Econometric-analysis-The課件

1、Applied EconometricsWilliam GreeneDepartment of EconomicsStern School of BusinessApplied Econometrics1. The Paradigm of EconometricsEconometrics: ParadigmTheoretical foundationsMicroeconometrics and macroeconometricsBehavioral modeling: Optimization, labor supply, demand equations, etc.Statistical f

2、oundationsMathematical ElementsModel building the econometric modelMathematical elementsThe underlying truth is there one?What is bias in model estimation?Why Use This Framework?Understanding covariationUnderstanding the relationship:Estimation of quantities of interest such as elasticitiesPredictio

3、n of the outcome of interestControlling future outcomes using knowledge of relationshipsMeasurement as Observation Population Measurement TheoryCharacteristicsBehavior PatternsChoicesInference Population Measurement EconometricsCharacteristicsBehavior PatternsChoicesModel Building in EconometricsRol

4、e of the assumptionsSharpness of inferencesParameterizing the modelNonparametric analysisSemiparametric analysisParametric analysisApplication: Is there a relationship between investment and capital stock? (10 firms, 20 years)Nonparametric RegressionWhat are the assumptions?What are the conclusions?

5、Semiparametric RegressionInvestmenti,t = a + b*Capitali,t + ui,tMedianui,t | Capitali,t = 0Parametric RegressionInvestmenti,t = a + b*Capitali,t + ui,tui,t | Capitalj,s N0,2 for all i,j,s,tEstimation PlatformsThe “best use” of a body of dataThe accretion of knowledgeModel basedKernels and smoothing

6、methods (nonparametric)Moments and quantiles (semiparametric)Likelihood and M- estimators (parametric)Methodology based (?)Classical parametric and semiparametricBayesian strongly parametricClassical Inference Population Measurement EconometricsCharacteristicsBehavior PatternsChoicesImprecise infere

7、nce about the entire population sampling theory and asymptoticsBayesian Inference Population Measurement EconometricsCharacteristicsBehavior PatternsChoicesSharp, exact inference about only the sample the posterior density.Empirical ResearchIterative search for information about the structureSpecifi

8、cation searchesStepwise modeling, data mining, etc.Leamer on specification searching and significance levelsJudge, et al. on sequential inference and pretestingHendry on the encompassing principle “general to specific”Classical estimation and inferenceData StructuresObservation mechanismsPassive, no

9、nexperimentalActive, experimentalThe natural experimentData typesCross sectionPure time seriesPanel longitudinal dataFinancial dataEconometric ModelsLinear; static and dynamicDiscrete choiceCensoring and truncationStructural models and demand systemsEstimation Methods and ApplicationsLeast squares e

10、tc. OLS, GLS, LAD, quantileMaximum likelihoodFormal MLMaximum simulated likelihoodRobust and M- estimationInstrumental variables and GMMBayesian estimation Markov Chain Monte Carlo methodsTrends in EconometricsSmall structural models vs. large scale multiple equation modelsParametric vs. non- and se

11、miparametric methodsRobust methods GMM (paradigm shift?)Unit roots, cointegration and macroeconometricsNonlinear modeling and the role of softwareBehavioral and structural modeling vs. “covariance analysis” pervasiveness of the econometrics paradigmCourse Objective Develop the tools needed to read a

12、bout with understanding and to do empirical research in economics using the current body of techniques.PrerequisitesA previous course that used regressionMathematical statisticsMatrix algebraWe will do some proofs and derivations We will also examine empirical applicationsThe Course OutlineTopicsSeq

13、uenceTimingNo class on: Tuesday September 30 Thursday, October 9 Thursday, November 27 ReadingsMain text: Greene, W., Econometric Analysis, 6th Edition, Prentice Hall, 2008.A few articlesNotes on the course website: /wgreene/Econometrics /Econometrics.htmCourse ApplicationsSoftwareLIMDEP/NLOGIT prov

14、ided, supportedSAS, Stata optional, your choiceGauss, Matlab, othersLab sessions: Fridays? To be determinedProblem setsSoftwareQuestions and review as requestedTerm Paper Application (more details later)Course RequirementsProblem sets: approximately 6 (15%)Midterm, in class (30%)Final exam (40%)Term

15、 paper/project: Application of method(s) developed in class to a live data set. (15%)EnthusiasmApplied EconometricsWilliam GreeneDepartment of EconomicsStern School of BusinessApplied Econometrics2. Regression and ProjectionStatistical RelationshipObjective: Characterize the stochastic relationship

16、between a variable and a set of related variables Context: An inverse demand equation, P = + Q + Y, Y = income. Q and P are two obviously related random variables. We are interested in studying the relationship between P and Q.By relationship we mean (usually) covariation. (Cause and effect is probl

