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Part 2: Basic Econometrics [ 1/48] Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business Part 2: Basic Econometrics [ 2/48] Part 2: Basic Econometrics [ 3/48] Part 2: Basic Econometrics [ 4/48] Trends in Econometrics Small structural models vs. large scale multiple equation models Non- and semiparametric methods vs. parametric Robust methods – GMM (paradigm shift? Nobel prize) Unit roots, cointegration and macroeconometrics Nonlinear modeling and the role of software Behavioral and structural modeling vs. “reduced form,” “covariance analysis” Identification and “causal” effects Pervasiveness of an econometrics paradigm Part 2: Basic Econometrics [ 5/48] Estimation Platforms Model based Kernels and smoothing methods (nonparametric) Semiparametric analysis Parametric analysis Moments and quantiles (semiparametric) Likelihood and M- estimators (parametric) Methodology based (?) Classical – parametric and semiparametric Bayesian – strongly parametric Part 2: Basic Econometrics [ 6/48] Objectives in Model Building Specification: guided by underlying theory Estimation: coefficients, partial effects, model implications Statistical inference: hypothesis testing Prediction: individual and aggregate Model assessment (fit, adequacy) and evaluation Model extensions Modeling framework Functional forms Interdependencies, multiple part models Heterogeneity Endogeneity Exploration: Estimation and inference methods Part 2: Basic Econometrics [ 7/48] Regression Basics The “MODEL” Modeling the conditional mean – Regression Other features of interest Modeling quantiles Conditional variances or covariances Modeling probabilities for discrete choice Modeling other features of the population Part 2: Basic Econometrics [ 8/48] Application: Health Care German Health Care Usage Data, 7,293 Individuals, Varying Numbers of Periods Data downloaded from Journal of Applied Econometrics Archive. They can be used for regression, count models, binary choice, ordered choice, and bivariate binary choice. There are altogether 27,326 observations. The number of observations ranges from 1 to 7. (Frequencies are: 1=1525, 2=2158, 3=825, 4=926, 5=1051, 6=1000, 7=987). Variables in the file are DOCTOR = 1(Number of doctor visits > 0) HOSPITAL = 1(Number of hospital visits > 0) HSAT = health satisfaction, coded 0 (low) - 10 (high) DOCVIS = number of doctor visits in last three months HOSPVIS = number of hospital visits in last calendar year PUBLIC = insured in public health insurance = 1; otherwise = 0 ADDON = insured by add-on insurance = 1; otherswise = 0 HHNINC = household nominal monthly net income in German marks / 10000. (4 observations with income=0 were dropped) HHKIDS = children under age 16 in the household = 1; otherwise = 0 EDUC = years of schooling AGE = age in years MARRIED = marital status Part 2: Basic Econometrics [ 9/48] Part 2: Basic Econometrics [ 10/48] Part 2: Basic Econometrics [ 11/48] Unobserved factors are “controlled for.” Part 2: Basic Econometrics [ 12/48] Household Income Kernel Density Estimator Histogram Part 2: Basic Econometrics [ 13/48] Regression – Income on Education ---------------------------------------------------------------------Ordinary least squares regression ............ LHS=LOGINC Mean = -.92882 Standard deviation = .47948 Number of observs. = 887 Model size Parameters = 2 Degrees of freedom = 885 Residuals Sum of squares = 183.19359 Standard error of e = .45497 Fit R-squared = .10064 Adjusted R-squared = .09962 Model test F[ 1, 885] (prob) = 99.0(.0000) Diagnostic Log likelihood = -559.06527 Restricted(b=0) = -606.10609 Chi-sq [ 1] (prob) = 94.1(.0000) Info criter. LogAmemiya Prd. Crt. = -1.57279 --------+------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------Constant| -1.71604*** .08057 -21.299 .0000 EDUC| .07176*** .00721 9.951 .0000 10.9707 --------+------------------------------------------------------------Note: ***, **, * = Significance at 1%, 5%, 10% level. ---------------------------------------------------------------------- Part 2: Basic Econometrics [ 14/48] Specification and Functional Form ---------------------------------------------------------------------Ordinary least squares regression ............ LHS=LOGINC Mean = -.92882 Standard deviation = .47948 Number of observs. = 887 Model size Parameters = 3 Degrees of freedom = 884 Residuals Sum of squares = 183.00347 Standard error of e = .45499 Fit R-squared = .10157 Adjusted R-squared = .09954 Model test F[ 2, 884] (prob) = 50.0(.0000) Diagnostic Log likelihood = -558.60477 Restricted(b=0) = -606.10609 Chi-sq [ 2] (prob) = 95.0(.0000) Info criter. LogAmemiya Prd. Crt. = -1.57158 --------+------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------Constant| -1.68303*** .08763 -19.207 .0000 EDUC| .06993*** .00746 9.375 .0000 10.9707 FEMALE| -.03065 .03199 -.958 .3379 .42277 --------+------------------------------------------------------------- Part 2: Basic Econometrics [ 15/48] Interesting Partial Effects ---------------------------------------------------------------------Ordinary least squares regression ............ LHS=LOGINC Mean = -.92882 Standard deviation = .47948 Number of observs. = 887 Model size Parameters = 5 Degrees of freedom = 882 Residuals Sum of squares = 171.87964 Standard error of e = .44145 Fit R-squared = .15618 Adjusted R-squared = .15235 Model test F[ 4, 882] (prob) = 40.8(.0000) Diagnostic Log likelihood = -530.79258 Restricted(b=0) = -606.10609 Chi-sq [ 4] (prob) = 150.6(.0000) Info criter. LogAmemiya Prd. Crt. = -1.62978 --------+------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------Constant| -5.26676*** .56499 -9.322 .0000 EDUC| .06469*** .00730 8.860 .0000 10.9707 FEMALE| -.03683 .03134 -1.175 .2399 .42277 AGE| .15567*** .02297 6.777 .0000 50.4780 AGE2| -.00161*** .00023 -7.014 .0000 2620.79 --------+------------------------------------------------------------- E[ Income | x] Age 2 Age Age2 Age Part 2: Basic Econometrics [ 16/48] Function: Log Income | Age Partial Effect wrt Age Part 2: Basic Econometrics [ 17/48] A Statistical Relationship A relationship of interest: Number of hospital visits: H = 0,1,2,… Covariates: x1=Age, x2=Sex, x3=Income, x4=Health Causality and covariation Theoretical implications of ‘causation’ Comovement and association Intervention of omitted or ‘latent’ variables Temporal relationship – movement of the “causal variable” precedes the effect. Part 2: Basic Econometrics [ 18/48] Endogeneity A relationship of interest: Number of hospital visits: H = 0,1,2,… Covariates: x1=Age, x2=Sex, x3=Income, x4=Health Should Health be ‘Endogenous’ in this model? What do we mean by ‘Endogenous’ What is an appropriate econometric method of accommodating endogeneity? Part 2: Basic Econometrics [ 19/48] Models Conditional mean function: E[y | x] Other conditional characteristics – what is ‘the model?’ Conditional variance function: Var[y | x] Conditional quantiles, e.g., median [y | x] Other conditional moments Conditional probabilities: P(y|x) What is the sense in which “y varies with x?” Part 2: Basic Econometrics [ 20/48] Using the Model Understanding the relationship: Estimation of quantities of interest such as elasticities Prediction of the outcome of interest Control of the path of the outcome of interest Part 2: Basic Econometrics [ 21/48] Application: Doctor Visits German individual health care data: N=27,236 Model for number of visits to the doctor: Poisson regression (fit by maximum likelihood) E[V|Income]=exp(1.412 - .0745 income) OLS Linear regression: g*(Income)=3.917 - .208 income H istogr am 11152 for N um ber of D octor V isits 8364 Frequency 5576 2788 0 0 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0 4 4 4 8 5 2 5 6 6 0 6 4 6 8 7 2 7 6 8 0 8 4 8 8 9 2 9 61 0 0 104 108 112 116 120 DOCV I S Part 2: Basic Econometrics [ 22/48] Conditional Mean and Linear Projection Projection and Conditional Mean Functions 4 .3 1 Function 3 .4 0 This area is outside the range of the data 2 .4 8 Most of the data are in here 1 .5 7 .6 6 -. 