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1、Econometric Analysis of Panel Data,William Greene Department of Economics Stern School of Business,Econometric Analysis of Panel Data,14. Nonlinear Models And Nonlinear Optimization,Agenda,Nonlinear Models Estimation Theory for Nonlinear Models Estimators Properties M Estimation Nonlinear Least Squa
2、res Maximum Likelihood Estimation GMM Estimation Minimum Distance Estimation Minimum Chi-square Estimation Computation Nonlinear Optimization Nonlinear Least Squares Newton-like Algorithms; Gradient Methods (Background: JW, Chapters 12-14, Greene, Chapters 16-18),What is a Model?,Unconditional chara
3、cteristics of a population Conditional moments: Eg(y)|x: median, mean, variance, quantile, correlations, probabilities Conditional probabilities and densities Conditional means and regressions Fully parametric and semiparametric specifications Parametric specification: Known up to parameter Paramete
4、r spaces Conditional means: Ey|x = m(x, ),What is a Nonlinear Model?,Model: Eg(y)|x = m(x,) Objective: Learn about from y, X Usually “estimate” Linear Model: Closed form; = h(y, X) Nonlinear Model Not wrt m(x,). E.g., y=exp(x + ) Wrt estimator: Implicitly defined. h(y, X, )=0, E.g., Ey|x= exp(x),Wha
5、t is an Estimator?,Point and IntervalClassical and Bayesian,Parameters,Model parameters The parameter space Interior of the parameter space Estimators of parameters The true parameter(s),The Conditional Mean Function,M Estimation,Classical estimation method,An Analogy Principle for M Estimation,Esti
6、mation,Identification,Continuity,Consistency,Asymptotic Normality of M Estimators,Asymptotic Normality,Asymptotic Normality,Asymptotic Variance,Estimating the Variance,Nonlinear Least Squares,Application - Income,German Health Care Usage Data, 7,293 Individuals, Varying Numbers of Periods Variables
7、in the file are Data downloaded from Journal of Applied Econometrics Archive. This is an unbalanced panel with 7,293 individuals. They can be used for regression, count models, binary choice, ordered choice, and bivariate binary choice. This is a large data set. There are altogether 27,326 observati
8、ons. 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). Note, the variable NUMOBS below tells how many observations there are for each person. This variable is repeated in each row of the data for the person. (Downloaded from the JA
9、E Archive) 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,Income Data,Exponential Model,Conventional Variance Estimator,S
10、ometimes omitted.,Estimator for the M Estimator,Computing NLS,Reject; hhninc=0$ Calc ; b0=log(xbr(hhninc)$ Nlsq ; labels=a0,a1,a2,a3; start=b0,0,0,0; fcn=exp(a0+a1*educ+a2*married+a3*age); lhs=hhninc;output=3$ Name ; x=one,educ,married,age$ Create; thetai = exp(xb); ei=hhninc=thetai ; gi=ei*thetai ;
11、 gi2=gi*gi ; hi=hhninc*thetai$ Matrix; varM = * xgi2x * $ Matrix; stat(b,varm)$,Iterations,Begin NLSQ iterations. Linearized regression. Iteration= 1; Sum of squares= 854.681775 ; Gradient= 90.0964694 Iteration= 2; Sum of squares= 766.073500 ; Gradient= 2.38006397 Iteration= 3; Sum of squares= 763.7
12、57721 ; Gradient= .300030163E-02 Iteration= 4; Sum of squares= 763.755005 ; Gradient= .307466962E-04 Iteration= 5; Sum of squares= 763.754978 ; Gradient= .365064970E-06 Iteration= 6; Sum of squares= 763.754978 ; Gradient= .433325697E-08 Iteration= 7; Sum of squares= 763.754978 ; Gradient= .514374906
13、E-10 Iteration= 8; Sum of squares= 763.754978 ; Gradient= .610586853E-12 Iteration= 9; Sum of squares= 763.754978 ; Gradient= .724960231E-14 Iteration= 10; Sum of squares= 763.754978 ; Gradient= .860927011E-16 Iteration= 11; Sum of squares= 763.754978 ; Gradient= .102139114E-17 Iteration= 12; Sum of
14、 squares= 763.754978 ; Gradient= .118640949E-19 Iteration= 13; Sum of squares= 763.754978 ; Gradient= .125019054E-21 Convergence achieved,NLS Estimates,+-+ | User Defined Optimization | | Nonlinear least squares regression | | Model was estimated Mar 18, 2005 at 11:17:37PM | | LHS=HHNINC Mean = .352
15、1352 | | Standard deviation = .1768699 | | Residuals Sum of squares = 763.7550 | +-+ +-+-+-+-+-+ |Variable | Coefficient | Standard Error |b/St.Er.|P|Z|z | +-+-+-+-+-+Conventional EstimatesA0 -1.89118955 .01879455 -100.624 .0000A1 .05471841 .00102649 53.306 .0000A2 .23756387 .00765477 31.035 .0000A3 .00081033 .00026344 3.076 .0021 +-+-+-+-+-+Recomputed variances using results for M Estimation.B_1 -1.89118955 .01910054 -99.012 .0000B_2 .05471841 .00115059 47.557 .0000B_3 .23756387 .00842712 28.190 .0000B_4 .00081033 .00026137 3.100 .0019,
