1、Analysis of Cross Section and Panel DataYan ZhangSchool of Economics, Fudan UniversityCCER, Fudan UniversityIntroductory EconometricsA Modern ApproachYan ZhangSchool of Economics, Fudan UniversityCCER, Fudan UniversityAnalysis of Cross Section and Panel DataPart 3. Some Advanced TopicsChap 13. Pooli

2、ng Cross Sections across TimevData StructurePooled Cross Section; Panel DatavIndependently Pooled Cross Sectionthey consist of independently sampled observations.與單一隨機樣本的差別：在不同時點對總體抽樣可能導致觀測點不與單一隨機樣本的差別：在不同時點對總體抽樣可能導致觀測點不是同分布的（是同分布的（not identically distributed.）Different intercept and slopesPolicy an

3、alysisvPanel DataThe same unitswe cannot assume that the observations of longitudinal data are independently distributed across time.Special models and methodsDifferencing (remove time-constant, unobserved attributes of the units.)Pooled Cross SectionsPooling cross sections from different years;Effe

4、ctively analyzing the effects of a new govt. policy;Similar to a standard cross section, except that we often need to account for secular differences in the variables across the time.Panel or Longitudinal DataThe same cross sectional members;To control certain unobserved Characteristic of cross sect

5、ions;To study the importance of lags in behavior or the result of decision making13.1 Pooling Independent Cross Sections across TimevIncrease the sample sizevDummy Variablesthe population may have different distributions in different time periodsdifferent intercept and slopesvYear dummy: including d

6、ummy variables for all but one year, where the earliest year in the sample is usually chosen as the base year.The pattern of coef. on the year dummies The change of the coef. of the key variable over timepolicy analysisExample 13.1 Has the pattern of womens fertility Changed?vFactors on Womens Ferti

7、lity over Time?age; education; religion; regiondependent variable: fertility rates; different period vData: FERTIL1.RAW, which is similar to that used by Sander (1994), comes from the National Opinion Research Centers General Social Survey for the even years from 1972 to 1984vInterpretationsbase yea

8、r: 1972education: .128(4)=.512.turning point of agevheteroskedasticity of error term over time? B-P test; WLSveduc? interaction effects (P. 13.7, IV) Has the effect of education on fertility rates changed over time?The Chow Test for Structural Change Across TimevOne form of the test obtains the sum

9、of squared residuals from the pooled estimation as the restricted SSR. The unrestricted SSR is the sum of the SSRs for the two separately estimated time periods. vAnother way: interacting each variable with a year dummy for one of the two years and testing for joint significance of the year dummy an

10、d all of the interaction terms.vUsually, after an allowance for intercept difference, certain slope coefficients are tested for constancy by interacting the variable of interest with year dummies.E.g. 13.2 Changes in the return to education and the gender wage gapvEconometric Model:vnominal vs. real

11、 valueProvided the dollar amounts appear in logarithmic form and dummy variables are used for all time periods (except, of course, the base period), the use of aggregate price deflators will only affect the intercepts; none of the slope estimates will change.vChow Test:What happens if we interact al

12、l independent variables with y85 in equation (13.2)?13.2 Policy Analysis with Pooled Cross Sectionsvnatural experiments: occurs when some exogenous eventoften a change in government policychanges the environment in which individuals, families, firms, or cities operate. control group: not affected by

13、 the policychangetreatment group: thought to be affected by the policy change.vMethods:to control for systematic differences between the control and treatment groups, we need two years of data, one before the policy change and one after the change.the difference-in-differences estimator: Example 13.

