On Variances of Conditional Linear Least‐Squares Search Parameter Estimates

Abstract

[Y t -rix.; ... , X lk ; {3)]2. 1=1In practice the maximum likelihood estimator /:J can be found either by solving a set of nonlinear normal equations or by using a search algorithm. In certain important applications the function F will have a form so that conditional on values of p < m of the m parameters in the {3 vector, and on the X's, the function is linear in the remaining parameters.Notable examples of models that have this property include the model that results from a Box-Cox power transformation of explanatory variables asNote that (3) is linear in its parameters, given A = {3k+I' The constant elasticity of likelihood (CES) production function, equation (4), is a second example. After logarithmic transformation it is linear in {33and In f30 if {31 and f32 are given. Just's (l974a,b, 1977) generalized adaptive expectations model is a third example. In its basic form, it can be written asThe variable Y t above is acreage harvested in year t , PI is price received for the crop in year t, and P/(f33) and V/({33) are the mean and variance, respectively, of the subjective distribution of year t + 1 prices as perceived in year t. A consistent finite approximation to (5) is defined asAmong the class of models that are nonlinear in the parameters, some lend themselves to a simple estimation method requiring repeated application of linear least squares. However, there is confusion over methods for computing standard errors of such estimates.In this paper we define a simple method for estimating parameters of certain nonlinear models and clarify the interpretation of results. This method is based on conditional linear least squares. It has been used by Just (l974a,b, 1977) to estimate a generalized adaptive-expectations supply model. We show that a method commonly used to estimate asymptotic standard errors of such parameter estimates systematically overstates precision. Factors that affect the degree of overstatement are delineated. A method of calculating correct precision measures is contrasted empirically with the incorrect procedure. Conditional Linear Least SquaresA wide range of stochastic models used in applied research can be written in the form (1) Y t = F(X/l, X/2, ... , X tk ; {3) + VI' It is assumed that the disturbances in (1) are independently and identically distributed normal random variables, and, under this assumption, nonlinear squares and maximum likelihood procedures applied to estimate {3 in (1) are equivalent. Note that the natural logarithm of the likelihood function associated with (1) takes the form n n (2) In L({3, cr-) = -T In 27T -T In cr-1 n --::2 L [Y t -F(X/l, ... , X l k ; {3)]2, 2u-t=1 so that for any cr-> 0, In L is maximized when {3 is chosen to minimize the residual sum of squares, Edmund A. where where co P t-{({33) = {33 L (l -f33)iP t -i -l , and i=O t-tb-J P*t-I ({33) = {33 I (l -{33) iPt-i-J' i=O

Agricultural arbitrage and risk preferences

“…Many studies have applied portfolio theory (Sharpe 1970) to explain acreage allocation in production agriculture (e.g., Behrman 1968, Estes, Blakeslee, and Mittelhammer 1981, Just 1974, Lin, Dean, and Moore, 1974, Lin 1977. These applications and many since have been primarily applied to crop acreage decisions assuming linear technology and most are in a static setting.…”