Spatial Econometric Analysis Using GAUSS 8 Kuan-Pin Lin Portland State University

Panel Data Analysis A Review Model Representation N-first or T-first representation Pooled Model Fixed Effects Model Random Effects Model Asymptotic Theory N→∞, or T→∞ N→∞, T→∞ Panel-Robust Inference

Panel Data Analysis A Review The Model One-Way (Individual) Effects: Unobserved Heterogeneity Cross Section and Time Series Correlation

Panel Data Analysis A Review N-first Representation Dummy Variables Representation T-first Representation

Panel Data Analysis A Review Notations

Pooled (Constant Effects) Model

Fixed Effects Model u i is fixed, independent of e it, and may be correlated with x it.

Fixed Effects Model Classical Assumptions Strict Exogeneity: Homoschedasticity: No cross section and time series correlation: Extensions: Panel Robust Variance-Covariance Matrix

Random Effects Model Error Components u i is random, independent of e it and x it. Define the error components as  it = u i + e it

Random Effects Model Classical Assumptions Strict Exogeneity  X includes a constant term, otherwise E(u i |X)=u. Homoschedasticity Constant Auto-covariance (within panels)

Random Effects Model Classical Assumptions (Continued) Cross Section Independence Extensions: Panel Robust Variance-Covariance Matrix

Fixed Effects Model Estimation Within Model Representation

Fixed Effects Model Estimation Model Assumptions

Fixed Effects Model Estimation: OLS Within Estimator: OLS

Fixed Effects Model Estimation: ML Normality Assumption

Fixed Effects Model Estimation: ML Log-Likelihood Function Since Q is singular and |Q|=0, we maximize

Fixed Effects Model Estimation: ML ML Estimator

Fixed Effects Model Hypothesis Testing Pool or Not Pool F-Test based on dummy variable model: constant or zero coefficients for D w.r.t F(N-1,NT-N-K) F-test based on fixed effects (unrestricted) model vs. pooled (restricted) model

Fixed Effects Model Hypothesis Testing Based on estimated residuals of the fixed effects model: Heteroscedasticity Breusch and Pagan (1980) Autocorrelation: AR(1) Breusch and Godfrey (1981)

Random Effects Model Estimation: GLS The Model

Random Effects Model Estimation: GLS GLS

Random Effects Model Estimation: GLS Feasible GLS Based on estimated residuals of fixed effects model

Random Effects Model Estimation: ML Log-Likelihood Function

Random Effects Model Estimation: ML where

Random Effects Model Estimation: ML ML Estimator

Random Effects Model Hypothesis Testing Pool or Not Pool Test for Var(u i ) = 0, that is For balanced panel data, the Lagrange-multiplier test statistic (Breusch-Pagan, 1980) is:

Random Effects Model Hypothesis Testing Pool or Not Pool (Cont.)

Random Effects Model Hypothesis Testing Fixed Effects vs. Random Effects EstimatorRandom Effects E(u i |X i ) = 0 Fixed Effects E(u i |X i ) =/= 0 GLS or RE-OLS (Random Effects) Consistent and Efficient Inconsistent LSDV or FE-OLS (Fixed Effects) Consistent Inefficient Consistent Possibly Efficient

Random Effects Model Hypothesis Testing Fixed effects estimator is consistent under H 0 and H 1 ; Random effects estimator is efficient under H 0, but it is inconsistent under H 1. Hausman Test Statistic

Random Effects Model Hypothesis Testing Alternative Hausman Test Estimate the random effects model F Test that  = 0

Random Effects Model Hypothesis Testing Heteroscedasticity H 0 : θ 2 =0 | θ 1 =0 H 0 : θ 1 =0 | θ 2 =0 H 0 : θ 2 =0, θ 1 =0

Random Effects Model Hypothesis Testing Heteroscedasticity (Cont.) Based on random effects model with homoscedasticity:

Random Effects Model Hypothesis Testing Heteroscedasticity (Cont.)

Random Effects Model Hypothesis Testing Heteroscedasticity (Cont.) Baltagi, B., Bresson, G., Pirotte, A. (2006) Joint LM test for homoscedasticity in a one-way error component model. Journal of Econometrics, 134,

Random Effects Model Hypothesis Testing Autocorrelation: AR(1) Based on random effects model with no autocorrelation: LM test statistic is tedious, see Baltagi, B., Li, Q. (1995) Testing AR(1) against MA(1) disturbances in an error component model. Journal of Econometrics, 68,

Random Effects Model Hypothesis Testing Joint Test for AR(1) and Random Effects Based on OLS residuals: Marginal Test for AR(1) & Random Effects

Random Effects Model Hypothesis Testing Robust LM Tests for AR(1) and Random Effects Because

Panel Data Analysis An Example: U. S. Productivity The Model (Munnell [1988]):

Panel Data Analysis An Example: U. S. Productivity Productivity Data 48 Continental U.S. States, 17 Years: STATE = State name, ST_ABB = State abbreviation, YR = Year, 1970,...,1986, PCAP = Public capital, HWY = Highway capital, WATER = Water utility capital, UTIL = Utility capital, PC = Private capital, GSP = Gross state product, EMP = Employment, UNEMP = Unemployment rate

U. S. Productivity Baltagi (2008) [munnell.1, munnell.2]munnell.1munnell.2 Panel Data Model ln(GSP) =   +   ln(Public) +  2 ln(Private) +  3 ln(Labor) +  4 (Unemp) +  Fixed Effectss.e Random Effectss.e   3 4 0 F(47,764) =75.82LM(1) = 4135 Hausman LM(4) = 905.1

Panel Data Analysis Another Example: China Provincial Productivity Cobb-Douglass Production Function ln(GDP) =  +  ln(L) +  ln(K) +  Fixed Effectss.e. Random Effectss.e    F(29,298) = LM(1) = Hausman LM(2) = 48.4

References B. H. Baltagi, Econometric Analysis of Panel Data, 4th ed., John Wiley, New York, W. H. Greene, Econometric Analysis, 6th ed., Chapter 9: Models for Panel Data, Prentice Hall, C. Hsiao, Analysis of Panel Data, 2nd ed., Cambridge University Press, J. M. Wooldridge, Econometric Analysis of Cross Section and Panel Data, The MIT Press, 2002.