Lecture08-2011 - Outline Analysis of Covariance Pooling Assumptions Estimation of Production Functions Fixed Effects in Panel Data Lecture VIII

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Unformatted text preview: Outline Analysis of Covariance Pooling Assumptions Estimation of Production Functions: Fixed Effects in Panel Data : Lecture VIII Charles B. Moss1 1 University of Florida September 20, 2011 Charles B. Moss Estimation of Production Functions: Fixed Effects in Panel Dat Outline Analysis of Covariance Pooling Assumptions 1 Analysis of Covariance Individual Effects Time Effects Modified Ordinary Least Squares 2 Pooling Assumptions Empirical Procedure Computations Corn Estimates Different Intercepts - Same Slope Same Slopes - Same Intercept Testing for Pooling Dummy Variable Formulation Sweeping the Data Charles B. Moss Estimation of Production Functions: Fixed Effects in Panel Dat Outline Analysis of Covariance Pooling Assumptions Individual Effects Time Effects Modified Ordinary Least Squares Analysis of Covariance Looking at a representative regression model yit = α∗ + β ￿ xit + γ ￿ zit + uit i = 1, · · · N t = 1, · · · T (1) 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 . 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 . Charles B. Moss Estimation of Production Functions: Fixed Effects in Panel Dat Outline Analysis of Covariance Pooling Assumptions Individual Effects Time Effects Modified Ordinary Least Squares Individual Effects 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 yit − yi ,t −1 = β ￿ (xit − xi ,t −1 ) + γ ￿ (zit − zi ,t −1 ) + (uit − ui ,t −1 ) (2) Since zit = zi ,t −1 = zi yit − yi ,t −1 = β ￿ (xit − xi ,t −1 ) + (uit − ui ,t −1 ) Charles B. Moss i = 1, · · · N t = 2, · · · T (3) Estimation of Production Functions: Fixed Effects in Panel Dat Outline Analysis of Covariance Pooling Assumptions Individual Effects Time Effects Modified Ordinary Least Squares Time Effects 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 yit − yi = β ￿ (xit − xi ) + γ ￿ (zit − zi ) + (uit − ui ) ¯ ¯ ¯ ¯ (4) Again zit = zi ¯ yit − yi = β ￿ (xit − xi ) + (uit − ui ) ¯ ¯ ¯ T T T ￿ ￿ 1 1 1￿ yi = ¯ yit xi = ¯ xit ui = ¯ uit T T T t =1 Charles B. Moss t =1 (5) t =1 Estimation of Production Functions: Fixed Effects in Panel Dat Outline Analysis of Covariance Pooling Assumptions Individual Effects Time Effects Modified Ordinary Least Squares Modified Ordinary Least Squares 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. Charles B. Moss Estimation of Production Functions: Fixed Effects in Panel Dat Outline Analysis of Covariance Pooling Assumptions Empirical Procedure Computations Corn Estimates Different Intercepts - Same Slope Same Slopes - Same Intercept Testing for Pooling Dummy Variable Formulation Sweeping the Data Pooling Assumptions Next, starting from the pooled estimates ∗ yit = α∗ + β ￿ xit + uit (6) Case I: Heterogeneous intercepts (αi ￿= α) and homogeneous slopes (βi = β ) yit = αi∗ + β ￿ xit + uit (7) Case II: Heterogeneous slopes and intercepts (αi ￿= α, βi ￿= β ) yit = αi∗ + βi￿ xit + uit Charles B. Moss (8) Estimation of Production Functions: Fixed Effects in Panel Dat Outline Analysis of Covariance Pooling Assumptions Empirical Procedure Computations Corn Estimates Different Intercepts - Same Slope Same Slopes - Same Intercept Testing for Pooling Dummy Variable Formulation Sweeping the Data 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. Charles B. Moss Estimation of Production Functions: Fixed Effects