14、4 Effects of Worker Compensation Laws on DurationvProblem: its effects on durationinfluenced: high-income workercontrol group (low) and treatment group (high)vMeyer, Viscusi and Durbin (1995)INJURY.RAWlog(durat); fchnge; highearn; age; gender; marital status; industry; type of injury13.3 Two-period

15、Panel Data AnalysisvTwo types of unobserved factors affecting the dependent v. in the panel data：keep constant: unobserved effect (fixed effect)vary over time: idiosyncratic error (time-varying error)vEstimationpooled cross sections; drawback:Heterogeneity bias: Therefore, even if we assume that the

16、 idiosyncratic error uit is uncorrelated with xit, pooled OLS is biased and inconsistent if ai and xit are correlated.In most applications, the main reason for collecting panel data is to allow for the unobserved effect, ai, to be correlated with the explanatory v.-s.first-differenced equation First

17、-Differenced Equationv vKey assumptions:strict exogeneity: dui is uncorrelated with dxi.first-differenced estimator dxi must have some variation across i.(13.17) satisfies the homoskedasticity assumption.E.g. 13.5 Sleeping vs. WorkingvSLP75_81.RAWv 13.5 Differencing with More than Two Time periodsvD

18、ata Structure (fixed effect & time-varying error)vKey Assumption (strict exogeneity):That is, the explanatory variables are strictly exogenous after we take out the unobserved effect, ai.vCases when strict exogeneity be false:If xitj is a lagged dependent variable. If we have omitted an importan

19、t time-varying variableMeasurement error in one or more explanatory variablesDifferencingvDifferencing:vWhen T is small relative to N, we should include a dummy variable for each time period to account for secular changes that are not being modeled.vThe total number of observations is N(T-1) if the

20、data sets are balanced. The differences for t=1 should be missing values for all N cross-sectional observations.Serial Correlation in the First-Differenced EquationvOnly when uit follows a random walk will uit be serially uncorrelated.vIf we assume the uit are serially uncorrelated with constant var

21、iance, then the correlation between uit and ui,t1 can be shown to be 0.5. vIf uit follows a stable AR(1) model, then uit will be serially correlated. Test Serial Correlation in the First-Differenced EquationvMethods: (AR(1)vZero Assumption:vSteps:First, we estimate (13.31) by pooled OLS and obtain t

22、he residuals,Then, we run the regression again with ri,t1 as an additional explanatory variable.The coefficient on ri,t1 is an estimate of , and so we can use the usual t statistic on ri,t1 to test H0: 0.Correct for the AR(1) Serial CorrelationvUnfortunately, standard packages that perform AR(1) cor

23、rections for time series regressions will not work. Standard Cochrane-Orcutt or Prais-Winsten methods will treat the observations as if they followed an AR(1) process across i and t; this makes no sense, as we are assuming the observations are independent across i.vCorrections to the OLS standard er

24、rors that allow arbitrary forms of serial correlation (and heteroskedasticity) can be computed when N is large (and N should be notably larger than T ). vIf there is no serial correlation in the errors, the usual methods for dealing with heteroskedasticity are valid.Chap 14 Advanced Panel Data Metho

25、dsvTwo Methods for Estimating Unobserved Effects Panel Data Model:Fixed Effects EstimationRandom Effects Estimation14.1 Fixed Effects EstimationvAn alternative Methods to eliminate the fixed effectsFixed Effects Transformation (Within Transformation): for each i, average this equation over time:Subs

26、tracting:vFixed Effects Estimator (Within Estimator)vUnbiasedness: Under a strict exogeneity assumption on the explanatory variables, the fixed effects estimator is unbiased: roughly, the idiosyncratic error uit should be uncorrelated with each explanatory variable across all time periods.vThe other

27、 assumptions needed for a straight OLS analysis to be valid are that the errors uit are homoskedastic and serially uncorrelated (across t)vthe degrees of freedom for the fixed effects estimator: df = NTNk= N(T1)k.vThe goodness-of-fit: The R-squared obtained from estimating (14.5) is interpreted as t

28、he amount of time variation in the yit that is explained by the time variation in the explanatory variables. Other ways of computing R-squared are possible, one of which we discuss later.Notes on some explanatory v.-s in Fixed Effects EstimationvWe cannot include variables such as gender or whether

29、a city is located near a river as any explanatory variable that is constant over time for all i gets swept away by the fixed effects transformationvAlthough time-constant variables cannot be included by themselves in a fixed effects model, they can be interacted with variables that change over time

