THEORETICAL AUTOCORRELATIONS j Cov yt yt j Var yt E yt E yt yt j E yt E yt E yt 2 j 1 2 L 0 1 and Cov yt yt j is often denoted by j while Var yt if often denoted by 0 Note that j j and j j and because of this symmetry the theoretical autocorrelation function and the sample autocorrelation function below only need be examined over the positive lags j 1 2L SAMPLE AUTOCORRELATIONS T rj y t j 1 t y yt j y T y t 1 t y 2 j 1 2 L The rj are consistent estimators of the theoretical autocorrelation coefficients j Under the assumption that yt follows a white noise process the standard errors of these rj are approximately equal to 1 T Thus under the null hypothesis that yt follows a white noise process roughly 95 of the rj should fall within the range of 1 96 T If more than 5 of the rj fall outside of this range then most likely yt does not follow a white noise process THEORETICAL PARTIAL AUTOCORRELATIONS jj Cov yt yt j yt 1 L yt j 1 Var yt yt 1 L yt j 1 E yt E yt yt 1 L yt j 1 yt j E yt yt 1 L yt j 1 E yt E yt yt 1 L yt j 1 2 the correlation between yt and yt j j 1 2 L after netting out the effects the intervening values yt L yt j 1 have on both of them SAMPLE PARTIAL AUTOCORRELATIONS jj are calculated using the formulas for the theoretical autocorrelations for a given ARMA p q model see my ACF PACF Table doc Word document for the formulas but replacing all of the theoretical autocorrelations j with the above sample autocorrelations rj and all of the unknown Box Jenkins coefficients i i with their corresponding estimates obtained by the method of moments or some other i i method The jj are consistent estimators of the theoretical partial autocorrelations jj Under the assumption that yt follows a white noise process the standard errors of these are approximately equal to 1 T Thus under the null hypothesis that y follows a jj t white noise process roughly 95 of the jj should fall within the range of 1 96 T If more than 5 of the jj fall outside of this range then most likely yt does not follow a white noise process GOODNESS OF FIT MEASURES 1 AIC Akaike Information Criterion AIC 2 L a t2 2 K where K p q 1 L a t2 the log of the likelihood function of the BoxJenkins ARMA p q model a t the residual at time t for the Box Jenkins model and the log likelihood function L a t2 is a monotonically decreasing function of the sum of squared residuals a t2 In other words the smaller larger L a t2 is and vice versa a 2 t is the 2 SBC Schwartz Bayesian Criterion SBC 2 L a t2 K ln n SBC 2 L a t2 K ln n where n is the number of residuals computed for the model In terms of choosing a Box Jenkins model the smaller these goodness of fit measures the better That is we prefer the Box Jenkins model that has the smallest AIC and SBC measures Notice that as you add coefficients to the Box Jenkins model i i the fit of the model as measured by the sum of squared residuals a 2 t always decreases and therefore adding coefficients always increases the log likelihood L a t2 of the BoxJenkins model To offset the tendency for adding coefficients to a model just to improve its fit the above goodness of fit information criteria each include a penalty term For the AIC criterion the penalty term is 2K while for the SBC measure the penalty term is Kln T Thus with these criteria as one adds coefficients to the Box Jenkins model the improvement in fit coming from reduction in the sum of squared residuals will eventually be offset by the penalty term moving in the opposite direction The goodness of fit criteria are then intended to keep us from building large order BoxJenkins models just to improve the fit just to find that such large order models don t forecast very well Shibata 1976 has shown that for a finite order AR process the AIC criterion asymptotically overestimates the order with positive probability Thus an estimator of the AR order p based on AIC will not be consistent By consistent we mean that as the sample size goes to infinity the correct order of an AR p Box Jenkins model will be correctly chosen with probability one In contrast the SBC criterion is consistent in choosing the correct order of an AR p model Often these two criteria choose the same Box Jenkins model as being the best model However when there is a difference in choice the AIC measure invariably implies a Box Jenkins model of bigger order K p q 1 than the order of the model implied by the SBC criterion In other words the SBC criterion tends to pick the more parsimonious model when there is a split decision arising from using these criteria Personally I prefer to rely on the SBC criterion in the case of split decisions A TEST FOR WHITE NOISE RESIDUALS and thus the Box Jenkins model s completeness H0 Residuals of Estimated Box Jenkins model are white noise i e uncorrelated at all lags Other things held constant the estimated Box Jenkins model is adequate H1 Residuals of Estimated Box Jenkins model are not white noise In this case a better model can be found by adding more parameters to the model The chi square test used to test for white noise residuals is calculated as m n n 2 2 m j 1 rj2 a t n j where n k rj a t a a t t k t 1 n a t 1 2 t n number of residuals and a t is the time t residual of the Box Jenkins model This statistic was suggested by Ljung and Box 1978 and is called the Ljung Box chi square statistic for testing for white noise residuals The null hypothesis above is accepted if the observed chi square statistic is small i e has a probability value greater than 0 05 and is rejected if the chi square statistic is large i e has a probability value less than 0 05 As far as the choice of the number of lags m to use I would suggest m 12 for quarterly data and m 24 for monthly data to increase the power of the test given the frequency with which the data is observed CONSTRUCTION OF THE P Q BOX In this class we will be constructing a P Q Box of the form 0 Q 1 2 0 P 1 2 where represents the following numbers in each cell AIC SBC m2 and the p value of the …