1、Applied Econometrics,William Greene Department of Economics Stern School of Business,Applied Econometrics,22. The Generalized Method of Moments,The Method of Moments,Estimating a Parameter,Mean of Poisson p(y)=exp(-) y / y! Ey= . plim (1/N)iyi = . This is the estimator Mean of Exponential p(y) = exp

2、(- y) Ey = 1/ . plim (1/N)iyi = 1/,Normal Distribution,Gamma Distribution,The Linear Regression Model,Instrumental Variables,Maximum Likelihood,Behavioral Application,Identification,Can the parameters be estimated? Not a sample property Assume an infinite sample Is there sufficient information in a

3、sample to reveal consistent estimators of the parameters Can the moment equations be solved for the population parameters?,Identification,Exactly Identified Case: K population moment equations in K unknown parameters. Our familiar cases, OLS, IV, ML, the MOM estimators Is the counting rule sufficien

4、t? What else is needed? Overidentified Case Instrumental Variables The Covariance Structures Model Underidentified Case Multicollinearity Variance parameter in a probit model,Overidentification,Overidentification,Underidentification,Multicollinearity: The moment equations are linearly dependent. Ins

5、ufficient Variation in Observable Quantities,The Data Competing Model A Competing Model B Which model is more consistent with the data?,Q,P,Q,P,Q,P,Underidentification Model/Data,Underidentification - Math,Agenda,The Method of Moments. Solving the moment equations Exactly identified cases Overidenti

6、fied cases Consistency. How do we know the method of moments is consistent? Asymptotic covariance matrix. Consistent vs. Efficient estimation A weighting matrix The minimum distance estimator What is the efficient weighting matrix? Estimating the weighting matrix. The Generalized method of moments e

7、stimator - how it is computed. Computing the appropriate asymptotic covariance matrix,Inference,Testing hypotheses about the parameters: Wald test A counterpart to the likelihood ratio test Testing the overidentifying restrictions,The Method of Moments,Moment Equation: Defines a sample statistic tha

8、t mimics a population expectation: The population expectation orthogonality condition: E mi () = 0. Subscript i indicates it depends on data vector indexed by i (or t for a time series setting),The Method of Moments - Example,Application,Method of Moments Solution,create ; y1=y ; y2=log(y)$ calc ; m

9、1=xbr(y1) ; ms=xbr(ys)$ sample ; 1 $ mini ; start = 2.0, .06 ; labels = p,l ; fcn= (l*m1-p)2 + (m2 - psi(p)+log(l) 2 $ +-+ | User Defined Optimization | | Dependent variable Function | | Number of observations 1 | | Iterations completed 6 | | Log likelihood function .5062979E-13 | +-+ +-+-+ |Variabl

10、e | Coefficient | +-+-+ P 2.41060361 L .07707026,Nonlinear Instrumental Variables,There are K parameters, yi = f(xi,) + i. There exists a set of K instrumental variables, zi such that Ezi i = 0. The sample counterpart is the moment equation (1/n)i zi i = (1/n)i zi yi - f(xi,) = (1/n)i mi () = () = 0

11、. The method of moments estimator is the solution to the moment equation(s). (How the solution is obtained is not always obvious, and varies from problem to problem.),The MOM Solution,MOM Solution,Variance of the Method of Moments Estimator,Example 1: Gamma Distribution,Example 2: Nonlinear Least Sq

12、uares,Variance of the Moments,Properties of the MOM Estimator,Consistent? The LLN implies that the moments are consistent estimators of their population counterparts (zero) Use the Slutsky theorem to assert consistency of the functions of the moments Asymptotically normal? The moments are sample mea

13、ns. Invoke a central limit theorem. Efficient? Not necessarily Sometimes yes. (Gamma example) Perhaps not. Depends on the model and the available information (and how much of it is used).,Generalizing the Method of Moments Estimator,More moments than parameters the overidentified case Example: Instr

14、umental variable case, M K instruments,Two Stage Least Squares,How to use an “excess” of instrumental variables (1) X is K variables. Some (at least one) of the K variables in X are correlated with . (2) Z is M K variables. Some of the variables in Z are also in X, some are not. None of the variable

15、s in Z are correlated with . (3) Which K variables to use to compute ZX and Zy?,Choosing the Instruments,Choose K randomly? Choose the included Xs and the remainder randomly? Use all of them? How? A theorem: (Brundy and Jorgenson, ca. 1972) There is a most efficient way to construct the IV estimator

16、 from this subset: (1) For each column (variable) in X, compute the predictions of that variable using all the columns of Z. (2) Linearly regress y on these K predictions. This is two stage least squares,2SLS Algebra,Method of Moments Estimation,FOC for MOM,Computing the Estimator,Programming Progra

17、m No all purpose solution Nonlinear optimization problem solution varies from setting to setting.,Asymptotic Covariance Matrix,More Efficient Estimation,Minimum Distance Estimation,MDE Estimation: Application,Least Squares,Generalized Least Squares,Minimum Distance Estimation,Optimal Weighting Matri

18、x,GMM Estimation,GMM Estimation,A Practical Problem,Testing Hypotheses,Application: Dynamic Panel Data Model,Applied Econometrics,William Greene Department of Economics Stern School of Business,Applied Econometrics,23. Discrete Choice Modeling,A Microeconomics Platform,Consumers Maximize Utility (!)