17、ematic.)Bivariate Distribution - Model for a Relationship Between Two Variables We might posit a bivariate distribution for Q and P, f(Q,P) How does variation in P arise? With variation in Q, and Random variation in its distribution. There exists a conditional distribution f(P|Q) and a conditional m

18、ean function, EP|Q. Variation in P arises because of Variation in the mean, Variation around the mean, (possibly) variation in a covariate, Z.ImplicationsRegression is the conditional meanThere is always a conditional meanIt may not equal the structure implied by a theoryWhat is the implication for

19、least squares estimation?LS always estimates regressionsLS does not necessarily estimate structuresStructures may not be estimable they may not be identified.Conditional MomentsThe conditional mean function is the regression function.P = EP|Q + (P - EP|Q) = EP|Q + E|Q = 0 = E. Proof: (The Law of ite

20、rated expectations)Variance of the conditional random variable = conditional variance, or the scedastic function.A “trivial relationship” may be written as P = h(Q) + , where the random variable =P-h(Q) has zero mean by construction. Looks like a regression “model” of sorts, but h(Q) is only EP|Q fo

21、r one specific function.An extension: Can we carry Y as a parameter in the bivariate distribution? Examine EP|Q,YSample Data (Experiment)50 Observations on P and QShowing Variation of P Around EPVariation Around EP|Q(Conditioning Reduces Variation)Means of P for Given Group Means of QAnother Conditi

22、oning VariableEP|Q,Y=1EP|Q,Y=0Conditional Mean FunctionsNo requirement that they be linear (we will discuss what we mean bylinear)No restrictions on conditional variancesProjections and RegressionsWe explore the difference between the linear projection and the conditional mean functiony = + x + wher

23、e x, E(|x) = 0 Cov(x,y) = Cov(x,) + Cov(x,x) + Cov(x,) = 0 + Var(x) + 0 So, = Cov(x,y) / Var(x) E(y) = + E(x) + E() = + E(x) + 0 = Ey - Ex.Regression and Projection Does this mean Ey|x = + x? No. This is the linear projection of y on XIt is true in every bivariate distribution, whether or not Ey|x i

24、s linear in x.y can always be written y = + x + where x, = Cov(x,y) / Var(x) etc.The conditional mean function is H(x) such thaty = H(x) + v where Ev|H(x) = 0.Data from a Bivariate PopulationThe Linear Projection Computed by Least SquaresLinear Least Squares ProjectionTrue Conditional Mean FunctionT

25、rue Data Generating MechanismApplication: Doctor VisitsGerman Individual Health Care data: N=27,236Model for number of visits to the doctor:True EV|Income = exp(1.412 - .0745*income)Linear regression: g*(Income)=3.917 - .208*incomeConditional Mean and Projection Notice the problem with the linear ap

26、proach. Negative predictions.Most of the data are in hereThis area is outside the range of the data Classical Linear Regression ModelThe model is y = f(x1,x2,xK,1,2,K) + = a multiple regression model (as opposed to multivariate). Emphasis on the “multiple” aspect of multiple regression. Important ex

27、amples: Marginal cost in a multiple output setting Separate age and education effects in an earnings equation.Form of the model Ey|x = a linear function of x. (Regressand vs. regressors)Dependent and independent variables. Independent of what? Think in terms of autonomous variation.Can y just change

28、? What causes the change? Very careful on the issue of causality. Cause vs. association. Modeling causality in econometricsModel Assumptions: GeneralitiesLinearity means linear in the parameters. Well return to this issue shortly.Identifiability. It is not possible in the context of the model for tw

29、o different sets of parameters to produce the same value of Ey|x.Conditional expected value of the deviation of an observation from the conditional mean function is zeroForm of the variance of the random variable around the conditional mean is specifiedNature of the process by which x is observed.As

30、sumptions about the specific probability distribution.Linearity of the Modelf(x1,x2,xK,1,2,K) = x11 + x22 + + xKKNotation: x11 + x22 + + xKK = x. Boldface letter indicates a column vector. “x” denotes a variable, a function of a variable, or a function of a set of variables. There are K “variables”

31、on the right hand side of the conditional mean “function.” The first “variable” is usually a constant term. (Wisdom: Models should have a constant term unless the theory says they should not.) Ey|x = 1*1 + 2*x2 + + K*xK. (1*1 = the intercept term).LinearitySimple linear model, Ey|x=xQuadratic model: Ey|x= + 1x + 2x2Loglinear model, Elny|lnx= + k lnxkkSemilog, Ey|x= + k lnxkkTranslog: Elny|lnx= + k lnxkk + (1/2) k l kl lnxk lnxl All are