FISHERY BULLETIN: VOL 78, NO. 4 



zt = at — O^at-i - 020-1-2 -■■■— Oqat-q 



or: 



Zt = a - 0,B - 628^ -...- OqB'^)at. 



A model that has both moving average and auto- 

 regressive parameters is a mixed autoregressive 

 moving average model, whose representation in 

 terms of the backshift operator is: 



(1 - (hiB - 626'' -...- (bpBPMl - B)'^xt 



a - e^B - 02B'- 



6qB'^)at. 



MODEL IDENTIFICATION 



The first step in the Box-Jenkins modeling 

 process is to use properties of the data to tenta- 

 tively identify a model. Even if a multivariate 

 model (i.e., a model based on catch and effort) is 

 the ultimate goal, univariate models of each series 

 are constructed first. Often the univariate model 

 produces forecasts that are almost as accurate as 

 the multivariate model forecast. 



My procedure was to identify, estimate, and 

 check a series of models based on the data from 

 January 1964 through July 1977. These models 

 were used to forecast the already observed catch 



and effort for the period August 1977-December 

 1978. The models with the best "fit" were then 

 reestimated to make the forecast for 1979. To 

 make clear the feedback nature of identification, 

 estimation, and checking in Box-Jenkins models, 

 results from models fixed to 163 and 180 mo of data 

 are intermingled, but clearly labeled. 



A tentative model can be identified by esti- 

 mating the autocorrelation and partial autocor- 

 relation functions for each series. These are shown 

 in Figures 3 and 4. Significant is the undamped 

 sinusoidal behavior of each, with a period of 12 mo. 

 Failure of both the autocorrelation and partial 

 autocorrelation functions to go to zero is a sign 

 of a nonstationary series, and the need for dif- 

 ferencing. The 12-mo period suggested a yearly 

 seasonal model, so that twelfth differences were 

 taken, i.e., 2? = (1 - B^^)xt. 



The estimated autocorrelation and partial auto- 

 correlation functions for the differenced catch and 

 effort series are given in Tables 1 and 2. Following 

 guidelines in appendix 9.1 in Box and Jenkins 

 (1976), seasonal models with period s of the form: 



Zt = (1 - OiB - e2B^)a - eiB'}at da) 



or zt= a - BiB - diB^) 



(1 - OiB' - Q2B^')at (lb) 



-05 





12 



24 30 



LAG IN MONTHS 



36 



42 



48 



Figure 3. — Estimated total catch autocorrelation function for the catch of skipjack tuna near Oahu, Hawaii, 1964-78. 



890 