in Panel Dat Empirical Procedure Computations Corn Estimates Different Intercepts - Same Slope Same Slopes - Same Intercept Testing for Pooling Dummy Variable Formulation Sweeping the Data Outline Analysis of Covariance Pooling Assumptions Computations T T 1￿ 1￿ yi = ¯ yit xi = ¯ xit T T t =1 t =1 ˆ ˆ¯ β = WXX ,i WXY ,i αi = yi − βi xi i = 1, · · · N ˆ ¯ WXX ,i = T ￿ t =1 (xit − xi ) (xit − xi )￿ WXY ,i = ¯ ¯ WYY ,i = T ￿ t =1 T ￿ t =1 (xit − xi ) (yit − yi ) ¯ ¯ (yit − yi )2 ¯ −1 ￿ RSSi = WYY ,i − WXY ,i WXX ,i WXY ,i N ￿ S1 = RSSi Charles B. Moss i =1Estimation of Production Functions: Fixed Effects in Panel Dat Outline Analysis of Covariance Pooling Assumptions Empirical Procedure Computations Corn Estimates Different Intercepts - Same Slope Same Slopes - Same Intercept Testing for Pooling Dummy Variable Formulation Sweeping the Data Corn Estimates Variable Nitrogen Phosphorous Potash Nitrogen Phosphorous Potash Nitrogen Phosphorous Potash Covariance Matrix Nitrogen Phosphorous Potash Illinois 1.2823 0.7194 1.5488 0.7160 0.6410 1.0156 1.5427 1.0174 2.0326 Indiana 1.0346 0.2489 0.7220 0.2348 0.3717 0.2320 0.7268 0.2448 0.6072 Pooled 2.3168 0.9683 2.2708 0.9508 1.0128 1.2475 2.2695 1.2622 2.6398 Charles B. Moss X ￿Y β α 0.7415 0.2204 0.7894 0.7985 - 0.9813 0.2734 3.7917 0.6577 -0.0913 0.4587 0.4386 -0.8905 0.5894 3.6162 1.3992 0.1291 1.2481 0.5924 - 0.9335 0.4008 3.9789 3.8851 Estimation of Production Functions: Fixed Effects in Panel Dat Outline Analysis of Covariance Pooling Assumptions Empirical Procedure Computations Corn Estimates Different Intercepts - Same Slope Same Slopes - Same Intercept Testing for Pooling Dummy Variable Formulation Sweeping the Data Different Intercepts - Same Slope −1 ˆ ˆ￿ ¯ βW = WXX WXY xi∗ = yi − βW xi i = 1, · · · N ¯ ¯ N N ￿ ￿ WXX = WXX ,i WXY = WXY ,i i =1 WYY = S2 = WYY − Charles B. Moss N ￿ i =1 (10) WYY ,i i =1 −1 ￿ WXY WXX WXY Estimation of Production Functions: Fixed Effects in Panel Dat Empirical Procedure Computations Corn Estimates Different Intercepts - Same Slope Same Slopes - Same Intercept Testing for Pooling Dummy Variable Formulation Sweeping the Data Outline Analysis of Covariance Pooling Assumptions Same Slopes - Same Intercept −1 ˆ β = TXX TXY α∗ = y − β x ˆ ¯ ˆ¯ NT ￿￿ TXX = (xit − x ) (xit − x )￿ ¯ ¯ i =1 t =1 TXY = NT ￿￿ (xit − x ) (yit − y )￿ ¯ ¯ i =1 t =1 NT ￿￿ TYY = i =1 t =1 (yit − y ) ¯ (11) 2 NT NT 1 ￿￿ 1 ￿￿ y= ¯ yit x = ¯ xit NT NT i =1 t =1 i =1 t =1 −1 ￿ S3 = TYY − TXY TXX TXY Charles B. Moss Estimation of Production Functions: Fixed Effects in Panel Dat Outline Analysis of Covariance Pooling Assumptions Empirical Procedure Computations Corn Estimates Different Intercepts - Same Slope Same Slopes - Same Intercept Testing for Pooling Dummy Variable Formulation Sweeping the Data Testing for Pooling Testing first for pooling both the slope and intercept terms ∗ ∗ ∗ H3 : α 1 = α 2 = · · · α N β 1 = β 2 = · · · β N ( S3 − S1 ) [(N − 1) (K + 1)] F3 = ∼ F ([N − 1] [K + 1] , NT − N [K + 1]) S1 [NT − N (K + 1)] (12) Charles B. Moss Estimation of Production Functions: Fixed Effects in Panel Dat Outline Analysis of Covariance Pooling Assumptions Empirical Procedure Computations Corn Estimates Different Intercepts - Same Slope Same Slopes - Same Intercept Testing for Pooling Dummy Variable Formulation Sweeping the Data If this hypothesis is rejected, we then test for homogeneity of the slopes, but heterogeneity of the constants H1 : β 1 = β 2 = · · · β N ( S2 − S1 ) [(N − 1) K ] F1 = ∼ F ([N − 1] K , NT − N [K + 1]) S1 [NT − N (K + 1)] (13) Charles B. Moss Estimation of Production Functions: Fixed Effects in Panel Dat Outline Analysis of Covariance Pooling Assumptions Empirical Procedure Computations Corn Estimates Different Intercepts - Same Slope Same Slopes - Same Intercept Testing for Pooling Dummy Variable Formulation Sweeping the Data Dummy Variable Formulation y1 e y2 0 Y = ··· = ··· y3 0 0 ∗ e∗ α1 + α2 + · · · ··· 0 x1 u1 x u + 2 β + 2 ··· ··· xN uN Charles B. Moss 0 0∗ α ··· N e (14) Estimation of Production Functions: Fixed Effects in Panel Dat Empirical Procedure Computations Corn Estimates Different Intercepts - Same Slope Same Slopes - Same Intercept Testing for Pooling Dummy Variable Formulation Sweeping the Data Outline Analysis of Covariance Pooling Assumptions x 1i 1 yi 1 yi 2 x1 i 2 yi = · · · xi · · · yit x1iT x2 i 1 x2 i 2 ··· x2iT ··· ··· .. . ··· ui￿ ∈ M1×T ui￿ ￿ xKi 1 xKi 2 ￿ ￿ e ∈ M1 × T e ￿ = 1 1 · · · ··· xKiT ui 1 ui 2 · · · uiN ￿ ￿ ￿ ￿ ￿ E [ui ] = 0 E ui ui￿ = σ 2 IT E ui uj￿ = 0 i ￿= j Charles B. Moss (15) Estimation of Production Functions: Fixed Effects in Panel Dat Outline Analysis of Covariance Pooling Assumptions Empirical Procedure Computations Corn Estimates Different Intercepts - Same Slope Same Slopes - Same Intercept Testing for Pooling Dummy Variable Formulation Sweeping the Data Given this formulation, we know the OLS estimation of yit = αi∗ + β xit + uit (16) The OLS estimation of α and β are obtained by minimizing S= N ￿ i =1 ui￿ ui = N ￿ i =1 (yi − e αi∗ − xi β )￿ (yi − e αi∗ − xi β ) (17) Charles B. Moss Estimation of Production Functions: Fixed Effects in Panel Dat Outline Analysis of Covariance Pooling Assumptions Empirical Procedure Computations Corn Estimates Different Intercepts - Same Slope Same Slopes - Same Intercept Testing for Pooling Dummy Variable Formulation Sweeping the Data Sweeping the Data ˆ βCV αi∗ = yi − β ￿ xi i = 1, · · · N ˆ ¯ ¯ T T 1￿ 1￿ yit xi = xit yi = ¯ ¯ T T t =1 ￿N T ￿ ￿ N t =1 ￿ T ￿￿ ￿￿ = (xit − xi ) (xit − xi )￿ ¯ ¯ (xit − xi ) (yit − yi ) ¯ ¯ i =1 t =1 i =1 t =1 Charles B. Moss (18) Estimation of Production Functions: Fixed Effects in Panel Dat Outline Analysis of Covariance Pooling Assumptions 100 0 1 0 0 0 1 000 1− 1 4 −1 4 −1 4 −1 4 Empirical Procedure Computations Corn Estimates Different Intercepts - Same Slope Same Slopes - Same Intercept Testing for Pooling Dummy Variable Formulation Sweeping the Data 1 Q = IT − ee ￿ T 0 1111 0 1 1 1 1 1 − = 0 4 1 1 1 1 1 1111 −1 −1 −1 4 4 4 1 − 1 −1 −1 4 4 4 −1 1 − 1 −1 4 4 4 −1 −1 1 − 1 4 4 4 Charles B. Moss (19) Estimation of Production Functions: Fixed Effects in Panel Dat Outline Analysis of Covariance Pooling Assumptions ￿ Empirical Procedure Computations Corn Estimates Different Intercepts - Same Slope Same Slopes - Same Intercept Testing for Pooling Dummy Variable Formulation Sweeping the Data ￿ 1 − 1 ￿x1 − 1￿ 2 − 1 x3 − 1 x4 4 4x 4 4 ￿ ￿ − 1 x1 + 1 − 1 x 2 − 1 x 3 − 1 x4 1￿ 4￿ 4 4 ￿ I4 − ee = 4 − 1 x1 − 1 x 2 + 1 − 1 x 3 − 1 x4 4 4 4 4￿ 4 ￿ 1 − 4 x1 − 1 x 2 − 1 x 3 + 1 − 1 x4 4 4 4 x1 − 1 ( x 1 + x 2 + x3 + x4 ) 4 x − 1 ( x + x 2 + x3 + x4 ) = 2 4 1 x3 − 1 ( x 1 + x 2 + x3 + x4 ) 4 x4 − 1 ( x 1 + x 2 + x3 + x4 ) 4 Charles B. Moss (20) Estimation of Production Functions: Fixed Effects in Panel Dat ...
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This note was uploaded on 02/01/2012 for the course AEB 6184 taught by Professor Staff during the Fall '09 term at University of Florida.

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