# 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

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