# Cointegration and models of exchange rate determination

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International Journal of Forecasting 3 (1987) 43-51 North-Holland 43 COINTEGRATION AND MODELS OF EXCHANGE RATE DETERMINATION Richard T. BAILLIE Michigan State University, East Lansing, MI 48824, USA David D. SELOVER University of California, Saz~ Diego, CA 92093, USA Abstract: The application of new techniques in testing for cointegration indicate the inappropriate- ness of the pure monetary model to explain movements in the nominal exchange rate. In general the fundamental variables are found to be integrated of different orders and there is a lack of cointegration between the exchange rate variables in the monetary model and relative prices. Estimation of other dynamic models are found to give rise to parameter estimates which do not support the monetary model. The results are broadly consistent across five countries. These results imply that it is not worthwhile to forecast from the monetary model and its main variants. Keywords: Monetary model, Non-stationarity, Cointegration, Purchasing power parity, Tests for unit roots. 1. Introduction The purpose of this paper is to consider the econometric validity of some popular models of exchange rate determination that are widely used in forecasting nominal exchange rates. The models include the pure flexible price monetary model of Frenkel (1976) and Bilson (1978), the sticky price monetary model of Dombusch (1976) and the real interest rate differential model of Frankel (1979). We first present results on the models estimated by conventional procedures for Canada, France, Japan, United Kingdom and West Germany vis a vis the US dollar. Several statistical problems are apparent, many of which have been noted by previous authors. However, recent work by Engle and Granger (1986) on the cointegration of a group of dynamic variables gives extra insight into the reasons why the simple models of exchange rate determination have broken down and failed to give rise to successful forecasting. Analysis along these lines indicates the different degrees of integrability and the lack of cointegration between the variables in the models. Thus there is no statistical evidence in support of a long run relationship consistent with the pure monetary model or its simple extensions. Estimation of more general relationships with the fundamental variables integrated to differing orders generally fails to render coefficient estimates consistent with the monetary model. Finally, it is also seen that there is a lack of cointegration between the nominal exhange rate and relative prices, so that purchasing power parity becomes of little use as a long run concept. Overall the results indicate grounds for believing the pure monetary model to be m&specified as a long run relationship and consequently suggests it is inappropriate to use for forecasting purposes. 44 R. T. Badlie. D. D. Selover / Cointegrution crnd ndels of deternwutron 2. Estimation of the monetary model Many of the simple models of exchange rate determination can be expressed in terms of the relationship ~,=Pl(m,-mf)+P*(Yt-~,*)+P3(r,-r**)+P4~t(Pr+1-Pt*+1)+~ut~ (1) where s is the natural logarithm of the nominal exchange rate, m and y represent the natural logarithms of domestic money supply and real output respectively, r is the domestic short term interest rate, E, p, + 1 is the expected domestic rate of inflation; asterisks denote foreign quantities and u, is a stochastic disturbance term. In general it woul be expected that & = 1, & < 0, p, > 0 with the sign of & being unclear since the standard Keynesian assumption of a rise in domestic interest rates leading to a currency appriciation implies & < 0, while a rise in the domestic interest rate due to inflationary expectations would imply & > 0. Also, in the Dornbusch (1976) sticky price model & = 0 and /3, > 0. Many previous authors have considered estimation of eq. (1). In particular Frankel (1979) developed a theory of the real interest differential and found empirical support for this mode1 for the West German deutschemark between 1974 and 1978. Driskill and Sheffrin (1981) noted the possible endogeneity of the short term interest rate differential variable and estimated (1) by instrumental variables; Haynes and Stone (1981) relaxed the restriction of domestic and foreign variables having opposite signs and the elasticities of domestic and foreign money supplies being plus and minus unity. Both Driskill and Sheffrin (1981) and Haynes and Stone (1981) found empirical evidence to support their criticisms of Frankel (1979). Recently, Boothe and Glassman (1986) have argued that the non-stationarity