VAR Order Selection and Checking the Model Adequacy

In the previous chapter, we have assumed that we have given a K-dimensional multiple time series % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa % aaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamyEamaaBaaa % leaacaWGubaabeaakiaacYcacaaMe8Uaam4DaiaadMgacaWG0bGaam % iAaiaaysW7caWG5bWaaSbaaSqaaiaadshaaeqaaOGaeyypa0ZaaeWa % aeaacaWG5bWaaSbaaSqaaiaaigdacaWG0baabeaakiaacYcacqWIMa % YscaGGSaGaamyEamaaBaaaleaacaWGlbGaamiDaaqabaaakiaawIca % caGLPaaadaahaaWcbeqaaOGamai4gkdiIcaacaGGSaaaaa!54F7! $$y_1 , \ldots ,y_T ,\;with\;y_t = \left( {y_{1t} , \ldots ,y_{Kt} } \right)^\prime , $$ which is known to be generated by a VAR(p) process, 4.1.1 % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa % aaleaacaWG0baabeaakiabg2da9iaadAhacqGHRaWkcaWGbbWaaSba % aSqaaiaaigdacaWG5bGaamiDaiabgkHiTiaaigdaaeqaaOGaey4kaS % IaeSOjGSKaey4kaSIaamyqamaaBaaaleaacaWGWbaabeaakiaadMha % daWgaaWcbaGaamiDaiabgkHiTiaadchaaeqaaOGaey4kaSIaamyDam % aaBaaaleaacaWG0baabeaakiaacYcaaaa!4CF4! $$ y_t = v + A_{1yt - 1} + \ldots + A_p y_{t - p} + u_t , $$ and we have discussed estimation of the parameters % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maai % ilaiaadgeadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiil % aiaadgeadaWgaaWcbaGaamiCaaqabaGccaGGSaGaaGjbVlaadggaca % WGUbGaamizaiaaysW7cqGHris5daWgaaWcbaGaamyDaaqabaGccqGH % 9aqpcaWGfbWaaeWaaeaacaWG1bWaaSbaaSqaaiaadshaaeqaaOGabm % yDayaafaWaaSbaaSqaaiaadshaaeqaaaGccaGLOaGaayzkaaGaaiOl % aaaa!5048! $$ \nu ,A_1 , \ldots ,A_p ,\;and\;\sum _u = E\left( {u_t u'_t } \right). $$ In deriving the properties of the estimators, a number of assumptions were made. In practice, it will rarely be known with certainty whether the conditions hold that are required to derive the consistency and asymptotic normality of the estimators. Therefore statistical tools should be used in order to check the validity of the assumptions made. In this chapter, some such tools will be discussed.