19、 Fundamental Choice Problem: Maximize U(x1,x2,) subject to prices and budget constraints A Crucial Result for the Classical Problem: Indirect Utility Function: V = V(p,I) Demand System of Continuous Choices The Integrability Problem: Utility is not revealed by demands,Theory for Discrete Choice,Theo

20、ry is silent about discrete choices Translation to discrete choice Existence of well defined utility indexes: Completeness of rankings Rationality: Utility maximization Axioms of revealed preferences Choice sets and consideration sets consumers simplify choice situations Implication for choice among

21、 a set of discrete alternatives Commonalities and uniqueness Does this allow us to build “models?” What common elements can be assumed? How can we account for heterogeneity? Revealed choices do not reveal utility, only rankings which are scale invariant,Modeling the Binary Choice Ui,suv = suv + Psuv

22、 + suvIncome + i,suv Ui,sed = sed + Psed + sedIncome + i,sed Chooses SUV: Ui,suv Ui,sed Ui,suv - Ui,sed 0 (SUV-SED) + (PSUV-PSED) + (SUV-sed)Income + i,suv - i,sed 0 i - + (PSUV-PSED) + Income,Choosing Between Two Alternatives,What Can Be Learned from the Data? (A Sample of Consumers, i = 1,N),Are t

23、he attributes “relevant?” Predicting behavior Individual Aggregate Analyze changes in behavior when attributes change,Application,210 Commuters Between Sydney and Melbourne Available modes = Air, Train, Bus, Car Observed: Choice Attributes: Cost, terminal time, other Characteristics: Household incom

24、e First application: Fly or Other,Binary Choice Data,Choose Air Gen.Cost Term Time Income 1.0000 86.000 25.000 70.000 .00000 67.000 69.000 60.000 .00000 77.000 64.000 20.000 .00000 69.000 69.000 15.000 .00000 77.000 64.000 30.000 .00000 71.000 64.000 26.000 .00000 58.000 64.000 35.000 .00000 71.000

25、69.000 12.000 .00000 100.00 64.000 70.000 1.0000 158.00 30.000 50.000 1.0000 136.00 45.000 40.000 1.0000 103.00 30.000 70.000 .00000 77.000 69.000 10.000 1.0000 197.00 45.000 26.000 .00000 129.00 64.000 50.000 .00000 123.00 64.000 70.000,An Econometric Model,Choose to fly iff UFLY 0 Ufly = +1Cost +

26、2Time + Income + Ufly 0 -(+1Cost + 2Time + Income) Probability model: For any person observed by the analyst, Prob(fly) = Prob -(+1Cost + 2Time + Income) Note the relationship between the unobserved and the outcome,+1Cost + 2TTime + Income,Modeling Approaches,Nonparametric “relationship” Minimal Ass

27、umptions Minimal Conclusions Semiparametric “index function” Stronger assumptions Robust to model misspecification (heteroscedasticity) Still weak conclusions Parametric “Probability function and index” Strongest assumptions complete specification Strongest conclusions Possibly less robust. (Not nec

28、essarily),Nonparametric,P(Air)=f(Income),Semiparametric,MSCORE: Find bx so that sign(bx) * sign(y) is maximized. Klein and Spady: Find b to maximize a semiparametric likelihood of G(bx),MSCORE,Klein and Spady Semiparametric,Note necessary normalizations. Coefficients are not very meaningful.,Paramet

29、ric: Logit Model,Logit vs. MScore,Logit fits worse MScore fits better, coefficients are meaningless,Parametric Model Estimation,How to estimate , 1, 2, ? Its not regression The technique of maximum likelihood Proby=1 = Prob -(+1Cost + 2Time + Income) Proby=0 = 1 - Proby=1 Requires a model for the pr

30、obability,Completing the Model: F(),The distribution Normal: PROBIT, natural for behavior Logistic: LOGIT, allows “thicker tails” Gompertz: EXTREME VALUE, asymmetric, underlies the basic logit model for multiple choice Does it matter? Yes, large difference in estimates Not much, quantities of intere

