Econometrics-I-15

# Efficiency improvement gains to gls none if identical

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Efficiency improvement. Gains to GLS: None if identical regressors - NOTE THE CAPM ABOVE! Implies that GLS is the same as OLS. This is an application of a strange special case of the GR model. “If the K columns of X are linear combinations of K characteristic vectors of , in the GR model, then OLS is algebraically identical to GLS.” We will forego our opportunity to prove this theorem. This is our only application. (Kruskal’s Theorem) Efficiency gains increase as the cross equation correlation increases (of course!). ™  28/45

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Part 15: Generalized Regression Applications The Identical X Case Suppose the equations involve the same X matrices. (Not just the same variables, the same data. Then GLS is the same as equation by equation OLS. Grunfeld’s investment data are not an example - each firm has its own data matrix. The 3 equation model on page 313 with Berndt and Wood’s data give an example. The three share equations all have the constant and logs of the price ratios on the RHS. Same variables, same years. The CAPM is also an example. (Note, because of the constraint in the B&W system (same δ parameters in more than one equation), the OLS result for identical Xs does not apply.) ™  29/45
Part 15: Generalized Regression Applications Estimation by FGLS Two step FGLS is essentially the same as the groupwise heteroscedastic model. (1) OLS for each equation produces residuals e i. (2) S ij = (1/n) eiej then do FGLS Maximum likelihood estimation for normally distributed disturbances: Just iterate FLS. (This is an application of the Oberhofer-Kmenta result.) ™  30/45

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Part 15: Generalized Regression Applications Inference About the Coefficient Vectors Usually based on Wald statistics. If the estimator is maximum likelihood, LR statistic T(log| S restricted| - log| S unrestricted|) is a chi-squared statistic with degrees of freedom equal to the number of restrictions. Equality of the coefficient vectors: (Historical note: Arnold Zellner, The original developer of this model and estimation technique: “An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests of Aggregation Bias (my emphasis). JASA, 1962, pp. 500-509. What did he have in mind by “aggregation bias?”  How to test the hypothesis? ™  31/45
Part 15: Generalized Regression Applications Application A Translog demand system for a 3 factor process: (To bypass a transition in the notation, we proceed directly to the application) Electricity, Y, is produced using Fuel, F, capital, K, and Labor, L. Theory: The production function is Y = f(K,L,F). If it is smooth, has continuous first and second derivatives, and if(1) factor prices are determined in a market and (2) producers seek to minimize costs (maximize profits), then there is a “cost function” C = C(Y,PK,PL,PF).

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