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APPENDIX 



Because the estimation of the extended Box-Cox functional form is carried out by max- 

 imum likelihood procedures, the log-likelihood function for the extended Box-Cox func- 

 tional form and estimation methods are briefly presented in this appendix. 



Under the assumptions that S and B are nonstochastic and the error term is normally 

 and independently distributed with zero mean and constant variance, aj, the log-likelihood 

 function of model (5) can be expressed following Spitzer (1982): 



Lia, A, M, e, o\) = -T/2(ln 2n + In o^) - i2af)-He'e) + (A - 1)2 In A (12) 



where T is the number of observations. To reduce the dimension of the estimation prob- 

 lem, the parameter Oj can be eliminated from Equation (12) to derive the concentrated 

 log-likelihood function as follows: 



L{q, a, ^i, 6) = -r/2(ln 2n + In d^) + (A - 1)1 In A 



where o'l = {llT)e e. 



When heteroscedasticity is present, the concentrated log-likelihood function for model 

 (5) and the error term expressed in model (6) can be expressed as 



L{a, A, M, Q) = -T/2(ln 2ti + In d^ + 1) 



- I In (/?i + /J.S'^' + /?35'«') + (A - 1)2 In A (14) 



where d'^ = (l/T)e'V~^e and F is a nxn matrix (n is the number of observations) in which 

 off-diagonal elements are zeros and diagonal elements are /?i -i- P-yS^^^ + P^B'^l 



The maximum log-likelihood parameter estimates for (a, A, /u, 6, and ft) can be obtained 

 by nonlinear least squares methods or iterated ordinary (weighted) least squares procedures. 

 Seaks and Layson (1983) provide an example of the iterated ordinary (weighted) least 

 squares method using the Time Series Processor (TSP) computer package for estimating 

 Box-Cox flexible functional form with standard econometric problems; i.e., heteroscedas- 

 ticity and autocorrelation. 



As Spitzer (1984) pointed out, the ordinary least squares method underestimates the 

 variance of the error term while the first derivative only gradient estimation methods (e.g., 

 Marquardt) overestimate the variance. In order to compress the range of under- and 

 overestimation of the error variance, Spitzer suggested that the dependent variable be 

 divided by its geometric mean. This scaling process will then eliminate the last term in 

 the concentrated log likehhood function in Equations (12)-(14). 



662 



