Also, we have assumed that there is no risk premium. We did add a constant to each equation considered, but it never proved significant in estimation. However, since the risk premium may be variable, it is not clear that this constant adequately captures this. See Gregory and McCurdy (1984).

6 Notice that the first difference of equation (1) is:

- - (E St - Et. 18t) = (it* - - - i' ) + ut - (1t" Clearly this equation is quite different from (1') with a different rational expectation solution. Also, since the equation contains an integrated moving average error term, the joint estimation problem is much more complex. Hence we consider only (1').

7 We note that the Box-Pierce portmanteau tests used in the substitution method also require ergodicity and stationarity. However, other tests such as Dufour (1981) are valid in the pres- ence of heteroskedasticity.




Although we have chosen to model the interest rate differential as an autoregressive process, it is not our intention to suggest that Canadian interest rates are determined in a purely 'mechanical' way. To some extent and degree, the policy of the Bank of Canada over the 1970s has been directed towards managing the nominal interest rates. While it is true that monetary policy in Canada has consistently focused upon controlling the nominal interest rates, the justifications and the intended goals and benefits from doing so have changed. For example, in the early 1970s conventional wisdom suggested that prudent management of interest rates would promote smoothly functioning financial markets and lead to healthy investment. However, in the mid-1970s attention was directed towards the growth rate of certain monetary aggregates and a new argument for controlling interest rates was advanced. In particular, targets for growth in the narrowly defined money supply (MI) were to be achieved using interest rates as the policy instrument (see Bank of Canada Annual Report 1975). Nevertheless, even for the period of money targeting, there appeared to have been departures from stated policy. For instance, as noted by Courchene (1981), the Bank of Canada in 1978 became especially concerned with the dramatic depreciation of the Canadian dollar and took defensive action.

For our estimation period, it would have been extremely difficult to take into account all the considerations used by the bank in establishing an interest rate. However, since Canadian interest rates have always tended to drift towards those that have prevailed in the United States, a close approximation is provided by modelling the interest rate differential as a stationary autoregressive process. Evidence in Gregory and Raynauld (1985) suggests that the lagged interest rate differentials are important determinants of Canadian interest rates. Also, in the empirical application, we added a constant to equation (2). This allows a constant difference to persist between the interest rates in the long run. However, as in Gregory and Raynauld (1985), the constant never proved significant in estimation for equation (2) and in the first difference form analysis.

Assuming that zt is part of the agents' information sets so that Etzt = zt, the solution to the partial difference equation (1) is (see Gourieroux, Laffont, and Monfort, 1982):

St = c(L)zt + ut,

where

CO= 1-+- I 2 (Pp

Ci ((j+ I + Pj +2 + *-+ Op) + Cj= ,P J+ - .2 , p- 1p c o = ,2--- f




with 1 - ol- 2 - -p 0.8 In this formulation we are assuming that zt

is exogenous. However, this may not be so severe, since we could regard zt as the autoregressive moving average solution from the final form equations of a dynamic simultaneous equation model.9 Also, since the spot rate is obtained as a linear filter of the zt and since a finite order linear filter maps a stationary process into another stationary process, then this modelling implies that it will not be possible for the interest rate differential process to be stationary when the spot rate is not.

Rational agents take into account the process governing the interest rate differential when forecasting future spot exchange rates. This places testable restrictions upon the manner in which current and lagged interest rate differentials influence exchange rate movements. The unrestricted model corresponding to equation (3) is:

p-I (4) St = 2 aizt-i + Ut-

i=O

Notice that there are p non-linear restrictions placed upon the parameters of equations (2) and (4).

As we argued earlier, allowing an independent and identically distributed error term into the interest rate parity equation (1) negates standard testing using the substitution method. To see this, consider the decomposition of the future spot rate into two orthogonal components:

St+l -EtSt+1 + qt+P (5)

where It+, = (st+l - Etst+?) and is serially independent. For the present model under rational expectations this is

Xt+ 1 Ut+l + Et+ _ (6)

The substitution method involves calculating an ex post residual wt as

Wt = St - Zt(7)

and testing whether this is serially independent using portmanteau tests such as the modified Q test in Ljung and Box (1976). However, for our model we may obtain an explicit form for wt, which is

Wt = Ut - qt+ 1

Et ?1 (8) = U t Ut+ Itl_ 8 1 01 - *-- - - )n

8 Gourieroux, Laffont and Monfort (1982) have demonstrated that restricting {st) to be a stationary series implies that the martingale part of the general solution is a constant sequence.

