Student

OXFORD BUtXETlN OF ECONOMICS AND STATISTICS. 52,2 (1990) 0305-9049 S3.00

MAXIMUM LIKELIHOOD ESTIMATION AND INFERENCE ON COINTEGRATION - WITH APPUCATIONS TO THE DEMAND FOR MONEY
Soren Johamen, Katarina Jtiselius
I. INTRODUCTION

LI. Background Many papers have over the last few years been devoted to the estitnation and testing of long-run relations under the heading of cointegration.Granger (1981), Granger and Weiss (1983), Engle and Granger (1987), Stock (1987), Phillips and Oullaris (1986), (1987), Johansen(1988b), (1989), Johansenand Juselius (1988), canonical analysis. Box and Tiao (1981), Velu, Wichem and Reinsel (1987), Pena and Box (1987). reduced rank regression, Velu, Reinsel and Wichem (1986), and Ahn and Reinsel (1987), common trends. Stock and Watson (1987), regressionwith integrated regressors, Phillips (1987), Phillips and Park (1986a), (1988b), (1989), as weU as under the heading testing for unit roots, see for instance Sims, Stock, and Watson (1986). There is a special issue of this BULLETIN (1986) dealing mainly with cointegration and a special issue of the Journal of Economic Dynamics and Cotitrol (1988) deeding with the same problems. We start with avector autoregressive model (cf. (1.1) below) and formulate the hypothesis of cointegration as the hypothesis of reduced rank of the longrun impact matrix II = afi'. The main purpose of this paper is to demonstrate the method of maximum likelihood in connection with two examples. The results concern the calculation of the maximum likelihood estimators and likelihood ratio tests in the model forcointegration under linear restrictions on the cointegration vectors 0 and weights a. These results are modifications of die procedure ^ven in Johansen (1988b) and apply the multivariate technique of partial canonical correlations, see Anderson (1984) or Tso (1981). For ii^erence we apply the results of Johamen (1989) on the asymptotic distribution of thelikelUuKxl ratio test. These disttibutiom aregivai in terms of a multivmate Brownian motion process and are tabidated in the Appendix. Inferences on a aiydfiimder linear restrictions can be amducted using the usual x^ distribution as an approximation to the distribution of likelihood ratio test. We also apply the limiting distribution of the tnaximum liketifaood estimator to a Wald test for hypotheses about a and 0. 169

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BULLETINI.I The Statistical Model Consider the model H,:X,=n,X,_,-l-... + ntX,_,-l-^ + 4»D,+e,,(/ = l,...,r), (1.1)

where £,,...,6^ are IINp(O, A) and X_ji.n,...,Xo we fixed. Here the variables D, are centered seasonal dummies which sum to zero over a full year. We assume that we have quarterly data, such that we include three dummies and a constant term. The unrestricted parameters (^, * , II,,...,n;t, A) are estimated on the basis of T observations from a vector autoregressive process. For a /ndimensional process with quarterly data this gives Tp observations and /> -I- 3p + Arp^ +/'(p +1 )/2 parameters. In general, economic time series are non-stationary processes, and VARsystems like (1.1) have usually been expressed in first differenced form. Unless the difference operator is also appliedto the error process and explicitly taken account of, differencing implies loss of information in the data. Using A = 1 - L, where L is the lag operator, it is convenient to rewrite themodd(l.l)as AX, = r,AX,_| + ...-l-rk_,AX,_4 + i + IIX,_t-l-/( + *D,-l-e,, where r,= - ( i - n , - . . . - n , ) , and Notice that model (L2) is expressed as a traditional first difference VARmodel except for theterm IIX,_^. It is the main purpose of this paper to investipte whether the coefficient matrix II contains information about long-run relationships between the variables in the data vector. There are t h r ^ possible cases: (i) Rank(Il)=p, i.e. the matrix II has full rank, indicating that the vector process X, is stationary. (ii) Rank(Il)=0, i.e. the matrix 11 is the null matrix Mid (1.2)...