of variables was a problem in Frankels study and they carry out work in differences. Before considering the concept of cointegration and its importance in this issue we first consider direct estimation of (1) by OLS, since this simple procedure throws light on the question of interest. The exact details and sources of the data are given in the Appendix and cover the period March 1973 until December 1983; so that the period where the US dollar rapidly appreciated vis B vis several major currencies is omitted. Table 1 gives the results of the OLS regressions and although many coefficient estimates are significant, the sign and magnitude of the estimates is frequently substan- tially different to that expected; so that, for example, the money supply differential variable is in every case significantly different from unity. Table 2 estimates the same equation assuming a first order autoregressive AR(l) error process on u,. Now most of the coefficients are insignificant and most importantly, for later ideas on cointegration, exhibit an AR(l) error process with a close to unit root. Overall the qualitative results for this extended data set are fairly similar to those obtained by Driskill and Sheffrin (1981), Haynes and Stone (1981) and Boothe and Glassman (1986) for West German deutschemark and are unsupportive of the monetary model. A further possibility is that it is more appropriate to consider multinational models of exchange rate determination to allow for common macroeconomic international effects. This idea has been acknowledged in related, but different exchange rate studies by Edwards (1983) and Ha&o (1984) who use the fact that the change in level of one exchange rate will be useful information in predicting the levels of other exchange rates. Some insight can be obtained from the correlation matrix of residuals from the models estimated for the five countries in table 2. The correlation matrix is presented in table 3 and confirms the existence of generally quite high correlations particularly between France and West Germany. An obvious model which captures intercountry effects and simple dynamics consistent with the single equation estimation is to estimate a five country model as a SURE system with a first order vector autoregressive (VAR) error process. To simplify specifica- R. T. Baiilie, D. D. Selover / Cointegration and models of determination 45 Table 1 OLS estimates of eq. (1). country Parameter estimates of a constant UK - 1.1222 (3.881) Japan - 7.1967 (11.938) West Germany -0.1855 (2.118) Canada 2.2580 (6.253) France - 1.2366 (32.86) Cm, - 47 0.6691 (5.899) - 0.3137 (2.618) - 1.1404 (8.213) - 0.9298 (6.439) 0.6886 (6.533) (Y, - ?a 0.4762 (1.827) 0.2769 (1.587) - 0.4806 (1.641) - 1.4435 (9.531) - 1.4285 (4.292) (or - rr*) 0.0102 (2.078) 0.0102 (3.042) 0.0135 (3.096) 0.0086 (2.830) 0.0276 (5.834) UP,+,- P;,,, R2 &, Durbin-Watson - 0.0020 0.44 0.1164 0.102 (0.248) 0.0286 0.59 0.0889 0.204 (3.806) 0.0080 0.51 0.0969 0.369 (1.434) -0.0015 0.56 0.0600 0.343 (0.119) 0.0559 0.68 0.1105 0.416 (4.542) a r-statistics are in parentheses. Table 2 Estimation of eq. (1) with AR(l) errors: u, = PU,_~ + c, Country Parameter estimates of a constant (m, - m:) (Y, - Y,*) (5-0 E,(P,+I - P:,,, R2 B 1 ac UK 0.2067 (0.576) Japan West Germany Canada - 6.0163 (15.12) - 0.9294 (7.556) -0.1883 (1.399) France - 1.7967 (7.032) 0.1618 (1.300) - 0.0969 (1.254) 0.0589 (0.629) 0.0340 (0.751) -0.1011 (1.054) - 0.2278 (1.720) 0.1298 (0.627) - 0.975 (0.610) 0.1039 (1.266) - 0.0714 (0.558) - 0.0002 (0.099) - 0.0044 (1.692) - 0.0023 (1.339) - 0.0038 (2.198) - 0.0007 (0.279) 0.0003 0.2053 0.9819 0.0289 (0.054) 0.0055 0.9209 0.9675 0.0323 (0.755) - 0.0094 0.2274 0.9794 0.334 (1.163) 0.0074 0.0534 0.9896 0.0130 (2.333) - 0.0021 0.1417 0.9940 0.0327 (0.327) a r-statistics are in parentheses. Table 3 Correlation matrix of residuals from estimated models in table 2. UK Japan West Germany Canada Japan 0.4326 West Germany 0.4503 0.4680 Canada 0.1725 0.0706 0.1491 France 0.4982 0.4982 0.8063 0.1981 46 R. T. Budlie, D. D. Selowr / Cointegratiotl ond models of deternzination Table 4 SURE estimates of (1) with a VAR(l) error process country Parameter estimates of a constant (m, - m:) ( Y, - Y; ) (rr ~ r,+) E,(P,+, - !J:+,, 6 UK - 2.0828 (9.743) Japan - 5.2835 (10.52) West Germany ~ 0.5737 (8.757) 1.0424 (12.56) 0.0648 (0.650) - 0.4657 (4.4X2) Canada 2.3546 ~ 0.9675 (7.268) (7.47X) France - 1.2083 (34.33) 0.8288 (8.453) 0.3093 (1.831) 0.4577 (3.308) - 0.0469 (0.243) - 1.4316 (10.39) - 0.9313 (3.100) 0.0050 (1.510) 0.0053 (1.970) 0.0072 (2.547) 0.0101 (3.673) 0.0133 (3.129) 0.0214 (3.762) 0.938 0.0355 (5.794) 0.913 0.0067 (1.524) 0.911 0.0028 (0.262) 0.802 0.0499 (4.497) 0.840 a t-statistics in parentheses tion it is assumed that there is a diagonal coefficient matrix on the lagged error process vector, so that the error from each countrys model only depends on its own lag. The results are shown in table 4, are qualitatively similar to the previous results and again give rise to near unit roots on each countrys autoregressive error process. This last property has also been apparent in other studies, e.g., Frankel(l981). 