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Linear Estimation of Production Functions in Panel Data Analysis

This lecture discusses the use of fixed effects in estimating production functions in panel data. It covers how to address bias in OLS estimations when covariates are not observed and how repeated observations in panel data can help eliminate such biases. It explores different scenarios of heterogeneous intercepts and slopes and presents empirical procedures to test hypotheses about regression coefficients. The lecture also delves into the estimation of different slopes and intercepts, testing for homogeneity, and using dummy-variable formulations for OLS estimation.

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Linear Estimation of Production Functions in Panel Data Analysis

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  1. Estimation of Production Functions: Fixed Effects in Panel Data Lecture VIII

  2. Analysis of Covariance • Looking at a representative regression model • It is well known that ordinary least squares (OLS) regressions of y on x and z are best linear unbiased estimators (BLUE) of α, β, and γ

  3. However, the results are corrupted if we do not observe z. Specifically if the covariance of x and z are correlated, then OLS estimates of the β are biased. • However, if repeated observations of a group of individuals are available (i.e., panel or longitudinal data) they may us to get rid of the effect of z.

  4. For example if zit = zi (or the unobserved variable is the same for each individual across time), the effect of the unobserved variables can be removed by first-differencing the dependent and independent variables

  5. Similarly if zit = zt (or the unobserved variables are the same for every individual at a any point in time) we can derive a consistent estimator by subtracting the mean of the dependent and independent variables for each individual

  6. OLS estimators then provide unbiased and consistent estimates of β. • Unfortunately, if we have a cross-sectional dataset (i.e., T = 1) or a single time-series (i.e., N = 1) these transformations cannot be used.

  7. Next, starting from the pooled estimates • Case I: Heterogeneous intercepts (αi ≠ α) and a homogeneous slope (βi = β).

  8. Case II: Heterogeneous slopes and intercepts (αi ≠ α , βi ≠ β )

  9. Empirical Procedure • From the general model, we pose three different hypotheses: • H1: Regression slope coefficients are identical and the intercepts are not. • H2: Regression intercepts are the same and the slope coefficients are not. • H3: Both slopes and the intercepts are the same.

  10. Estimation of different slopes and intercepts

  11. Estimation of different intercepts with the same slope

  12. Estimation of homogeneous slopes and intercepts

  13. Testing first for pooling both the slope and intercept terms:

  14. If this hypothesis is rejected, we then test for homogeneity of the slopes, but heterogeneity of the constants

  15. Dummy-Variable Formulation

  16. Given this formulation, we know the OLS estimation of • The OLS estimation of α and β are obtained by minimizing

  17. Sweeping the data

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