or habit formation. And the exponentially distri- 

 buted lag is but one of a large class of more com- 

 plex lag models (Griliches 1967; Kmenta 1971; 

 Rao and Miller 1971). 



Application of the partial adjustment model to 

 the demand Equation (2) results in the following: 



4 



Q*t = ^0 + ^a.x* + a,ql^ + u, (8) 



where the coefficients a, can be interpreted in 

 terms of the coefficients of Equation (2) as follows: 



at = yb,;i = 1, 



. .4 



a. 



(1- y). 



STATISTICAL PROCEDURES 



For a given value of the transformation 

 parameter, k, the coefficients of either the 

 equilibrium model [Equation (2)] or the partial 

 Adjustment model [Equation (8)] can be esti- 

 mated by the ordinary least squares method. Two 

 statistical issues requiring further development, 

 however, are the selection of the "best" value for 

 A, and the test for significance of the lagged ad- 

 justment parameter. An appropriate procedure 

 for estimation of K was first suggested by Box and 

 Cox (1964). The procedure is more clearly 

 explained in the linear regression context by 

 Kmenta (1971) and is reviewed by Chang (1977). 

 For a fixed value of A, the linear regression proce- 

 dure yields an estimate of the error variance &^. 

 Box and Cox showed that the maximized log 

 likelihood is, except for a constant, 



^max (^) = -(N/2) log d-2 (A) + (A - 1)S log 9,. (9) 



A maximum likelihood estimate of A can, there- 

 fore, be found by searching through successive 

 values of A to maximize Equation (9). The use of 

 this likelihood function implies, of course, that the 

 error terms conform to full normal theory as- 

 sumptions, i.e., that the w, are independently 

 normally distributed with zero mean and con- 

 stant variance. An approximate 100% (1 - a) 

 confidence region for A is defined by 



■^max *^^) ~ L 



(A) < 1/2 x,Ha) (10) 



where XiHa) represents the value of the chi- 

 square distribution with 1 df (Box and Cox 1964). 



272 



FISHERY BULLETIN: VOL. 78, NO. 2 



Serial correlation in the errors of the regression 

 model raises problems in the interpretation of the 

 test statistics for the nonlagged variables and the 

 lagged adjustment parameter, and contradicts 

 the assumptions of the log likelihood function. 

 Careful examination of the hypotheses and 

 statistics regarding the residuals of the regres- 

 sion equation is clearly necessary. Existence of 

 serial correlation in the errors of the static de- 

 mand model can be tested with the Durbin- 

 Watson statistic. If no serial correlation is appar- 

 ent in the residuals, then neither the distributed 

 lag model nor the serial correlation model need be 

 considered. If serial correlation is present in the 

 residuals of the static model, then the problem is 

 to distinguish between the distributed lag model 

 and the serial correlation model. 



Griliches (1967) showed that the serial correla- 

 tion and lagged adjustment models cannot be dis- 

 tinguished by a simple ^test on the adjustment 

 parameter. For example, if errors generated by a 

 first order Markov process, i.e., e^ = se,_i + Uj, 

 occur in a regression equation, the coefficients of 

 the lagged variables may be judged significant 

 by the usual ^-test even though there is no real 

 lagged response in the underlying structural 

 relationship. Similarly, it can be shown that se- 

 rially correlated residuals will occur if a non- 

 lagged model is mistakenly fit to data from an 

 inherently dynamic process. 



Although there is no fully satisfactory method 

 for determining which model is the truth, 

 Griliches (1967) developed a provisional test. 

 Briefly, the serial correlation model is 



9, =«o +}^^xu +^t 



(11a) 

 (lib) 



where s is a positive fraction and u, is a nonse- 

 rially correlated error term. From Equation 

 (lla),e,.i =Qt-i -o-Q- S,a,:<;,,.i;sothate^ = s(9m - 

 Uq - S(a,x„_i) + "r Substituting this into Equation 

 (11a) yields 



% =<1 -s)ao + S(a^A:,, - 6,x,,.,)+ s g^ ^ + u,. (12) 



When Equation ( 12) is computed, the serial corre- 

 lation model implies that a^s = -6, for each i. 

 Griliches suggested that the first-order serial cor- 

 relation model be rejected if these four equalities 

 do not appear to hold. Thus, there are four 

 h3^otheses of the following form: 