3. Testing for cointegration Since (1) and versions of it are essentially long run relationships specified from appropriate economic theory the concept of cointegration is important in assessing the statistical validity of the preceding models. Engle and Granger (1986) define a series X, with no deterministic component and with a covariance stationary, invertible moving average representation after differencing d times to be integrated of order d, i.e., x, - Z(d). If two variables x, and y, are both Z(d), then it will generally be true that a linear combination z, = x, - uy, (2) will also be Z(d). However, it may happen that z, - Z(d - b) where b > 0 and x, and y, are then said to be cointegrated of order d, b. A paricularly important case is where X, and y, are both Z(l), but (2) is Z(0). If x, represents a g dimensional vector of random varables and all the components are Z(d); then if there exists a vector LU # 0 such that z,=ax,-Z(d-b), then (Y is known as a cointegrating vector. For the basic monetary model (1) to be a long run equilibrium relationship it is necessary that z, in (3) be Z(O), so that z, will rarely drift from zero and R. T. Buillie, D. D. Selover / Cointegrution und models of deternmution 47 equilibrium will occasionally occur. A VAR representation in first differenced form for x, will then only be valid if it is of the form A(L) = -yz,_l + u,, (4) where u, is a stationary process, y is gxr in dimension, r I g - 1 and A(L) is a gxg matrix with elements that are polynomials in the lag operator. Engle and Granger (1986) and Granger (1985) discuss ways of dealing with the g = 2 case, although precise inferential procedures in higher order models are currently unresolved. In order to determine the degree of integratability of the variables in (1) we employ the conventional Augmented Dickey Fuller (ADF) statistic used by Engle and Granger (1986) for this purpose and also discussed by Fuller (1976, pp. 366-382). The basic test and variants of it are discussed by Dickey and Fuller (1979 and 1981). The hypothesis of a unit root in an autoregression is tested by means of estimating the model and testing Ho: p = 0, versus H,: /? > 0, so that the null hypthesis implies a unit root. It should be noted that the critical values of the above one sided test statistic based upon the standard t-statistic (5) crucially depends upon the sample size and whether or not a constant term is included. For a sample size of n = 100 at the l%, 5% and 10% significance levels the appropriate test statistic is -2.60, - 1.95 and - 1.61 respectively; see Fuller (1976, p. 373). When a constant is included in (5) the test statistic at the same significance levels is - 3.51, - 2.89 and - 2.58 respectively. The value of p in (5) is chosen on the basis of being sufficiently large so that C, is a close approximation to white noise. An unnecessarily large value of p and the consequent inclusion of insigificant lagged Ax, variables would reduce the power of the test. The results of applying the ADF test to the variables in the model are given in table 5. It should be noted that small positive values of the test statistic are suggestive that a variable is Z(1). However, the distribution of the statistic against different alternatives is unclear and inferences must be made with caution. Some general patterns that emerge are that the nominal exchange rate, relative money supply differentials (excluding the UK), and nominal long term interest rate differentials to proxy expected inflation differentials are generally Z(1). Similar results for exchange rates have been previously obtained by Meese and Singleton (1982). While real output differentials, excluding Japan Table 5 Augmented Dickey Fuller tests. country Variables a S, cm, - mt) (Y, - YF) (r, - r?) K(P,+, - P:i 1) (P,Y P:) UK - 1.74 - 2.51 (1) - 1.85 - 2.72 (2) -0.88 (2) - 2.01 (1) Japan - 0.44 - 0.54 (2) - 1.19 (2) - 2.03 (1) -0.11 (1) 0.47 West Germany -0.36 -0.39 (2) ~ 2.42 - 1.72 (2) -0.34 (1) - 1.71 (2) Canada 0.62 -0.23 (1) - 2.09 - 2.23 (1) ~ 1.32 (2) 0.66 (4) France 1.77 2.53 (2) - 2.21 (1) - 2.45 (1) - 1.30 (2) 1.21 (3) a The 1% and 5% significance levels of the above test statistics are - 1.95 and - 2.60 respectively. The number in parentheses besides