FISHERY BULLETIN: VOL. 86. NO. 4 



RESULTS 



In order to test the hypothesis of density inde- 

 pendence for salmon utilizing the extended Box-Cox 

 flexible functional form, the two data sets analyzed 

 by McCari and Rettig (1983) were also used here. 

 The first data set contains total Hokkaido hatchery 

 chum salmon fry releases and brood year adult 

 returns for the years 1950 through 1969 (Moberly 

 and Lium 1977). The second data set pertains to 

 Oregon coho salmon for the years 1960 through 

 1980 (Oregon Department of Fish and Wildlife 

 1982). This latter data set was also analyzed by 

 Clark and McCarl. 



Hokkaido Chum Salmon Results 



Due to the lack of data on body size and other 

 factors affecting the survival rate of fry, adult pro- 

 duction (in thousands) is estimated with the single 

 explanatory variable, number of fry released (in 

 millions). As explained in the Appendix, the depen- 

 dent variable is divided by its geometric mean of 

 3,332,440. The iterated weighted least squares 

 method produces the following maximum log-likeli- 

 hood results with the ^statistics given in parenthe- 

 ses below the coefficients: 



^(-0.4) ^ _ 1,254 -l,756.3S'-i-4) (7) 



(-4.07) (4.07) 



R- = 0.48, Durbin-Watson = 1.5, Log-likelihood 

 = -7.89. 



The weight used to remove heteroscedasticity is 

 5(1.06) ^ (51.06 _ i)/i.o6. This implies that as the 

 number of fry is increased by 1%, the standard 

 deviation of the adult production increases by 1.06%, 

 which is much smaller than the 2.5% reported by 

 McCarl and Rettig (1983). 



The above results suggest that the output elas- 

 ticity" of fry is 1,756.3A''^S "i-^. When evaluated at 

 the mean values oiA and S (which are 1.1278 and 

 288.36, respectively), a percentage increase in the 

 number of fry increases the adult production by 

 0.66%, implying density dependence. In addition, 



^Equation (7) can be written as 



{A.-^-^ - l)/(-0.4) = -1254 + 1756.3(S-'-^ 

 A-"^" = 0.8 -h 501.85-^'*. 



l)/(-1.4), or 



Output elasticity £„^ = (%M)/(%A5) = d.A\dS) {SI A) = 

 l.TSe.SS-^M"" dtJdS = -2,458.8S-^-M"'^ < 0. 



658 



this output elasticity is a decreasing function of fry 

 releases. In contrast to these results, McCarl and 

 Rettig (1983) reported a constant output elasticity 

 of 1.09 which was not found to be statistically dif- 

 ferent from 1.0, supporting density independence. 

 The hypothesis of density independence was for- 

 mally tested by estimating the linear relationship 

 between A and S (i.e., the power transformations 

 for A and S are restricted to be one) by using the 

 weighted least squares method (S was treated as the 

 weight) with the following results: 



A = 0.18 + 0.0034S 

 (0.6) (2.84) 



(8) 



R" = 0.31, Durbin-Watson = 1.04, Log-likelihood 

 = -12.56. 



The weighted least squares method produces a 

 Durbin-Watson value of 1.04 which is below the 

 lower limit of its critical value, suggesting the possi- 

 ble existence of autocorrelation. However, the 

 Durbin-Watson value is well known to be below the 

 lower limit (or above the upper limit) which could 

 be the cause by model misspecification or autocorre- 

 lated error terms. Because the use of improper 

 functional form is a model misspecification, the 

 extended Box-Cox functional form needs to be 

 explored before assuming the existence of an 

 autocorrelation problem in light of low (or high) 

 Durbin-Watson statistics. The extended Box-Cox 

 results have a Durbin-Watson statistic of 1.50 (im- 

 plying no autocorrelation), and hence, it is concluded 

 that the low Durbin-Watson value in Equation (8) 

 is a result of incorrect functional form. Since the 

 Durbin-Watson statistic is for detecting first-order 

 autocorrelation, the least squares procedure de- 

 scribed in Pagan (1974) was applied to test higher- 

 order autocorrelation. It is concluded that the Box- 

 Cox results are free from autocorrelation problems, 

 first or higher orders. 



The hypothesis of density independence can be 

 tested by comparing the log-likelihood values of 

 Equations (7) and (8). The test statistic of twice the 

 difference between the log-likelihood functions 

 under the two specifications follows a chi-square 

 distribution with the number of degrees of freedom 

 equal to the number of restrictions (Theil 1971). This 

 test procedure is similar to the Akaike Information 

 Criterion (Akaike 1974) and has the advantage of 

 testing the significance of the difference between 

 the log-likelihood functions of different model spe- 

 cifications. It is concluded that the density-indepen- 