31、st are more stable.,Underlying Probability Distributions for Binary Choice,Estimated Binary Choice (Probit) Model,+-+ | Binomial Probit Model | | Maximum Likelihood Estimates | | Dependent variable MODE | | Weighting variable None | | Number of observations 210 | | Iterations completed 6 | | Log lik

32、elihood function -84.09172 | | Restricted log likelihood -123.7570 | | Chi squared 79.33066 | | Degrees of freedom 3 | | ProbChiSqd value = .0000000 | | Hosmer-Lemeshow chi-squared = 46.96547 | | P-value= .00000 with deg.fr. = 8 | +-+ +-+-+-+-+-+-+ |Variable | Coefficient | Standard Error |b/St.Er.|

33、P|Z|z | Mean of X| +-+-+-+-+-+-+ Index function for probability Constant .43877183 .62467004 .702 .4824 GC .01256304 .00368079 3.413 .0006 102.647619 TTME -.04778261 .00718440 -6.651 .0000 61.0095238 HINC .01442242 .00573994 2.513 .0120 34.5476190,Estimated Binary Choice Models,LOGIT PROBIT EXTREME

34、VALUE Variable Estimate t-ratio Estimate t-ratio Estimate t-ratio Constant 1.78458 1.40591 0.438772 0.702406 1.45189 1.34775 GC 0.0214688 3.15342 0.012563 3.41314 0.0177719 3.14153 TTME -0.098467 -5.9612 -0.0477826 -6.65089 -0.0868632 -5.91658 HINC 0.0223234 2.16781 0.0144224 2.51264 0.0176815 2.028

35、76 Log-L -80.9658 -84.0917 -76.5422 Log-L(0) -123.757 -123.757 -123.757,+1Cost + 2Time + (Income+1),Effect on Predicted Probability of an Increase in Income,( is positive),Marginal Effects in Probability Models,ProbOutcome = some F(+1Cost) “Partial effect” = F(+1Cost) / ”x” (derivative) Partial effe

36、cts are derivatives Result varies with model Logit: F(+1Cost) / x = Prob * (1-Prob) * Probit: F(+1Cost) / x = Normal density Scaling usually erases model differences,The Delta Method,Marginal Effects for Binary Choice,Logit Probit,Estimated Marginal Effects,Logit Probit Extreme Value,Marginal Effect

37、 for a Dummy Variable,Probyi = 1|xi,di = F(xi+di) =conditional mean Marginal effect of d Probyi = 1|xi,di=1=Probyi= 1|xi,di=0 Logit:,(Marginal) Effect Dummy Variable,HighIncm = 1(Income 50),+-+ | Partial derivatives of probabilities with | | respect to the vector of characteristics. | | They are com

38、puted at the means of the Xs. | | Observations used are All Obs. | +-+ +-+-+-+-+-+-+ |Variable | Coefficient | Standard Error |b/St.Er.|P|Z|z | Mean of X| +-+-+-+-+-+-+ Characteristics in numerator of ProbY = 1 Constant .4750039483 .23727762 2.002 .0453 GC .3598131572E-02 .11354298E-02 3.169 .0015 1

39、02.64762 TTME -.1759234212E-01 .34866343E-02 -5.046 .0000 61.009524 Marginal effect for dummy variable is P|1 - P|0. HIGHINCM .8565367181E-01 .99346656E-01 .862 .3886 .18571429,Computing Effects,Compute at the data means? Simple Inference is well defined Average the individual effects More appropria

40、te? Asymptotic standard errors. (Not done correctly in the literature terms are correlated!) Is testing about marginal effects meaningful?,Average Partial Effects,Elasticities,Elasticity = How to compute standard errors? Delta method Bootstrap Bootstrap the individual elasticities? (Will neglect var

41、iation in parameter estimates.) Bootstrap model estimation?,Estimated Income Elasticity for Air Choice Model,+-+ | Results of bootstrap estimation of model.| | Model has been reestimated 25 times. | | Statistics shown below are centered | | around the original estimate based on | | the original full

42、 sample of observations.| | Result is ETA = .71183 | | bootstrap samples have 840 observations.| | Estimate RtMnSqDev Skewness Kurtosis | | .712 .266 -.779 2.258 | | Minimum = .125 Maximum = 1.135 | +-+,Mean Income = 34.55, Mean P = .2716, Estimated ME = .004539, Estimated Elasticity=0.5774.,Odds Ra

43、tio Logit Model Only,Effect Measure? “Effect of a unit change in the odds ratio.”,Ordered Outcomes,E.g.: Taste test, credit rating, course grade Underlying random preferences: Mapping to observed choices Strength of preferences Censoring and discrete measurement The nature of ordered data,Modeling O

