The estimated residual sequences used in identifying the order of

the autoregressive components in the catch equations were generated by

equation (54). Each estimated residual sequence constituted a sta-

tionary time series with T = 19 observations. Due to the small sample

size, identification of the order of the autoregressive processes was

extremely difficult. The difficulty in identification arose for two

reasons. First, estimation of the autocorrelation coefficients incurs

biased estimates, with the bias of magnitude O(T-1). In that T is

small, significant bias existed in the estimated autocorrelation

coefficients. Secondly, many of the standard identification techniques

employ variance estimates inversely proportional to the series length.

The short length of the series, resulted in rather large standard

errors for the estimated autocorrelation coefficients.

 Four different identification procedures were utilized in testing

the estimated residual sequences. These tests are briefly presented

below. The interested reader may refer to the cited references for more

detailed discussion of these procedures. The results of this analysis,

taken as a whole, suggested a first order autoregressive process was

present in each of the estimated residual sequences.

 Bartlett Test


 A discussion of the Bartlett identification test can be found in

Granger and Newbold (1977). This test is somewhat ad hoc in nature.

Basically, this procedure involves comparing the estimated autocorrela-

tion and partial autocorrelation function with those generated by a

theoretical time series process. For example, given an autoregressive

process of order P, the autocorrelation function tails of according to