Estimation of Production Functions: Random Effects in Panel Data Lecture IX

Fall 2005Lecture IX2 Basic Setup Regression analysis typically assumes that a large number of factors affect the value of the dependent variable, while some of the variables are measured directly in the model the remaining variables can be summarized by a random distribution

Fall 2005Lecture IX3 When numerous observations on individuals are observed over time, it is assumed that some of the omitted variables represent factors peculiar to individual and time periods. Going back to the panel specification

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Fall 2005Lecture IX6 The variance of y it on x it based on the assumption above is Thus, this kind of model is typically referred to as a variance-component (or error-components) model.

Fall 2005Lecture IX7 Letting the panel estimation model can be written in vector form as

Fall 2005Lecture IX8 The expected value of the residual becomes

Fall 2005Lecture IX9 Using the basic covariance estimator Whether α i is fixed or random the covariance estimator is unbiased. However, if the α i is random the covariance estimator is not the best linear unbiased estimator (BLUE). Instead, a BLUE estimator can be derived using generalized least squares (GLS).

Fall 2005Lecture IX10 The Generalized-Least-Squares Estimator Because both u it and u is contain α i, they are correlated.

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Fall 2005Lecture IX12 A procedure for estimation

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Fall 2005Lecture IX14 This looks bad, but think about

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Fall 2005Lecture IX16 Solving this system yields

Fall 2005Lecture IX17 Using the inverse of a partitioned matrix

Fall 2005Lecture IX18 Where Where β b is the between estimator.

Fall 2005Lecture IX19 The variance of the estimator can be written as

Fall 2005Lecture IX20 3.Given that we dont know ψ a priori, we estimate