44、rdered Choices,Random Utility Uit = + xit + izit + it = ait + it Observe outcome j if utility is in region j Probability of outcome = probability of cell PrYit=j = F(j ait) - F(j-1 ait),Health Care Satisfaction (HSAT),Self administered survey: Health Care Satisfaction? (0 10),Continuous Preference S

45、cale,Ordered Probability Model,Ordered Probabilities,Five Ordered Probabilities,Coefficients,Effects in the Ordered Probability Model,Assume the k is positive. Assume that xk increases. x increases. j- x shifts to the left for all 5 cells. Proby=0 decreases Proby=1 decreases the mass shifted out is

46、larger than the mass shifted in. Proby=2 decreases same reason. Proby=3 increases. Proby=4 increases,When k 0, increase in xk decreases Proby=0 and increases Proby=J. Intermediate cells are ambiguous, but there is only one sign change in the marginal effects from 0 to 1 to to J,Ordered Probability M

47、odel for Health Satisfaction,+-+ | Ordered Probability Model | | Dependent variable HSAT | | Number of observations 27326 | | Underlying probabilities based on Normal | | Cell frequencies for outcomes | | Y Count Freq Y Count Freq Y Count Freq | | 0 447 .016 1 255 .009 2 642 .023 | | 3 1173 .042 4 1

48、390 .050 5 4233 .154 | | 6 2530 .092 7 4231 .154 8 6172 .225 | | 9 3061 .112 10 3192 .116 | +-+ +-+-+-+-+-+-+ |Variable | Coefficient | Standard Error |b/St.Er.|P|Z|z | Mean of X| +-+-+-+-+-+-+ Index function for probability Constant 2.61335825 .04658496 56.099 .0000 FEMALE -.05840486 .01259442 -4.6

49、37 .0000 .47877479 EDUC .03390552 .00284332 11.925 .0000 11.3206310 AGE -.01997327 .00059487 -33.576 .0000 43.5256898 HHNINC .25914964 .03631951 7.135 .0000 .35208362 HHKIDS .06314906 .01350176 4.677 .0000 .40273000 Threshold parameters for index Mu(1) .19352076 .01002714 19.300 .0000 Mu(2) .4995505

50、3 .01087525 45.935 .0000 Mu(3) .83593441 .00990420 84.402 .0000 Mu(4) 1.10524187 .00908506 121.655 .0000 Mu(5) 1.66256620 .00801113 207.532 .0000 Mu(6) 1.92729096 .00774122 248.965 .0000 Mu(7) 2.33879408 .00777041 300.987 .0000 Mu(8) 2.99432165 .00851090 351.822 .0000 Mu(9) 3.45366015 .01017554 339.

51、408 .0000,Ordered Probability Effects,+-+ | Marginal effects for ordered probability model | | M.E.s for dummy variables are Pry|x=1-Pry|x=0 | | Names for dummy variables are marked by *. | +-+ +-+-+-+-+-+-+ |Variable | Coefficient | Standard Error |b/St.Er.|P|Z|z | Mean of X| +-+-+-+-+-+-+ These ar

52、e the effects on ProbY=00 at means. *FEMALE .00200414 .00043473 4.610 .0000 .47877479 EDUC -.00115962 .986135D-04 -11.759 .0000 11.3206310 AGE .00068311 .224205D-04 30.468 .0000 43.5256898 HHNINC -.00886328 .00124869 -7.098 .0000 .35208362 *HHKIDS -.00213193 .00045119 -4.725 .0000 .40273000 These ar

53、e the effects on ProbY=01 at means. *FEMALE .00101533 .00021973 4.621 .0000 .47877479 EDUC -.00058810 .496973D-04 -11.834 .0000 11.3206310 AGE .00034644 .108937D-04 31.802 .0000 43.5256898 HHNINC -.00449505 .00063180 -7.115 .0000 .35208362 *HHKIDS -.00108460 .00022994 -4.717 .0000 .40273000 . repeat

54、ed for all 11 outcomes These are the effects on ProbY=10 at means. *FEMALE -.01082419 .00233746 -4.631 .0000 .47877479 EDUC .00629289 .00053706 11.717 .0000 11.3206310 AGE -.00370705 .00012547 -29.545 .0000 43.5256898 HHNINC .04809836 .00678434 7.090 .0000 .35208362 *HHKIDS .01181070 .00255177 4.628

55、 .0000 .40273000,Ordered Probit Marginal Effects,Multinomial Choice Among J Alternatives, Random Utility Basis Uitj = ij + i xitj + izit + ijt i = 1,N; j = 1,J(i); t = 1,T(i) Maximum Utility Assumption Individual i will Choose alternative j in choice setting t iff Uitj Uitk for all k j. Underlying assumptions Smoothness of utilities Axioms: Transitive, Complete, Monotonic,Utility Functions,The li