A Primer on Box-Cox Estimation

Abstract

/-3) + f32X2'+ . . + 8,/Xk k + E (2) can be specified and estimated. On occasion, neither a priori reasoning nor theory clearly dictate the correct functional form (transformation) which an additive model should assume. With the Box-Cox transformation, the functional form is dictated by the parameters, Xi, which are themselves estimated. Note that if X1 = 1 in (2), then y(l) enters the equation linearly; also, y enters (2) as In y, and y(-1) enters (2) as the reciprocal of y. Thus the estimation procedure itself chooses the transformations which best fit the data. Furthermore, hypothesis tests can be made on the estimated Ai in order to determine if alternative functional forms (transformations) are also consistent with the data. (See the appendix for a note on discriminating between functional forms.) Estimation of (2) requires the maximization of a nonlinear likelihood function which can be extremely complicated. Since computer programs for maximizing such complex functions may not be readily available, the estimation of generalized functional forms such as (2) may be impeded. It may not be generally recognized that estimation of the parameters of (2) can be accomplished in at least four different ways. This paper will look at four alternative ways of estimating the parameters f3j, Xi and o-2. Each approach can be made to yield identical parameter estimates, and identical estimates of the covariance matrix of the parameter estimates. In section II, the general problem will be addressed and the likelihood function derived. Section III will look at each estimation approach. Section IV will conclude the paper. Problems of estimation only are dealt with in this paper. Furthermore, the approximate normality of the error terms is assumed throughout. For a discussion of estimation methodology when the error terms are truncated normal, see Poirier (1978). For a discussion of the interpretation of estimated coefficients in Box-Cox models, see Poirier and Melino (1978) or Huang and Kelingos (1979).