MENDELSSOHN: USING BOX-JENKINS MODELS IN FISHERY DYNAMICS 



a Cobbs-Douglas production function. He con- 

 cluded, among other things, that natural fluctua- 

 tions in resource availablity are significant, but 

 did not include them in his analysis, nor did he 

 provide a means for forecasting future catch. The 

 National Marine Fisheries Service, using a re- 

 gression model based on the previous year's catch, 

 water temperature, and salinity at the start of the 

 year, makes yearly predictions that have been 

 mixed in accuracy. 



Box-Jenkins models are autoregressive-inte- 

 grated-moving-average models, or ARIMA mod- 

 els. These are linear, stochastic models that can 

 describe fairly complex behavior, in contrast to 

 Parrish and MacCall (1978) who use highly non- 

 linear equations to model the fluctuations in 

 fishery data. 



The modeling is based on the properties of 

 stationary time series. A time series xt is station- 

 ary if it has a constant mean, and if the covariance 

 between events xt, xt-s depends only on s and not 

 on t. Many series are stationary after removing a 



deterministic trend. Others are differenced in 

 order to achieve stationary. Also, transforming 

 the time series, particularly using the Box-Cox 

 family of transformations, often improves the 

 behavior of the time series. The initial step then is 

 to transform and difference the data as necessary 

 to achieve stationary. It is convenient to use the 

 backshift operator B-i, where B->xt - xt-j, to de- 

 note lagged variables. Given the new series zt 

 = (1 - B )xt, a mixture of autoregressive and 

 moving average models are sought. Autoregres- 

 sive models are models that depend on the past 

 history of the time series: 



Zt = (b\Zt-i + (b2Zt-2 +■■•+ fi>pZt-p + at 



in terms of the backshift operator: 



(1 - <i>iB - (i)2fi2 -...- (bpBP)zt = at 



while moving average models depend on past 

 values of the noise or error: 



to 



0. 



UJ 



m 



IT 



o 



jFMAMJJASONDJFMAMJJASONDJfMAMJJASONDJFMAMJJASONDJFMAMJJASOND 



MONTHS 

 Figure 2. — Number of fishing trips per month by the Hawaii skipjack tuna fleet 1964-78, near Oahu, Hawaii. 



889 