2 6 0 5 10 15 20 I NCOM E CONDM E AN P ROJ E CT N Notice the problem with the linear projection. Negative predictions. Part 2: Basic Econometrics [ 23/48] What About the Linear Projection? What we do when we linearly regress a variable on a set of variables Assuming there exists a conditional mean There usually exists a linear projection. Requires finite conditional variance of y. Approximation to the conditional mean? If the conditional mean is linear Linear projection equals the conditional mean Part 2: Basic Econometrics [ 24/48] Partial Effects What did the model tell us? Covariation and partial effects: How does the y “vary” with the x? Partial Effects: Effect on what????? For continuous variables δ(x)=E[y|x]/x, usually not coefficients For dummy variables (x,d)=E[y|x,d=1] - E[y|x,d=0] Elasticities: ε(x)=δ(x) x / E[y|x] Part 2: Basic Econometrics [ 25/48] Average Partial Effects When δ(x) ≠β, APE = Ex[δ(x)]= (x)f(x)dx x Approximation: Is δ(E[x]) = Ex[δ(x)]? (no) Empirically: Estimated APE = (1 /N)Ni=1ˆ (xi ) Empirical approximation: Est.APE = ̂(x) For the doctor visits model δ(x)= β exp(α+βx)=-.0745exp(1.412-.0745income) Sample APE = -.2373 Approximation = -.2354 Slope of the linear projection = -.2083 (!) Part 2: Basic Econometrics [ 26/48] APE and PE at the Mean δ(x)=E[y|x]/x, =E[x] δ(x) δ()+δ()(x-)+(1/2)δ()(x-)2 + E[δ(x)]=APE δ() + (1/2)δ()2x Implication: Computing the APE by averaging over observations (and counting on the LLN and the Slutsky theorem) vs. computing partial effects at the means of the data. In the earlier example: Sample APE = -.2373 Approximation = -.2354 Part 2: Basic Econometrics [ 27/48] The Canonical Panel Data Problem y x c c is unobserved individual heterogeneity How do we estimate partial effects in the presence of c? PE(x) = Ec δ(x,c)=Ec E[y|x,c]/x APE = E x Ec E[y|x,c]/x Part 2: Basic Econometrics [ 28/48] The Linear Regression Model y = X+ε, N observations, K columns in X, including a column of ones. Standard assumptions about X Standard assumptions about ε|X E[ε|X]=0, E[ε]=0 and Cov[ε,x]=0 Regression? If E[y|X] = X then X is the projection of y on X Part 2: Basic Econometrics [ 29/48] Estimation of the Parameters Least squares, LAD, other estimators – we will focus on least squares -1 b = (X'X) X'y s2 e'e /N or e'e /(N-K) Classical vs. Bayesian estimation of Properties Statistical inference: Hypothesis tests Prediction (not this course) Part 2: Basic Econometrics [ 30/48] Properties of Least Squares Finite sample properties: Unbiased, etc. No longer interested in these. Asymptotic properties Consistent? Under what assumptions? Efficient? Contemporary work: Often not important Efficiency within a class: GMM Asymptotically normal: How is this established? Robust estimation: To be considered later Part 2: Basic Econometrics [ 31/48] Least Squares Summary ˆ b, plim b = d N(b ) N{0, 2 [plim( X'X / N)]1 } a b N{, (2 / N)[plim( X'X / N)]1 } Estimated Asy.Var[b]=s 2 ( X'X ) 1 Part 2: Basic Econometrics [ 32/48] Hypothesis Testing Nested vs. nonnested tests y=b1x+e vs. y=b1x+b2z+e: Nested y=bx+e vs. y=cz+u: Not nested y=bx+e vs. logy=clogx: Not nested y=bx+e; e ~ Normal vs. e ~ t[.]: Not nested Fixed vs. random effects: Not nested Logit vs. probit: Not nested x is endogenous: Maybe nested. We’ll see … Parametric restrictions Linear: R-q = 0, R is JxK, J < K, full row rank General: r(,q) = 0, r = a vector of J functions, R (,q) = r(,q)/’. Use r(,q)=0 for linear and nonlinear cases Part 2: Basic Econometrics [ 33/48] Example: Panel Data on Spanish Dairy Farms N = 247 farms, T = 6 years (1993-1998) Units Mean Output Milk Milk production (liters) Input Cows # of milking cows 22.12 Input Labor # man-equivalent units Input Land Input Feed 131,107 Std. Dev. Minimum 92,584 14,410 Maximum 727,281 11.27 4.5 82.3 1.67 0.55 1.0 4.0 Hectares of land devoted to pasture and crops. 