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**Econometric Analysis of Panel Data**

William Greene Department of Economics Stern School of Business

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Dear Professor Greene, I have to apply multiplicative heteroscedastic models, that I studied in your book, to the analysis of trade data. Since I have not found any Matlab implementations, I am starting to write the method from scratch. I was wondering if you are aware of reliable implementations in Matlab or any other language, which I can use as a reference.

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**a “multi-level” modelling feature along the following lines**

a “multi-level” modelling feature along the following lines? My data has a “two level” hierarchical structure: I'd like to perform an ordered probit analysis such that we allow for random effects pertaining to individuals and the organisations they work for.

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Random Coefficients OrdProbs Model Dependent variable HSAT Log likelihood function Estimation based on N = 947, K = 14 Inf.Cr.AIC = AIC/N = 3.951 Unbalanced panel has 250 individuals Ordered probit (normal) model LHS variable = values 0,1,...,10 Simulation based on 200 Halton draws | Standard Prob. 95% Confidence HSAT| Coefficient Error z |z|>Z* Interval |Nonrandom parameters Constant| *** 16.05 .0000 AGE| *** .0000 EDUC| *** 4.33 .0000 |Scale parameters for dists. of random parameters Constant| *** 27.57 .0000 |Standard Deviations of Random Effects R.E.(01)| * 1.71 .0877 |Threshold parameters for probabilities Mu(01)| ** 2.53 .0113 ... Mu(09)| *** 39.20 .0000

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**Agenda Single equation instrumental variable estimation Panel data**

Exogeneity Instrumental Variable (IV) Estimation Two Stage Least Squares (2SLS) Generalized Method of Moments (GMM) Panel data Fixed effects Hausman and Taylor’s formulation Application Arellano/Bond/Bover framework

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**Structure and Regression**

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Exogeneity

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**An Experimental Treatment Effect**

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**Instrumental Variables**

Instrumental variable associated with changes in x, not with ε dy/dx = β dx/dx + dε /dx = β + dε /dx. Second term is not 0. dy/dz = β dx/dz + dε /dz. The second term is 0. β =cov(y,z)/cov(x,z) This is the “IV estimator” Example: Corporate earnings in year t Earnings(t) = β R&D(t) + ε(t) R&D(t) responds directly to Earnings(t) thus ε(t) A likely valid instrumental variable would be R&D(t-1) which probably does not respond to current year shocks to earnings.

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Least Squares

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The IV Estimator

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**A Moment Based Estimator**

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**Cornwell and Rupert Data**

Cornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 Years Variables in the file are EXP = work experience, EXPSQ = EXP2 WKS = weeks worked OCC = occupation, 1 if blue collar, IND = 1 if manufacturing industry SOUTH = 1 if resides in south SMSA = 1 if resides in a city (SMSA) MS = 1 if married FEM = 1 if female UNION = 1 if wage set by unioin contract ED = years of education LWAGE = log of wage = dependent variable in regressions These data were analyzed in Cornwell, C. and Rupert, P., "Efficient Estimation with Panel Data: An Empirical Comparison of Instrumental Variable Estimators," Journal of Applied Econometrics, 3, 1988, pp See Baltagi, page 122 for further analysis. The data were downloaded from the website for Baltagi's text.

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**Wage Equation with Endogenous Weeks**

logWage=β1+ β2 Exp + β3 ExpSq + β4OCC + β5 South + β6 SMSA + β7 WKS + ε Weeks worked is believed to be endogenous in this equation. We use the Marital Status dummy variable MS as an exogenous variable. Wooldridge Condition (5.3) Cov[MS, ε] = 0 is assumed. Auxiliary regression: For MS to be a ‘valid’ instrumental variable, In the regression of WKS on [1,EXP,EXPSQ,OCC,South,SMSA,MS, ] MS significantly “explains” WKS. A projection interpretation: In the projection XitK =θ1 x1it + θ2 x2it + … + θK-1 xK-1,it + θK zit , θK ≠ 0. (One normally doesn’t “check” the variables in this fashion.

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Auxiliary Projection | Ordinary least squares regression | | LHS=WKS Mean = | |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| Constant EXP EXPSQ OCC SOUTH SMSA MS

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**Application: IV for WKS in Rupert**

| Ordinary least squares regression | | Residuals Sum of squares = | | Fit R-squared = | | Adjusted R-squared = | |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Constant EXP EXPSQ D OCC SOUTH SMSA WKS

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**Application: IV for wks in Rupert**

| LHS=LWAGE Mean = | | Standard deviation = | | Residuals Sum of squares = | | Standard error of e = | | Fit R-squared = | | Adjusted R-squared = | | Not using OLS or no constant. Rsqd & F may be < 0. | |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Constant EXP EXPSQ D OCC SOUTH SMSA WKS OLS WKS

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**Generalizing the IV Estimator-1**

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**Generalizing the IV Estimator - 2**

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**Generalizing the IV Estimator**

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**The Best Set of Instruments**

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**Two Stage Least Squares**

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2SLS Estimator

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2SLS Algebra

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A General Result for IV We defined a class of IV estimators by the set of variables The minimum variance (most efficient) member in this class is 2SLS (Brundy and Jorgenson(1971)) (rediscovered JW, 2000, p )

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**GMM Estimation – Orthogonality Conditions**

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GMM Estimation - 1

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GMM Estimation - 2

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IV Estimation

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**An Optimal Weighting Matrix**

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The GMM Estimator

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GMM Estimation

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**Application - GMM NAMELIST ; x = one,exp,expsq,occ,south,smsa,wks$**

NAMELIST ; z = one,exp,expsq,occ,south,smsa,ms,union,ed$ 2SLS ; lhs = lwage ; RHS = X ; INST = Z $ NLSQ ; fcn = lwage-b1'x ; labels = b1,b2,b3,b4,b5,b6,b7 ; start = b ; inst = Z ; pds = 0$

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Application - 2SLS

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GMM Estimates

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2SLS GMM with Heteroscedasticity

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**Testing the Overidentifying Restrictions**

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**Inference About the Parameters**

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**Specification Test Based on the Criterion**

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**Extending the Form of the GMM Estimator to Nonlinear Models**

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**A Nonlinear Conditional Mean**

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**Nonlinear Regression/GMM**

NAMELIST ; x = one,exp,expsq,occ,south,smsa,wks$ NAMELIST ; z = one,exp,expsq,occ,south,smsa,ms,union,ed$ ? Get initial values to use for optimal weighting matrix NLSQ ; lhs = lwage ; fcn=exp(b1'x) ; inst = z ; labels=b1,b2,b3,b4,b5,b6,b7 ; start=7_0$ ? GMM using previous estimates to compute weighting matrix NLSQ (GMM) ; fcn = lwage-exp(b1'x) ; inst = Z ; labels = b1,b2,b3,b4,b5,b6,b7 ; start = b ; pds = 0 $ (Means use White style estimator)

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**Nonlinear Wage Equation Estimates NLSQ Initial Values**

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**Nonlinear Wage Equation Estimates 2nd Step GMM**

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IV for Panel Data

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