ESTIMATION OF EFFICIENT REGRESSION MODELS FOR APPLIED AGRICULTURAL ECONOMICS RESEARCH

numerically simulate the probability distribution from which they could have been drawn. Specifically, let

^γi be the kx 1 vector of maximum likelihood estimators for Γi, the vector of true population parameters

^ ^

underlying the normal (i=N, k=10) or the non-normal (i=NN, k=14) regression model, and CM [^ri ] be the
estimated covariance matrix for ^γi. Then, the joint probability distribution of ^γi is simulated by:

~

(9)     Si= Z x Chol( CM[Γi ]) + Γi,

where Z is an mxk matrix of independently distributed standard normal random variables, Chol(.) denotes the
~                                          ^

Cholesky decomposition, and ^γi the kx 1 vector of parameter estimates obtained from ^γi. Equation (9) yields

~                      ^ ^

an mxk matrix of random variables with mean ^γi and covariance matrix CM[^ri^]. Since, under a correct

^                                ^ ^                              ^

model specification, ^γi is a consistent estimator for Γi and CM[^ri^] is a consistent estimator for CM[^ri^], Si
is a theoretically correct probabilistic statement about Γi. Thus, the boundaries of a (1-α)% confidence
interval for the expected price under the normal (non-normal) model at time period t can be obtained by
extracting the m sets of simulated parameter values from Sn (Snn), using them to obtain m “predicted” price
values for time t, and finding the (a/2) x m^th and the [(1-a)+a/2] x m^th largest of these m price values.

Confidence intervals for the actual price realizations require simulation of m error term draws as well.
In the case of the normal regression model, these are obtained by extracting the m simulated values for the
standard deviation parameter (σc or σs) from Sn and multiplying them by m independent draws from a
standard normal random variable. In the case of the non-normal model, the m sets of simulated values for σc,
μ c and Θc (or σs, μ s and Θs) have to be extracted from Snn and coupled with m independent standard normal
draws. Then, m non-normal error term values are simulated by applying equation (2). The final step in
constructing the boundaries of a (1-α)% confidence interval for the actual price observations is to add the m
simulated error term values to the corresponding m price “predictions” and find the (a/2) x m^th and the [(1-

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