9 The author would like to thank an anonymous referee for this interpretation and also for pointing out an error in the solution of the model in an earlier version of the paper.




Clearly, wt is serially correlated even under the assumption E(utct) = 0 for all t.10 The ex post residual in such a test is composed of a linear combination of two error terms, one of which is a first-order moving average process with a unit root. Thus standard application of Q tests should indeed detect serial correlation, even when the interest rate parity and rational expectations assumptions are true. However, under this joint null hypothesis, it follows that E[wt wt -IjI = 0 for allj = 1 .... Thus we should find no serial correlation at lag length greater than one, and we may test this by the following statistic (see also, Box and Jenkins, 1970, 35): 1

Q- =RT - R, (9)

where RT = [r2, ..., r,] a (1 X r) row vector of the sample correlation A

coefficients and E is the estimated variance-covariance matrix:

var[rkI -(1 + 2r12) k _ 2

A v[rk, rk+1]

2 r 2

N

cov[rk, rk+21 =-rl k ' 2 N

c8v[rk, rk+?2 ? k _ 2 c$v[k r?I=0k ?2 s 3 3

If w has a constant variance, then Q' will be approximately distributed as X (m - 1) under the null hypothesis of rational expectations and interest rate parity as expressed by (1). Since this test is only valid under homoskedasticity, an alternative test to apply is the rank serial dependence tests of the Wilcoxon type developed in Dufour (1981).12

RESULTS: TESTS FOR UNIT ROOTS, SERIAL CORRELATION, AND THE

RATIONAL EXPECTATIONS RESTRICTION

In this study we use month-end data from 1972(1) to 1985(6), giving 162

10 The fact that the ex post residuals are correlated may be shown without explicit reference to a particular rational expectations solution. Also, if one considers the first difference of equation (1) (see fn 6) as distinct from an equation specified with variables expressed in first differences (equation (1') ) then the ex post error structure would be correlated up to lag length two.

11 Thanks go to Jean-Marie Dufour for suggesting this test. 12 In the empirical section of the paper we calculated these rank tests but found that in general

they were typically 'close' to the centre of the distribution under the null hypothesis of serial independence. Also we note that strictly speaking these are independence tests which are not directly applicable when first-order serial correlation is possible.




TABLE 1

Testing for unit roots and serial correlation

Unit roots Q tests on ex post residuals

Dates S Z Levels First difference

1972-85(6) R,= 0.038 R, = 0.01 Q = 0.2 X 10-5 Q = 0.2 X 10-4 R2 = 0.008 R2 = 0.17 Q' 0.2X105 Q'= 0.99

1972-6 R = 0.02 RI = 0.40 Q = 0.6 X 10-7 Q = 0.012 R2= 0.0001 R2 = 0.50 Q' =0.1 X 10--10 Q' = 0.99

1972-9 RI= 0.11 RI = 0.13 Q = 0.5 X 10-6 Q = 0.7 X 10-7 R2= 0.14 R2 = 0.09 Q' -0.1 X 10-10 Q' = 0.99

1980-5(6) RI = 0.18 RI = 0.02 Q = 0.85 Q = 0.01 R2 = 0.11 R2 = 0.0001 Q' 0.82 Q' 0.99

NOTE: These numbers are marginal significance levels. R 1 and R2 are tests for unit roots developed by Evans and Savin (1981) and Phillips (1985), respectively. The marginal significance levels for these tests are approximate; being based on inspection of table I in Evans and Savin (1981, 760). Q is the modified Box-Pierce statistic discussed in Ljung and Box (1978), and Q' is the adjusted statistic as defined in equation (9).

observations. The data were kindly supplied by the Bank of Canada and are available upon request from the author. The spot rate is a closing rate and the interest rates are Canadian and American thirty-day commercial paper rates. The interest rates and spot rates are carefully matched according to the rules discussed in Longworth, Boothe, and Clinton (1983).

In table 1 are the marginal significance levels of tests for unit roots in the spot exchange rate (S) and the interest rate differential (Z). Also in this table are the results for testing for serial correlation in the ex post residuals of the interest parity relation with variables defined in levels and in first differences. Before the results are discussed, it is worthwhile stressing that these marginal significance levels are calculated for each test in isolation; no adjustment is made for the fact that many tests are being jointly considered. Since these tests and others that follow are not independent, strict adherence to iigid rejection regions based upon some standard arbitrary level of confidence should be avoided. Assessment is in terms of general tendencies and trends.