the statistic is the value of p in (5); when not given it is zero. 48 R.T. Baillie, D.D. Selover / Cointegration and models of determination Table 6 Estimationof s,=Po+PlA(m,-m:)+P2(~,-~~)+P3(~-~*)+P~AE,(~~+~-~~l)+Ut. Country PO PI P2 P3 P-4 LM(12) a UK - 0.0062 (- 1.89) Japan 0.0044 (1.06) West Germany 0.0076 (1.59) Canada - 0.0020 (- 1.56) France - 0.0064 (- 2.09) 0.0689 (0.53) -0.1013 (- 1.33) 0.0681 (0.74) 0.0260 (0.57) -0.1361 (- 1.45) - 0.0140 ( - 0.29) - 0.0120 ( - 0.44) - 0.0040 ( - 0.05) 0.0085 (0.26) - 0.0799 (- 1.08) - 0.0012 (- 1.27) - 0.0013 (- 1.76) - 0.0023 (- 1.69) - 0.0002 ( - 0.42) - 0.0007 (-0.59) - 0.0002 15.03 (-0.05) 0.0021 7.34 (0.29) - 0.0138 13.45 (-1.71) 0.0064 4.82 (2.03) - 0.0028 8.67 ( - 0.48) a LM(12) is a Lagrange Multiplier score test for 12th order autocorrelation. and probably the UK, and short term interest rate differentials, excluding West Germany are I(0). Since the variables in (1) possess apparent different orders of cointegrability it follows that (1) cannot exist as a long run equilibrium. We also examined the fundamental variables separately for each country, rather than being in relative form vis a vis the US equivalent. The results were generally extremely similar to those reported in table 5 and are omitted for reasons of space. They can be obtained from the authors on request. However a possible long run relationship could exist between appropriately differenced versions of the variables in (1). Following the implications of the ADF tests, such models were estimated for each country and are reported in table 6. Once again the results were disappointing, with few parameter estimates approaching significance and with little support for the monetary model being apparent. One interesting feature of the work on cointegration by Engle and Granger (1986) is that the cointegrating vector LY in (3) can be estimated by OLS and the possible endogeneity of a right hand side variable is irrelevant. Essentially the OLS estimates of LY will be consistent and will have a variance of O(nm2), where n is the sample size; see Stock (1984). Thus the possible endogeneity of short term interest rates in (1) is irrelevant when it is desired to estimate a long run equilibrium or cointegrating relationship. It should be noted that an attempt at finding a cointegrating relationship has indirectly been obtained through the estimates of (1) reported in table 1. The fact that two I(0) variables are included is superfluous to the search for cointegration between the remaining Z(1) variables. The Table 7 Dickey Fuller statistics applied to residuals in eq. 1. country DF statistic UK Japan West Germany Canada France - 0.75 - 1.41 - 1.28 - 0.80 - 0.61 R. T. Buillie, D. D. Selooer / Cointegrarion and models of determindon 49 residuals from (1) had ADF tests applied to them, see table 7 and the fact that the unit root restriction could not be rejected, implies that no cointegrating relationship can be found between the nominal exchange rate and the two other Z(1) variables; money supply differentials and short term interest rate differentials. 4. The exchange rate and relative prices In the background behind the versions of the monetary model is the notion that purchasing power parity (PPP) is at least a long run phenomenon. From table 5 we note that the exchange rate and relative prices appear to be Z(1); although borderline rejection of the Z(1) relative prices hypothesis was obtained for the UK. Attempts at finding a cointegrating relationship between the nominal exchange rate and relative prices are reported in table 8 with the following model being estimated: The hypothesis that the residuals are Z(1) can only be rejected for France; so that for the other four countries, no cointegrating relationship can be found. Thus exchange rates and relative prices will apparently drift apart without bounds. While PPP has been a convenient assumption in many models, Pigott and Sweeney (1985) and others have noted that permanent departures from PPP can arise from changes in productivity and tastes, shifts in comparative advantage, etc., and have acknowledged the possibility of divergence from long run PPP. Other empirical studies by Adler and Lehman (1983) and Taylor (1986) provide confirmatory empirical evidence of this fact. 5. Conclusions The results presented in this paper appear to provide more dismal evidence on the inappropriate- ness of the monetary model. Since the fundamental variables have radically different trends and there is a lack of cointegration, it follows there is no long run relationship similar to the pure Table 8 Estimates of s, = a + p( p, - p:) + u, Country & DF Statistic il UK 0.6251 0.0119 - 1.17 (50.49) (9.47) Japan - 5.51 (-462.71) 0.0049 - 1.49 (3.40) West Germany - 8.002 (- 52.55) 0.0031 - 1.31 (2.52) Canada ~ 0.0522 (-6.94) 0.0173 - 1.35 (10.32) France - 1.5847 (- 198.11) 0.0248 - 2.23 (21.43) a DF is the Dickey Fuller statistic to test for a unit root in D,. 