LIN an-i WILLIAMS: INFLUENCE OF SMOLT ON ADULT SALMON 



adult production, implying density independence. 

 When the hypothesis of density independence is re- 

 jected, output elasticity is less than 1.0. Therefore, 

 there is an one-to-one correspondence between 

 the hypothesis of density independence and the 

 value of output elasticity. The purpose of using 

 the concept of output elasticity here is to facilitate 

 the discussion of the restriction inherent in model 

 (4). 



There is no theoretical support for imposing the 

 restriction of constant output elasticity, rather it 

 should be treated as a hypothesis to be tested. More 

 important, body size (B) is likely to have a positive 

 effect on the output elasticity, i.e., de^JdE > 0. In 

 other words, when body size of smolts is enlarged, 

 the improved ability of enduring unfavorable envi- 

 ronmental conditions should increase the incre- 

 mental return rate of adult salmon. But, a constant 

 output elasticity implies that body size and the out- 

 put elasticity are independent. 



The comparison between model (3) and model (4) 

 centers around the role of body size in the variability 

 of adult production. However, data on body size is 

 unavailable so that the comparison becomes em- 

 pirically irrelevant. Consequently, the difference 

 between these two m.odels, in essence, rests on 

 model specification. It should also be pointed out 

 that the estimate of h'''{S,B) is influenced by the 

 functional form of f{S,B) and vice versa, because 

 h'-{S,B) is the heteroscedastic error term to be 

 handled by the weighted least squares method. It 

 is, therefore, important to select a more general 

 functional form for the mean and variance of adult 

 production in testing the hypothesis of density in- 

 dependence and in estimating the variability in adult 

 production. 



The Box-Cox flexible functional form developed 

 by Box and Cox (1964) and extended by Zarembka 

 (1974) has been a popular tool for both discrimi- 

 nating among alternative functional forms and pro- 

 viding added flexible form in model specification 

 (Moschini and Meilke 1984). The extended Box-Cox 

 functional form for relating adult salmon produc- 

 tion to smolts and other explanatory variables X 

 (such as upwelling) can be expressed as 



^<^' = a. + a,SW + a.X'S' + 



where A*'*' = 



S^^^ = 



(A^ - 

 In A 



1)/A 



(5f^^ - I)/ IX 

 In 5 



(5) 



for A ?t 

 for A = 



for ^ 9^ 

 for /.i = 



v(e) ^ i i^Q  

 \\nX 



e ~ NID(0, o2). 



1)1 B 



for ^ 

 for = 



Model (5) includes the linear (A = /.< = = 1), 

 multiplicative-error or log-log (A = f^ = = 0), and 

 log-linear (A = 0, /i = = 1) functional forms as 

 special cases. Therefore, models (3) and (4) are 

 special cases of model (5), which allows both non- 

 constant output elasticity and nonzero effect of X 

 on the output elasticity.^ 



When the variability of adult salmon production 

 is affected by the values of its explanatory variables, 

 the error term has nonconstant variance, i.e., 

 heteroscedasticity. Zarembka (1974) demonstrated 

 that while the Box-Cox model is fairly robust to 

 departures from normality, it is sensitive to hetero- 

 scedasticity. Failure to correct for this problem can 

 generate misleading results (Lahiri and Egy 1981). 

 When heteroscedasticity is present, we can also 

 specify a Box-Cox functional form for the variance 

 of the error term in model (5) as the following: 



e ~ NID{Q, h{S;X) o|) 

 where h{S^) = p^S^'^ + [i^X^^K 



(6) 



The parameters, a,, /3,, A, ^x, 0, t, and ^ can be 

 estimated by maximum-likelihood algorithms (see 

 Appendix for a discussion of the log-likelihood func- 

 tion and estimation methods). The hypothesis of den- 

 sity independence can be tested by estimating the 

 model with the restriction that A = fi = 1 against 

 the unrestricted model. 



'The superior flexibility of model (5) compared with models (3) 

 and (4) is an important consideration in testing the hv'pothesis of 

 density independence in light of the following remarks on the com- 

 parison of models (1) and (2) in Peterman (1981, p. 1117): 



"This is not to say that model 2 is the 'true' form of natural 

 variability, because there are numerous other models that were 

 not tested here (many of these alternatives cannot be tested in 

 practice). ..." 



We also cannot claim that the extended Box-Cox functional form 

 can produce the "best" or "true" functional form. There exist other 

 flexible functional forms, such as Fourier (Gallant 1984), and the 

 literature is silent in the comparison of these flexible functional 

 forms. 



Even though the Box-Cox functional form was first proposed 

 in 1964, its application and investigation of its statistical proper- 

 ties have not received much attention until recently. Therefore, 

 there are numerous aspects of transformations that merit further 

 study (Box and Cox 1982). Lacking software support also makes 

 its application difficult. Nevertheless, the superior flexibility of the 

 Box-Cox functional form compared with other functional forms 

 used traditionally is evident and its application should be encour- 

 aged. 



657 