Marginal significance levels are tabulated for two tests for unit roots. The first (labelled RI in table 1) developed by Evans and Savin (1981) is appropriate to test p- 1 in the equation:

t= pt-I + et (10)

where Et is independently and identically distributed. The second test (identified in the table as R2) recently derived by Phillips (1985) also tests p = 1 but is valid for a wide class of weakly dependent and possibly heterogeneously distributed errors. Moreover this test is applicable even when the rest of the model is excluded from the test equation (providing that this omitted portion




can admit a Wold decomposition). Both test statistics have their asymptotic distribution calculated in Evans and Savin (1981, 760). However, since the tabulation is fairly coarse, the marginal significance levels reported in table I are approximate.13

There is agreement from both tests for unit roots, suggesting that when the full data set is used (1972(1)-1985(6) ) the evidence is against first differencing the spot exchange rate. This is despite a numerical estimate of p of 1.0093. Results from testing for unit roots for the interest rate differentials are conflicting: the Phillips's test appears to be in favour, while the Evans and Savin does not. Subsampling reveals that it is the early period for the spot rate which causes the rejection of the unit restriction for the full sample. In contrast, according to both tests, it is the latter part of the sample for which there is rejection of the unit root in the interest rate differential. Interestingly, for the period 1972-79 both tests confirm that the unit root restrictions are supported by the data for the spot rate and the interest rate differential.

Also in table I are the tests for serial correlation in the ex post residuals from the interest parity relation. The tests are calculated for both the levels (1) and the first differences (1') versions of the interest parity relation. Q is the standard Box-Pierce statistic (Ljung and Box, 1978) and Q' is the test given in equation (9) that allows for possible correlation of residuals at lag length one. The number of simple correlations used in both tests is arbitrarily set at twelve.

For the most part, these results confirm the findings of Cumby and Obstfeld (1981). That is, with spot rate and interest rate differentials defined in levels there is ample evidence of serial correlation in the ex post residuals for the full sample. Subsampling illustrates that the serial correlation occurs for the 1970s observations and does not for the 1980s. Using only the 1980s observations, the Q test does not detect serial correlation in the residuals. With the variables defined in levels, there is not much difference between the Q and Q' statistic, since the marginal significance levels are quite similar. However, the situation changes once variables are first differenced.

With the variables cast in first differences, the Q tests show that these ex post residuals are serially correlated for the entire sample and over various subsamples. Again, the 1980s residuals appear to have substantially less correlation. Once account is taken of possible correlations of the simple correlation coefficient at lag length one, the marginal significance levels increase dramatically. For example, the 1972-6 observations produces a marginal significance level of 0.99 for the Q' statistic compared with a Q test wvith a level of 0.012. In fact there is no evidence of serial correlation of lag lengths greater than one for all sample periods considered, once the variables are expressed as first differences.

The results of table 1 raise important questions with respect to the 13 The truncation length for calculation of Phillips's test in this study is twelve. This is roughly

the square root of the number of observations (Phillips, 1985, 15-19).




stationarity of the spot rate and the interest rate differential in levels. This conclusion is particularly relevant for the next part of the study. Estimating the model in levels (equation (1)) for any subperiod or full sample failed to produce a statistically satisfactory empirical specification. The estimated model was highly unstable showing explosive roots, serial correlation and heteroskedasticity. Durbin-Watson statistics of less than 0.50 were commonly observed. In view of the lack of success of the model estimated in levels, we consider only the results in first difference form (equation (1') ). Although the individual test results in table 1 on occasion suggest that this may not be strictly valid (at conventional levels) the overall impression is that the joint restriction of first differencing the spot rate and interest rate differential is amenable to the data. Upon subsampling we do not observe the outcome of both unit root restrictions being simultaneously rejected. Also, while the coefficients appear to be at times significantly different from one, numerically they are nevertheless 'fairly close' to one. In addition the results we discuss below do not seem to be very sensitive to using the estimated value of p from equation (10) and defining the transformed variables as

