FISHERY BULLETIN: VOL. 78, NO. 4 



fied, overfitting is tried, that is extra parameters 

 are added to see if they are found to be not 

 significantly different from zero. 



To insure that I found the simplest model 

 possible, I fitted first the model zt = (1 - diB) 

 (1 - QiB^^)at, and then added parameters as 

 seemed necessary based on the diagnostic check- 

 ing. The estimates for Model (1) for catch and 

 effort are given in Tables 3 and 4. Estimates 

 using two estimation techniques, one using back- 

 forecasting and one suppressing it, are presented. 

 Some programs do not have a backforecasting 

 feature; my experience is that the estimated 

 models obtained using backforecasting are far 

 superior, as can be seen in the tables presented. 



Table 3. 



Parameter 



-Parameter estimates for effort model, Model (1) 

 (see text). (Based on 180 observations.) 



Estimate 



supressing 



backtorecasting 



SE 



Estimate using 

 backforecasting 



SE 



Table 4. — Parameter estimates for catch model, Model (1) 

 (see text). (Based on 163 observations.) 



Parameter 



Estimate suppressing 

 backforecasting 



SE 



The estimated autocorrelation and partial auto- 

 correlation functions of the residuals from both 

 models are given in Tables 5 and 6. For the effort 

 series, there is no sign of a lack of fit, while for 

 the catch series terms of lag three or four are 

 suggested. An overspecified model: 



Zt 



= (1 - B^B^ - O2B'' - OsB"" - e^B^) 



(1 - QiB^'')at (2) 



was estimated for both the catch and effort time 

 series. The results are summarized in Tables 7 and 

 8. The estimated autocorrelation and partial auto- 



892 



correlation functions of the residuals (not shown) 

 show no sign of additional lags or trend. The test 

 statistic that the residual series are not signifi- 

 cantly different from white noise gave no reason 

 to doubt the models adequacy, and overfitting by 

 including a GaB^"* term found this term to be 

 nonsignificant. 



TRANSFER FUNCTION MODELS 



If both the catch time series, say yt, and the 

 effort time series, say xt, have been suitably 

 transformed so that the resulting series are sta- 

 tionary, a transfer function of the form: 



(1 - 81B - SaB^ -...- ^rB'')xt 



(ojo — oiiB — oiiB' 



o)sB^)yt-b + rjt 



can be estimated where rjt is not assumed to be 

 white noise, but itself can be modeled as an 

 autoregressive-moving average process of a^. 



The procedures for identifying and estimating a 

 transfer function model are similar to those for the 

 univariate model, except that attention is focused 

 on the estimated cross-correlation function be- 

 tween the "pre whitened" catch and effort series. 

 Series are prewhitened if they are reduced to the 

 residuals left from a given model. In this instance, 

 both series are prewhitened by the univariate 

 model for effort estimated in the preceding sec- 

 tion. The estimated correlation function, impulse 

 response function, and residual noise autocorrela- 

 tion function are given in Table 9. The estimated 

 autocorrelation function for the noise is similar 

 to the original univariate autocorrelations, sug- 

 gesting a noise model of the form: 



T,, = (1 - 0iB - diB"" - dzB^ - d^B^) 



(1 - Q^B^'')at. (3) 



Based on guidelines in Box and Jenkins ( 1976:386- 

 388) and knowledge of the fishery, two models 

 were hypothesized: 



a-B^'')yt= {ojo){l-B^'')xt+ y]t (4) 



and: {1 - h^B - h^B"") (l - B^^)yt 



= (ojo - wi5 - waB^) (1 - B'^)JCi + r]t. (5) 



Tables 10 and 11 summarize the estimates when 



