FISHERY BULLETIN: VOL. Sfi. NO. 4 



cant. Figure 1 shows the data set with estimated 

 relationships, Equations (7)-(9), superimposed. It is 

 evident that the Box-Cox specification produces a 

 relationship of much bigger curvature and better fit 

 than the multiplicative-error specification. 



Oregon Coho Salmon Results 



McCarl and Rettig (1983) suggested that aggre- 

 gated (wild and hatchery) adult coho salmon produc- 

 tion (in thousands) is affected by smolt releases (in 

 millions), S, and upwelling index, U. The flexible 

 functional form for Oregon coho salmon can be ex- 

 pressed as Equation (5). The model was estimated 

 by iterated ordinary least squares with the follow- 

 ing maximum log-likelihood results: 



A(-o*5) =-50.08 + 93.715(-20) + 0.59[/«'<" (10) 

 (-1.17) (1.08) (3.37) 



R^ = 0.51, Durbin-Watson = 2.07, Log-likelihood 

 = -1.66. 



To detect any violations of the assumption regard- 

 ing the homoscedastic error term, a series of tests 

 were conducted by running regressions of squared 

 residuals (e-) or logs of (e-) on the predicted values 

 of A or the explanatory variables S and U. The 

 regression of e^ on all explanatory variables is 

 known as the Breusch-Pagan-Godfrey test and the 

 regression of log(e") on all explanatory variables 

 are known as the Harvey test (White 1987). Five 

 tests were conducted using the chi-square distribu- 

 tion, and the assumption of homoscedastic error fails 

 to be rejected at a 5% significance level. The same 

 conclusion is reached when model (2) was fitted by 

 Peterman (1981). The Box-Cox results are also 

 found to be free from autocorrelation problems, first 

 or higher orders. 



Empirical results as summarized in Equation (10) 

 indicate that the number of smolt released con- 

 tributes positively to adult production at a 15% 

 significance level. Upwelling also positively affects 

 adult production at a 1% significance level. The Box- 

 Cox results produce a nonlinear relationship be- 

 tween adult production and smolt release and an 

 output elasticity of less than one, suggesting that 

 the null hypothesis should be rejected. To formally 

 test the hypothesis of density independence, the 

 power transformations for A and S are restricted 

 to be 1.0 and the Box-Cox functional form is re- 

 estimated with the following results: 



A = -0.58 + 0.000655 + 0.084[/(''^^' (11) 

 (-1.2) (0.1) (3.6) 



R'^ = OAl, Durbin-Watson = 2.09, Log-likelihood 

 = -6.54. 



By comparing the values of the log-likelihood 

 functions for Equations (10) and (11) and follow- 

 ing a chi-square test with 2 degrees of freedom, it 

 is concluded that the hypothesis of density in- 

 dependence for Oregon coho salmon can be rejected. 

 The same conclusion was reached by Peterman 

 and Routledge (1983) and McCarl and Rettig 

 (1983). 



CONCLUSION 



The findings of testing the hypothesis of density- 

 independent marine survival for salmon and of the 

 effect of the number of smolts released on the vari- 

 ability of adult production have important implica- 

 tions for fishery managers as noted in the literature. 

 If the hypothesis of density independence fails to 

 be rejected, there is no technical maximum^ for the 

 adult production from releasing smolts. A technical 

 maximum of adult production exists when the num- 

 ber of smolts has a positive and decreasing effect 

 on adult production. If the variability of adult pro- 

 duction is positively affected by the number of 

 smolts, it will be useful for fishery managers and 

 the fishing industry to know the form of variability 

 to evaluate the effectiveness of salmon hatchery 

 operations. Further, fishery managers can improve 

 management strategies by considering the trade-off 

 between the mean and variance of adult production. 

 The hypothesis of density independence has been 

 tested extensively for different sets of data with con- 

 flicting results. Functional form selection and data 

 deficiency have been suggested as the causes of con- 

 flicting findings. 



Results of this study confirm that functional form 

 selection is critical in testing the hypothesis of den- 

 sity independence and estimating the form of the 

 variability of adult production. By using the ex- 

 tended Box-Cox functional form, it is concluded that 

 there exists a density-dependent relationship be- 



^A "technical maximum" refers to the maximum adult produc- 

 tion in physical terms. This may not be an appropriate objective 

 for fishery managers to achieve, because the release of smolts at 

 technical maxima may not generate maximum benefits to the 

 fishing industry. Maximization of the return to hatchery operations 

 appears to be a more suitable objective of a single-attribute model 

 to be accomplished without considering the risk factor. 



660 