Yt = Yt- lot-1- In table 2 we report the results of testing and estimating the model with

variables expressed in first differences (equation (1') ). Since innovations or shocks to the interest rate differential process are likely to be correlated with shocks to the foreign exchange market, we have allowed for a contemporaneous correlation between E and ut in estimation.14 Both the unrestricted (equations (2) and (4) ) and restricted (equations (2) and (3)) models are estimated by full information maximum likelihood (FIML). The non-linear rational expectations restrictions are tested using likelihood ratio tests. That is, we take twice the difference of the log likelihood obtained from unrestricted and restricted systems estimation. Under the joint null hypothesis of interest rate parity and rational expectations this statistic is asymptotically distributed as a x2 with p degrees of freedom. With regard to the properties of this testing procedure, Monte Carlo evidence in Hoffman and Schinidt (1981) and Gregory and Veall (1985) suggest that the likelihood ratio tests of the non-linear rational expectations restrictions compared with the central chi square distribution have quite good finite sample properties. Reasonable test results can often be obtained for data sets with as few as fifty observations.

In order to estimate the model, we must specify the order of the autoregressive process for the interest rate differential. Since there is no widely accepted consensus as to how to select the appropriate order for a time series representation, we estimate the model for p = 7, 6, 3, and 1. Only the latter three are shown in table 2, since the results withp = 7 are usually quite silmilar to p = 6. To evaluate the empirical specification for each p, we calculated for the unrestricted model (the equations when the rational expectations 14 Equation (4) is exactly identified, since there is one predetermined variable, z, p, that does

not appear in this equation.




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restrictions are not imposed): (1) a test for serial correlation due to Godfrey (1978) labelled AUTO; and (2) a test for autoregressive conditional heteroskedasticity discussed in Engle (1982), identified as ARCH in the table. These tests are not generally applicable in models with dependent regressors and are intended solely for descriptive purposes. LR corresponds to a likelihood ratio test using the unrestricted model of: (1) p = 6 against p = 7; (2) p = 3 against p = 6 and (3) p = 1 against p = 3 with test values aligned in table 2 with the order of the autoregressive process under the appropriate null hypothesis.

Tests of the rational expectations restrictions are tabulated for the different ps using the entire stated sample period and the second half of this sample. The period 1972-6 comprises the first half with the second half starting with the 1977(1) observation and proceeding to the end of the stated sample. The choice of 1972-6 as the benchmark was made on the basis of the uniformity and consistency of the results over this period for all p. Stability tests are also given in table 2 using both unrestricted and restricted estimates with the split sample 1972-6 and the second half of the relevant period. In all cases comparisons of calculated statistics are made against central x2 distribution. The marginal significance levels are recorded for individual tests and therefore do not reflect the fact that many tests are being jointly considered.

The first set of results in table 2 are for the period 19724. In the unrestricted model there is no strong evidence of either serial correlation or ARCH errors. The tests indicate (LR) that the model can be reduced to p = 1 without a large drop in the log likelihood. For this period and for all p, the rational expectations restrictions from the interest rate parity condition are supported by the data. However, once we reestimate the model for the full sample (1972-85(6) ) the picture changes. Results forp = 3 andp = 1 indicate: (1) autocorrelation and ARCH errors are present in the interest rate differential equation; (2) the rational expectations restrictions are rejected for the full and second half observation; (3) a rejection of p = 3 or p = 1 against p = 6; and (4) instabilities in the restricted model. For p = 6, the full sample results show that while the unrestricted model appears to be statistically adequate, the rational expectations restrictions are rejected for the full-sample and the second-half observations.

Since the 1972-6 observations produced reasonable test results, we reestimated the model adding twelve monthly observations at a time and calculated the test statistic for the rational expectations restrictions. The largest sample period for which this null hypothesis is not rejected for either p = 6 or 3 or 1 is 1972-9 and the results are given in table 2. For this subsample, there is no evidence of serial correlation, ARCH errors, or instabilities for any of the equations. The data suggest that p can be reduced to 3 but not to 1. However, even for this period, the second half of the sample for p = 3 or 1, the rational expectations restrictions are rejected. This fact is reinforced once the 1980 observations are added with either p = 2 or 1 we see: (1) there is strong




evidence of misspecification according to AUTO or ARCH in both the spot and interest rate differential equation; (2) the restrictions that p = 1 against p = 3 and p = 3 against p = 6 are rejected; (3) the rational expectation restrictions are soundly rejected for the full period and the second half and (4) the unrestricted and restricted models are unstable using the break point at 1976(12).