12.99 6.17 2.0 45.1 Total amount of feedstuffs fed to dairy cows (Kg) 57,941 47,981 3,924.14 376,732 Part 2: Basic Econometrics [ 34/48] Application y = log output x = Cobb douglas production: x = 1,x1,x2,x3,x4 = constant and logs of 4 inputs (5 terms) z = Translog terms, x12, x22, etc. and all cross products, x1x2, x1x3, x1x4, x2x3, etc. (10 terms) w = (x,z) (all 15 terms) Null hypothesis is Cobb Douglas, alternative is translog = Cobb-Douglas plus second order terms. Part 2: Basic Econometrics [ 35/48] Translog Regression Model x H0:z=0 Part 2: Basic Econometrics [ 36/48] Wald Tests r(b,q)= close to zero? Wald distance function: r(b,q)’{Var[r(b,q)]}-1 r(b,q) 2[J] Use the delta method to estimate Var[r(b,q)] Est.Asy.Var[b]=s2(X’X)-1 Est.Asy.Var[r(b,q)]= R(b,q){s2(X’X)-1}R’(b,q) The standard F test is a Wald test; JF = 2[J]. Part 2: Basic Econometrics [ 37/48] Wald= b z - 0 {Var[b z - 0]}1 b z - 0 Close to 0? Part 2: Basic Econometrics [ 38/48] Likelihood Ratio Test The normality assumption Does it work ‘approximately?’ For any regression model yi = h(xi,)+εi where εi ~N[0,2], (linear or nonlinear), at the linear (or nonlinear) least squares estimator, however computed, with or without restrictions, 2 2 ˆ and logL( ˆ /N) (N/2)[1+log2+log ˆ ˆ ˆ ] This forms the basis for likelihood ratio tests. ˆunrestricted ) log L ( ˆrestricted )] 2[log L ( 2 ˆ d Nlog 2restricted 2 [ J ] ˆunrestricted Part 2: Basic Econometrics [ 39/48] Part 2: Basic Econometrics [ 40/48] Score or LM Test: General Maximum Likelihood (ML) Estimation A hypothesis test H0: Restrictions on parameters are true H1: Restrictions on parameters are not true Basis for the test: b0 = parameter estimate under H0 (i.e., restricted), b1 = unrestricted Derivative results: For the likelihood function under H1, logL1/ | =b1 = 0 (exactly, by definition) logL1/ | =b0 ≠ 0. Is it close? If so, the restrictions look reasonable Part 2: Basic Econometrics [ 41/48] Why is it the Lagrange Multiplier Test? Maximize logL() subject to restrictions r ()= 0 X such as Rβ - q = 0 or Z = 0 when = , R = (0:I) and q = 0. Z Use LM approach: Maximize wrt ( ) L* = logL() r(). L * / ˆ R logL(ˆ R ) / ˆ R r(ˆ R ) / ˆ R 0 FOC: 0 L * / r (ˆ R ) If = 0, the constraints are not binding and logL(ˆ R ) / ˆ R 0 so ˆ R ˆ U . If 0, the constraints are binding and logL(ˆ ) / ˆ 0. R Direct test: Test H 0 : = 0 Equivalent test: Test H 0LM : logL(ˆ R ) / ˆ R 0. R Part 2: Basic Econometrics [ 42/48] Computing the LM Statistic ˆ (1|0),i g N 2 e 1 x i 2 ei|0 , ei|0 =y i x ib, s 02 i=1 i|0 N s 0 zi ˆ (1|0) =Ni=1g ˆ (1|0),i = Ni=1 g N ˆ ˆ ˆ H (1|0) i=1 g(1|0),i g(1|0),i 1 x i 1 X'e0 1 0 e = 2 i|0 2 2 s 0 zi s 0 Z'e0 s 0 Z'e0 1 2 xi xi N i=1 4 ei|0 ' s0 zi zi Note this is the middle matrix in the White estimator. x and s 04 disappear from the product. Let w i0 ei|0 i . zi Then LM=i'W0 (W0 W0 ) 1 W0 i, which is simple to compute. The s 2 0 2 The derivation on page 64 of Wooldridge’s text is needlessly complex, and the second form of LM is actually incorrect because the first derivatives are not ‘heteroscedasticity robust.’ Part 2: Basic Econometrics [ 43/48] Application of the Score Test Linear Model: Y = X+Zδ+ε Test H0: δ=0 Restricted estimator is [b’,0’]’ Namelist ; X = a list… ; Z = a list … ; W = X,Z $ Regress ; Lhs = y ; Rhs = X ; Res = e $ Matrix ; list ; LM = e’ W * <W’[e^2]W> * W’ e $ Part 2: Basic Econometrics [ 44/48] Restricted regression and derivatives for the LM Test Part 2: Basic Econometrics [ 45/48] Tests for Omitted Variables ? Cobb - Douglas Model Namelist ; X = One,x1,x2,x3,x4 $ ? Translog second order terms, squares and cross products of logs Namelist ; Z = x11,x22,x33,x44,x12,x13,x14,x23,x24,x34 $ ? Restricted regression. Short. Has only the log terms Regress ; Lhs = yit ; Rhs = X ; Res = e $ Calc ; LoglR = LogL ; RsqR = Rsqrd $ ? LM statistic using basic matrix algebra Namelist ; W = X,Z $ Matrix ; List ; LM = e'W * <W’[e^2]W> * W'e $ ? LR statistic uses the full, long regression with all quadratic terms Regress ; Lhs = yit ; Rhs = W $ Calc ; LoglU = LogL ; RsqU = Rsqrd ; List ; LR = 2*(Logl - LoglR) $ ? Wald Statistic is just J*F for the translog terms Calc ; List ; JF=col(Z)*((RsqU-RsqR)/col(Z)/((1-RsqU)/(n-kreg)) )$ Part 2: Basic Econometrics [ 46/48] Regression Specifications Part 2: Basic Econometrics [ 47/48] Model Selection Regression models: Fit measure = R2 Nested models: log likelihood, GMM criterion function (distance function) Nonnested models, nonlinear models: Classical Akaike information criterion= – (logL – 2K)/N Bayes (Schwartz) information criterion = –(logL-K(logN))/N Bayesian: Bayes factor = Posterior odds/Prior odds (For noninformative priors, BF=ratio of posteriors) Part 2: Basic Econometrics [ 48/48] Remaining to Consider for the Linear Regression Model Failures of standard assumptions Heteroscedasticity Autocorrelation and Spatial Correlation Robust estimation Omitted variables Measurement error Endogenous variables and causal effects Part 2: Basic Econometrics [ 49/48] Appendix: 1. Computing the LM Statistic 2. Misc Results Part 2: Basic Econometrics [ 50/48] LM Test ˆ ˆ1|=b0 Ni=1g ˆ (1|=b0 ),i G'i g ˆ is a vector of derivatives. where each row of G ˆ1|=b0 is close to 0. Use a Wald statistic to assess if g By the information matrix equality, ˆ1|=b0 ] E[H1|=b0 ] where H is the Hessian. Var[g Estimate this with the sum of squares as usual: ˆ ˆ ˆ 1|=b Ni=1g ˆ ˆ H g G'G (1|=b0 ),i (1|=b0 ),i 0 Part 2: Basic Econometrics [ 51/48] LM Test (Cont.) ˆ1| =b0 Wald statistic = g -1 ˆ 1|=b g H ˆ1|=b0 0 -1 ˆ G'G ˆ ˆ G'i ˆ = iG -1 ˆ G'G ˆ ˆ G'i ˆ /N = N iG = NR 2 R 2 = the R 2 in the 'uncentered' regression of a column of ones on the vector of first derivatives. (N is equal to the total sum of squares.) This is a general result. We did not assume any specific model in the derivation. (We did assume independent observations.) Part 2: Basic Econometrics [ 52/48] Representing Covariation Conditional mean function: E[y | x] = g(x) Linear approximation to the conditional mean function: Linear Taylor series ĝ( x ) = g( x 0 ) + ΣKk=1 [gk | x = x 0 ](x k -x k0 ) = 0 + ΣKk=1k (x k -x k0 ) The linear projection (linear regression?) g*(x)= 0 Kk 1 k (x k -E[x k ]) 0 E[y] Var[x]}-1 {Cov[x ,y]} Part 2: Basic Econometrics [ 53/48] Projection and Regression The linear projection is not the regression, and is not the Taylor series. Example: f(y|x)=[1/λ(x)]exp[-y/λ(x)] λ(x)=exp(+x)=E[y|x] x~U[0,1]; f(x)=1, 0 x 1 Part 2: Basic Econometrics [ 54/48] For the Example: α=1, β=2 20.88 Conditional Mean 16.70 Linear Projection Linear Projection Variable 12.53 Taylor Series 8.35 4.18 .00 .00 .20 .40 .60 .80 X EY_X PROJECTN TAYLOR 1.00 Part 2: Basic Econometrics [ 55/48] www.oft.gov.uk/shared_oft/reports/Evaluating-OFTs-work/oft1416.pdf Part 2: Basic Econometrics [ 56/48] Econometrics Theoretical foundations Microeconometrics and macroeconometrics Behavioral modeling Statistical foundations: Econometric methods Mathematical elements: the usual ‘Model’ building – the econometric model

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