a mixture of damped exponentials and/or sine waves while the partial

autocorrelations cut off for lags greater than P. Thus, if the esti-

mated autocorrelation and partial autocorrelation functions exhibit this

behavior, the process is identified as Pth order autoregressive.

 Max x2 Test


 The Max x2 test was developed by McClave (1978). This procedure

involves step-up sequential hypothesis testing of estimated autoregres-

sive processes of increasing orders. The theoretical distribution of

the test statistic corresponds to that of the maximum order statistic of
 2
a sequence of k independent xI, random variables, where k is the order

of the autoregressive process being tested. This test is based mainly

on asymptotic results and as such is of limited value when sample size

is small.

 Akaike's Final Prediction Error (FPE) Test


 The Akaike's FPE test is based upon the fact that the variances of

autoregressive processes of increasing orders generate a monotonic

decreasing sequence (Akaike, 1969, 1970). By adjusting this sequence of

variance estimates for degrees of freedom (a monotonic increasing

adjustment), a U-shaped function of adjusted variances (FPE) as plotted

against the order parameters results. The order parameter corresponding

to the minimum of this function is then chosen as the order of the

autoregressive process in question.