where I, denotes the d x d identity matrix, and: The empirical levels of the portmanteau test statistics Q,(v) and Q},(v) for the PVAR models without and with parameter constraints are presented in Tables | and II, respectively. As expected, the test statistics Qj,(v) exhibited better empirical levels than Qy,(v): as in the PAR case, the factor correction proposed by McLeod (1994) improved the y* approximation for the test statistic O7, (9), offering generally better finite-sample properties than Q,/(v), particularly as M increases. We concentrate the rest of our discussion on Q},(v) only. Generally, the EMPIRICAL LEVELS (IN PERCENTAGE) OF THE PORTMANTEAU TEST STATISTICS Q yy (v) DEFINED BY EQN (34), AND ITS MODIFIED VERSION Qj/(v) DEFINED BY EQN (35), FOR THE PVAR MoDEL WITHOUT CONSTRAINTS ON THE PARAMETERS, GIVEN BY EQN (38) 7° distribution provided a satisfactory approximation for all lags, at both significance levels. The results for the models without and with parameter constraints were very comparable. For N = 200, some overrejection has been observed at season v = 3 for DGP, and DGP3, but in general the rejection rates at the 5% and 10% nominal levels are within two standard errors of 5% and 10%, respectively, or very close to these intervals. For N = 400, almost all corresponding empirical levels of Qj,(v), v = 1, 2,3, 4, lie within the 5% significant limits. EMPIRICAL LEVELS (IN PERCENTAGE) OF THE PORTMANTEAU TEST STATISTICS Q yy (v) DEFINED BY EQN (34), AND ITS MODIFIED VERSION Q},(v) DEFINED BY EQN (35), FOR THE PVAR MODEL WITH CONSTRAINTS ON THE PARAMETERS, GIVEN BY EQN (39) From this limited empirical study, the finite-sample performance of the test statistics seems rather reasonable, particularly for moderate to large sample sizes. Given the number of parameters involved in vector periodic time series, it is not really surprising that moderate to large sample sizes are needed in order to have satisfactory results. Overall, Q;;(v), v = 1,...,s and Qj, can be recommended for diagnosing PVAR models. TABLE I EMPIRICAL LEVELS (IN PERCENTAGE) OF THE GLOBAL PORTMANTEAU TEST Qiy AND Qi, DEFINED BY EQNS (36) AND (37) FOR THE PVAR MODEL WITHOUT CONSTRAINTS ON THE PARAMETERS, GIVEN BY En (38) TABLE IV EMPIRICAL LEVELS (IN PERCENTAGE) OF THE GLOBAL PORTMANTEAU TEST Qy AND Qj, DEFINED BY E@ns (36) AND (37) FOR THE PVAR MODEL WITH CONSTRAINTS ON THE PARAMETERS, GIVEN BY EQN (39) TABLE III In order to have stationarity, the data are transformed by applying the first difference of the logarithm for each variable, giving a bivariate time series of sample size equal to n = 111. Because we analyse quarterly data, the period v = 4 is naturally selected. The two time series are displayed in Liitkepohl (2005, Figure 17.1). A strong seasonal pattern is observed. LEAST SQUARES ESTIMATORS USED TO FIT THE WEST GERMAN DATA TO A BIVARIATE PVAR MODEL WITH v = 4, SUCH THAT THE AUTOREGRESSIVE ORDERS, OBTAINED BY THE BIC CRITERION DEfiNED BY Een (40), ARE GIVEN BY (p(1), p(2), p(3), p(4)) = (2.1.3.1), WITH CONSTRAINTS ®, ;;(1) = 0, ®, 9)(1) = 0, By 12(1) = 0, Bz o0(1) = 0, ®) 5)(2) = 0 AND ® 59(2) = 0; THE STANDARD ERRORS ARE GIVEN IN PARENTHESES