50 R.T. Baillie, D. D. Selouer / Cointegmtion crnd models of determinutlon monetary model, or conformable with long run PPP. Consequently, attempts at using these models to forecast will generally not be worthwhile and results such as those obtained by Meese and Rogoff (1983) are entirely to be expected. Appendix Monthly seasonally unadjusted data were obtained from the International Monetary Funds International Financial Statistics database tape for the United States, United Kingdom, France, West Germany, Japan and Canada. The period studied was from March 1973 until December 1983 which realized 130 observations. The series used were s: Y: P: i: m: E,P,+1 end of month exchange rates in terms of US dollars per unit of foreign currency, index of industrial production (1975 = 100) as a proxy for real income, consumer price index (1975 = 100) short-term interest rate as monthly average (call money rate or federal funds rate), money supply, Ml equivalent, (end of period) in billions of currency units, long-term interest rate, or bond yield as a proxy for the expected rate of inflation. References Adler, M. and B. Lehman, 1983, Deviations from purchasing power parity in the long run, Journal of Finance 38, 1471-1487. Bilson, J., 1978, The monetary approach to the exchange rate: some empirical evidence, IMF Staff Papers 25, 48-75. Boothe, P. and D. Glassman, 1985, Off the mark: post mortem of an exchange rate model, University of British Columbia discussion paper 85/14. Dickey. D.A. and W.A. Fuller, 1979, Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74. 427-431. Dickey, D.A. and W.A. Fuller, 1981, The likelihood ratio statistic for autoregressive time series with a unit root, Econometrica 49, 1057-1072. Dombusch, R., 1976, Expectations and exchange rate dynamics, Journal of Political Economy 84, 1161-1176. D&kill, R.A. and SM. Shefftin. 1981, On the mark: comment, American Economic Review 71, 1068-1074. Edwards, S., 1983, Floating exchange rates, expectations and new information, Journal of Monetary Economics 11. 321-336. Engle, R.F. and C.W.J. Granger, 1986, Cointegration and error correction: representation, estimation and testing, Econometrica, forthcoming. Frankel, J.A., 1979, On the mark: a theory of floating exchange rates based on real interest differentials, American Economic Review 69, 610-622. Frankcl, J.A., 1981, On the mark: comment, American Economic Review 71, 107551082. Frenkel, J., 1976, A monetary approach to the exchange range: doctrinal aspects and empirical evidence, Scandinavian Journal of Economics 78, 200-224. Fuller, W.A., 1976, Introduction to statistical time series (Wiley, New York). Granger, C.W.J., 1985, Developments in the study of cointegrated economic variables, UCSD Mimeo. Hal&o. C.S., 1984, A re-examination of purchasing power parity, Journal of International Economics 17. 265-277. Haynes, SE. and J.A. Stone, 1981, On the mark: comment, American Economic Review 71, 1060-1067. Meese, R.A. and K. Rogoff, 1983. Empirical exchange rate models of the 70s: do they fit out of sample?, Journal of International Economics 14, 3-24 Meese. R.A. and K.J. Singleton, 1982, On unit roots and the empirical modeling of exchange rates, Journal of Finance 37, 1029-1035. Pigott, C. and R.J. Sweeney, 1985, Testing the exchange rate implications of two popular monetary models, in: SW. Amott, R.J. Sweeney and T.D. Willet, eds.. Exchange rates, trade and the US economy (Ballinger Publications). Stock, J.H., 1984, Asymptotic properties of a least squares estimator of cointegrating vector, Mimeo. (Harvard University, Cambridge, MA). Taylor, M.P., 1986, Long run purchasing power parity, Mimeo, (University of Newcastle, Newcastle). R. T. Baillie, D. D. Selover / Cointegrution and models of determination 51 Biography: Richard T. BAILLIE has a Ph.D. from the London School of Economics and is currently Associate Professor in the Department of Economics at Michigan State University. His main interests are in econometric theory and applications in macroeconomics and international finance. He has published articles in Econometrica, Journal of the Ameri- can Statistical Association, Journal of Econometrics, Oxford Economic Papers and other professional journals. David D. SELOVER has a Masters degree in Economics from San Diego State University and is currently a Ph.D. candidate in Economics at the University of California San Diego.

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