59
After the ARIMA model has been identified and estimated, the model
should be checked to determine if the specification is correct. The
residuals (innovations) of an estimated ARIMA model are given as
5t 4>CB) 81CB)xt
If the model has been correctly specified, the residuals are white
noise; i.e., the residuals are not dependent on their own past. Thus,
the sample autocorrelation function of the residuals (rt) given as
. ยก h-*
r,
would be approximately zero for lags (k) greater than zero. If the
model is correctly specified, the residual autocorrelations are indepen
dent, normally distributed random variables with mean zero and variance
1/T, where T is the number of observations (Pindyck and Rubinfeld,
1981). A test is then performed using the statistic Q (Box and Pierce,
1970) given as
Q
for the first K residual autocorrelations. The Q statistic is dis
tributed as chi square with K-p-q degrees of freedom. If Q is greater
than the tabulated critical value, the hypothesis that the residuals are
white noise is rejected. In this case, an alternative ARIMA model is
selected and the procedure repeated