Order determination99found for the regression (1.1) with t = 1, . . . , T and k �1 lags. In the second step t is analysed in an auxillary regression for t = m + 1, . . . , T, where t is regressed on t� , . . . , t as well as the original regressors Xt� = (Xt� , . . . , Xt+1 ) and Dt . The original regressors are included to mimic the above likelihood analysis where Xt� , Dt are partialled out from Xt and Xt . A test based on the squared sample correlation of the variables in the auxillary regression is asymptotically equivalent to the likelihood ratio tests, so the degrees of freedom do not include the dimension of Xt� , Dt . In the multivariate case, p > 1, the test can be implemented in three ways, using either a simultaneous test, a marginal test or a conditional test. The joint test, is based on the test statistic tr(T R2 ), where R2 is the squared sample multiple correlation of t and (t� , . . . , t , Xt� , Dt ) . The other two tests are based on a q-dimensional subset of the p components of t . As the equations in the model equation (1.1) can be permuted there is no loss of generality in focussing on the st q components. Thus, partition t = t,1 t,2 , Xt = Xt,1 Xt,2 ,where t,1 and Xt,1 are q-dimensional. The marginal model consists of the st q equations of (1.1), that is Xt,1 given Xt� , Dt . The marginal test is then based on the squared sample multiple correla2 tion, Rmarg say, of t,1 and (t�,1 , . . . , t,1 , Xt� , Dt ). The conditional model consists of the st q equations of (1.1) given Xt,2 , that is Xt,1 given Xt,2 , Xt� , Dt . The conditional test is based on the squared sample 2 multiple correlation, Rcond say, of t,1 and (t�,1 , . . . , t,1 , Xt,2 , Xt� , Dt ). The following asymptotic result can be established. Theorem 2.7. Suppose Assumptions (2.1), (2.2), (2.5) are satisd and k0 < k. 2 2 Then tr(T R2 ) is asymptotically 2 (p2 m), while tr(T Rmarg ) and tr(T Rcond ) are as2 2 ymptotically (q m). Sometimes these test are implemented so that the auxillary regression is carried out for t = 1, . . . , T rather than t = m+1, . . . , T with the convention that 0 = = 1 = 0. Variants of the tests have been considered, in particular for the univariate case, by Durbin [7], Godfrey [8], Breusch [3] and Pagan [19]. Those variants have been argued to be score/Lagrange multiplier type tests and asymptotic theory has been established for the stationary case |(B)| < 1. 3. Proofs The likelihood ratio test statistic for testing Ak = 0 is given by LR (Ak = 0) = log det(︹1 ) k� = log det{Ip �︹1 (� � )}, k�(3.1) where and � represent the unrestricted and restricted maximum likelihood estimators for the variance matrix deed below. In the following st some notation is introduced. Then comes an asymptotic analysis of � and � � and ally proofs of the main theorems follow.
