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2.1. UNRESTRICTED VECTOR AUTOREGRESSIONS 25Q = plim ³ 1 P Tt=1qT tq´,beapositivedeÞnite 0 tmatrix of constants which ⇐(5)exists by the law of large numbers and the covariance stationarity assumption.Then, as T →∞√T ( ˆβ − β) → D N(0, Ω), (2.2)where Ω = Σ ⊗ Q −1 . The asymptotic distribution can be used to test ⇐(6)hypotheses about the β vector.Lag-Length DeterminationUnless you have a good reason to do otherwise, you should let the datadetermine the lag length p. If the q tare drawn from a normal distribution,the log likelihood function for (2.1) is −2ln|Σ| + c where c is aconstant. 3 If you choose the lag-length to maximize the normal likelihoodyou just choose p to minimize ln | ˆΣ p |,where ˆΣ p = 1 P Tt=p+1 ˆ²T −p tˆ² 0 tis the estimated error covariance matrix of the VAR(p). In applicationswith sample sizes typically available to international macroeconomists—100 or so quarterly observations—using the likelihood criterion typicallyresults in choosing ps that are too large. To correct for the upwardsmall-sample bias, two popular information criteria are frequently usedfor data-based lag-length determination. They are AIC suggested byAkaike [1], and BIC suggested by Schwarz [125]. Both AIC and BICmodify the likelihood by attaching a penalty for adding additional lags.Let k be the total number of regression coefficients (the a ij,r coef-Þcients in (2.1)) in the system. In our bivariate case k =4p. 4 Thelog-likelihood cannot decrease when additional regressors are included.Akaike [1] proposed attaching a penalty to the likelihood for addinglags and to choose p to minimizeAIC = 2 ln | ˆ Σ p | + 2kT .3 |Σ| denotes the determinant of the matrix Σ.4 This is without constants in the regressions. If constants are included in theVAR then k =4p +2.