As might be expected, the results with p = 6 are more robust to additional observations. None the less a large drop in the marginal significance level occurs from testing the rational expectations restrictions when the 1980 observations are added. Rejection of these restrictions happens with the addition of the 1981 observations. In table 3 we provide the coefficient estimates from the unrestricted and restricted model for p = 6 over the period 1972-9. We note that most of the estimated coefficients are individually not significantly different from zero (jointly they are).

Lastly in table 2, we estimate the model for the period 1980-5(6). Although the rational expectations restrictions are supported by the data, there is evidence that the autocorrelation in the interest rate differential equation is not adequately captured by p = 6, 3, or 1. At p = 7, there is no further indication of misspecification. However, the rational expectation restrictions are rejected with a marginal significance level of 0.008.

CONCLUDING REMARKS

The aim of this paper has been to examine the joint hypothesis of interest rate parity and rational expectations for Canada and United States using monthly data. Testing of this hypothesis proceeded in two directions. The first approach followed the substitution method in which the ex post residuals in levels and first differences from the interest parity relation were tested for serial correlation. Assuming that the parity condition is true up to a stochastic serially independent error term implied that standard tests for serial correlation are not appropriate. In these circumstances, correlation at lag length one can occur, and consequently an adjustment to the standard tests was required. Also tests for unit roots in both the spot exchange rate and interest rate differentials were calculated. From these tests we found that often the unit root restrictions were not supported by the data. However, for 'practical' purposes the coefficient estimates were 'close' to unity, suggesting that first differencing is generally a good approximation. Moreover this version of interest rate parity seems, at times, to be supported by the data.

The second approach involved specifying and estimating a simple model of exchange rate determination based upon interest rate parity and rational expectations. The testing investigated whether the cross-equation rational expectations restrictions were supported by the data. For this testing exercise, meaningful results occurred only if the variables were defined in first differences.




TABLE 3

Estimates of unrestricted and restricted model 1972-9 (p = 6)

Unrestricted estimates Restricted estimates

Zt St zt St -1 =-0.13 a0 = 3.63 01 = -0.091

(0.11) (0.56) (0.096) 02 =-0.064 a1 = 1.03 02= 0.0023

(0.13) (0.74) (0.12) 3 =-0.096 a2= 1.10 03= 0.16

(0.15) (0.72) (0.12) -04 = -0.066 a3 = 0.055 4)4 =-0.021

(0.10) (.074) (0.089) 05 = 0.0017 a4 = 0.24 45 = 0.038

(0.12) (0.62) (0.10) 46 = 0.1674 a6 = 0.43 46 = 0.22

(0.09) (0.69) (0.12) DW 1.97 DW 1.97 DW 2.01 DW 2.17

L 710.0 L = 707.0

NOTE: Standard errors are given in parentheses. DW iS the Durbin-Watson statistic intended solely for descriptive purposes, and L is the log of the likelihood excluding the constant.

The evidence presented in this paper suggests that in levels (equation (1)), there is little support for the joint hypothesis of interest rate parity and rational expectations for Canada and the United States over the 1970s. The serial correlation tests from the substitution method detect temporal dependence in the ex post residuals. This is true even when allowance is made for the potential correlation at lag length of one. However, testing for the 1980s in levels using both Q and Q' indicate an absence of any correlation in the residuals.

Casting the variables in first differences (equation (1') ), we found support of the joint hypothesis using the Q' test for all periods. Evidently most of the correlation occurred at lag length one. This conclusion is reaffirmed in the modelling approach in which we are able to retain the cross-equation rational expectation restrictions. In fact, these restrictions are not rejected for the 19724 period for any of the choices of the order of the autoregressive processes. On the other hand, estimating the model over the full sample indicated a rejection of the rational expectations restrictions. Rejections occurred primarily when the observations from the 1980s were added. According to several diagnostic tests and in view of the changing conclusions from the Q, Q', and tests of the rational expectations restrictions, we conclude that an important structural shift took place around the end of the 1970s. Indeed, we might reasonably expect the existence of such shifts. The 1980s have witnessed unprecedentedly high nominal interest rates with much larger interest rate differentials between Canada and the United States. Certainly an often cited event is the October 1979 change in the Federal Reserve Board operating procedure. Explaining the shifts and developing an exchange rate




model suitable to the 1980s is likely to require careful attention to the issue of interest rate formation in Canada and the United States. A worthwhile consideration in this pursuit is the possible existence of a risk premium in the interest rate parity equation.

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testing interest rate parity and rational expectations for canada and the united states

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