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Panel Data Estimation: Random Effects in Production Functions

This lecture discusses regression analysis in panel data models, focusing on the assumption of random factors affecting dependent variables. It explores variance-component models, covariance estimators, and the Generalized-Least-Squares Estimator along with the estimation procedure. The lecture emphasizes the unbiasedness of covariance estimators and the implications of fixed versus random effects in the model. Techniques such as partitioned matrices and estimating unknown parameters are explained.

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Panel Data Estimation: Random Effects in Production Functions

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

  2. 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 Lecture IX

  3. 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 Lecture IX

  4. Lecture IX

  5. Lecture IX

  6. The variance of yit on xit based on the assumption above is • Thus, this kind of model is typically referred to as a variance-component (or error-components) model. Lecture IX

  7. Letting the panel estimation model can be written in vector form as Lecture IX

  8. The expected value of the residual becomes Lecture IX

  9. Using the basic covariance estimator • Whether αi is fixed or random the covariance estimator is unbiased. • However, if the αi 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). Lecture IX

  10. The Generalized-Least-Squares Estimator • Because both uit and uis contain αi , they are correlated. Lecture IX

  11. Lecture IX

  12. A procedure for estimation Lecture IX

  13. Lecture IX

  14. This looks bad, but think about Lecture IX

  15. Lecture IX

  16. Solving this system yields Lecture IX

  17. Using the inverse of a partitioned matrix Lecture IX

  18. Where • Where βb is the between estimator. Lecture IX

  19. The variance of the estimator can be written as Lecture IX

  20. 3. Given that we don’t know ψa priori, we estimate Lecture IX

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