Estimation of Production Functions: Fixed Effects in Panel Data

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Estimation of Production Functions Fixed Effects
in Panel Data

• Lecture VIII

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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 a, ß, and ?

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• 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 ß
  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.

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• 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

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• 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

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• OLS estimators then provide unbiased and
  consistent estimates of ß.
• 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.

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• Next, starting from the pooled estimates
• Case I Heterogeneous intercepts (ai ? a) and a
  homogeneous slope (ßi ß).

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• Case II Heterogeneous slopes and intercepts (ai
  ? a , ßi ? ß )

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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.

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Estimation of different slopes and intercepts
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Covariance Matrices Covariance Matrices Covariance Matrices Covariance Matrices Covariance Matrices Covariance Matrices Covariance Matrices Covariance Matrices Covariance Matrices Covariance Matrices Covariance Matrices Covariance Matrices Covariance Matrices
X'X Nitrogen Nitrogen Phosphorous Potash X'Y beta alpha
Illinois Illinois Illinois Illinois Illinois Illinois Illinois Illinois Illinois Illinois Illinois Illinois Illinois
Nitrogen Nitrogen 1.2823 0.7194 1.5488 0.7415 0.7415 0.7985 0.7985 3.7917
Phosphorous Phosphorous 0.7160 0.6410 1.0156 0.2204 0.2204 -0.9813 -0.9813
Potash Potash 1.5427 1.0174 2.0326 0.7894 0.7894 0.2734 0.2734
Indiana Indiana Indiana Indiana Indiana Indiana Indiana Indiana Indiana Indiana Indiana Indiana Indiana
Nitrogen Nitrogen 1.0346 0.2489 0.7220 0.6577 0.4386 0.4386 3.6162
Phosphorous Phosphorous 0.2348 0.3717 0.2320 -0.0913 -0.8905 -0.8905
Potash Potash 0.7268 0.2448 0.6072 0.4587 0.5894 0.5894
Pooled Pooled Pooled Pooled Pooled Pooled Pooled Pooled Pooled Pooled Pooled Pooled Pooled
Nitrogen Nitrogen 2.3168 0.9683 2.2708 1.3992 0.5924 0.5924 3.9789
Phosphorous Phosphorous 0.9508 1.0128 1.2475 0.1291 -0.9335 -0.9335 3.8851
Potash Potash 2.2695 1.2622 2.6398 1.2481 0.4098 0.4098
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Estimation of different intercepts with the same
slope
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Estimation of homogeneous slopes and intercepts
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• Testing first for pooling both the slope and
  intercept terms

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

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Dummy-Variable Formulation
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• Given this formulation, we know the OLS
  estimation of
• The OLS estimation of a and ß are obtained by
  minimizing

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Sweeping the data
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