30、and, in particular, with year dummy variables.vWhen we include a full set of year dummiesthat is, year dummies for all years but the firstwe cannot estimate the effect of any variable whose change across time is constant.Example 14.2 The Return to Education over TimevFixed effects The Dummy Variable

31、 Regression: A traditional view of the fixed effects model is to assume that the unobserved effect, ai, is a parameter to be estimated for each i.The way we estimate an intercept for each i is to put in a dummy variable for each cross-sectional observation, along with the explanatory variables (and

32、probably dummy variables for each time period).vThe dummy variable regression gives exactly the same estimates of the j that we would obtain from the regression on time-demeaned data, and the standard errors and other major statistics are identical. Therefore, the fixed effects estimator can be obta

33、ined by the dummy variable regression.vThe R-squared from the dummy variable regression is usually rather high.vWhen T=2, FE and FD estimates and all test statistics are identicalvWhen T2, the FE and FD estimators are not the same.For large N and small T, the choice between FE and FD hinges on the r

34、elative efficiency of the estimators, and this is determined by the serial correlation in the idiosyncratic errors, uit.When T is large, and especially when N is not very large (for example, N=20 and T=30), we must exercise caution in using the fixed effects estimator. For large N and small T: FE or

35、 FD?vFor large N and small T, the choice between FE and FD hinges on the relative efficiency of the estimators, and this is determined by the serial correlation in the idiosyncratic errors, uit.When the uit are serially uncorrelated, fixed effects is more efficient than first differencing (and the S

36、.E reported from FE are valid).If uit follows a random walkwhich means that there is very substantial, positive serial correlationthen the difference is serially uncorrelated, and first differencing is better.In many cases, the uit exhibit some positive serial correlation, but perhaps not as much as

37、 a random walk. Then, we cannot easily compare the efficiency of the FE and FD estimators.We can test whether the differenced errors, , are serially uncorrelated as section 13.3 showed. If this seems to be the case, FD can be used. If there is substantial negative serial correlation in the uit , FE

38、is probably better. It is often a good idea to try both: if the results are not sensitive, so much the better.For large T: FE or FD?vWhen T is large, and especially when N is not very large (for example, N=20 and T=30), we must exercise caution in using the fixed effects estimator. they are extremel

39、y sensitive to violations of the assumptions when N is small and T is large. In the case of unit root, FD is better.fixed effects turns out to be less sensitive to violation of the strict exogeneity assumption, especially with large T. Some authors even recommend estimating fixed effects models with

40、 lagged dependent variables (which clearly violates Assumption FE.3 in the chapter appendix). When the processes are weakly dependent over time and T is large, the bias in the fixed effects estimator can be small.vUnbalanced Panels: have missing years for at least some cross-sectional units in the s

41、ample.vIf Ti is the number of time periods for cross-sectional unit i, we simply use these Ti observations in doing the time-demeaning.Any regression package that does fixed effects makes the appropriate adjustment for this loss of degree of freedom.vIf the reason a firm leaves the sample (called at

42、trition) is correlated with the idiosyncratic errorthose unobserved factors that change over time and affect profitsthen the resulting sample section problem (see Chapter 9) can cause biased estimators. Fortunately, FE means that, with the initial sampling, some units are more likely to drop out of

43、the survey, and this is captured by ai.14.2 Random Effects EstimationvRandom Effects Model: If the unobserved effect ai is uncorrelated with each explanatory variable,vThe usual pooled OLS can give consistent estimators of , but as its standard errors ignore the positive serial correlation in the co

44、mposite error term, they will be incorrect, as will the usual test statistics.vSolution: use GLS to solve the serial correlation problemRandom Effects Estimation: GLS transformationvGLS transformation to eliminate the serial correlation:quasi-demeaned datavEstimation of :where a is a consistent esti

45、mator of . These estimators can be based on the pooled OLS or fixed effects residuals.vRandom Effects Estimator: The feasible GLS estimator that uses in place ofRE, FE and PLSvPooled OLS:vRandom Effects Estimator:vFixed Effects Estimator:vThe transformation in (14.11) allows for explanatory variables that are constant over time, and this is one advantage of