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Econometric Analysis of Panel Data

Econometric Analysis of Panel Data

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Econometric Analysis of Panel Data

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  1. Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business

  2. Benefits of Panel Data • Time and individual variation in behavior unobservable in cross sections or aggregate time series • Observable and unobservable individual heterogeneity • Rich hierarchical structures • More complicated models • Features that cannot be modeled with only cross section or aggregate time series data alone • Dynamics in economic behavior

  3. Short Term Agenda for Simple Effects Models • Models with individual effects • Interpretation of models • Computation (practice) and estimation (theory) • Extensions • Nonstandard panels: Rotating, Pseudo-, Nested • Generalizing the regression model • Alternative estimators • Methods • Least squares: OLS, GLS, FGLS • MLE and Maximum Simulated Likelihood

  4. Fixed and Random Effects • Unobserved individual effects in regression: E[yit | xit, ci] Notation: • Linear specification: Fixed Effects: E[ci | Xi ] = g(Xi). Cov[xit,ci] ≠0 effects are correlated with included variables. • Random Effects: E[ci | Xi ] = μ; effects are uncorrelated with included variables. If Xi contains a constant term, μ=0 WLOG. Common: Cov[xit,ci] =0, but E[ci | Xi ] = μ is needed for the full model

  5. Convenient Notation • Fixed Effects – the ‘dummy variable model’ • Random Effects – the ‘error components model’ Individual specific constant terms. Compound (“composed”) disturbance

  6. Balanced and Unbalanced Panels • Distinction: Balanced vs. Unbalanced Panels • A notation to help with mechanics zi,t, i = 1,…,N; t = 1,…,Ti • The role of the assumption • Mathematical and notational convenience: • Balanced, n=NT • Unbalanced: • Is the fixed Ti assumption ever necessary? Almost never. (Baltagi chapter 9 is about algebra, not different models!) • Is unbalancedness due to nonrandomattrition from an otherwise balanced panel? This will require special considerations.

  7. An Unbalanced Panel: RWM’s GSOEP Data on Health Care N = 7,293 Households Some households exited then returned

  8. Exogeneity • Contemporaneous exogeneity • E[εit|xit,ci]=0  Not sufficient for regression • Doesn’t imply how to estimate β • Strict exogeneity – the most common assumption • E[εit|xi1, xi2,…,xiT,ci]=0 • Can use first difference or fixed effects • Cannot hold if xit contains lagged values of yit • Sequential exogeneity? • E[εit|xi1, xi2,…,xit,ci] = 0 • These assumptions are not testable. They are part of the model.

  9. Assumptions for Asymptotics • Convergence of moments involving cross section Xi. • N increasing, T or Ti assumed fixed. • “Fixed T asymptotics” (see text, p. 175) • Time series characteristics are not relevant (may be nonstationary) • If T is also growing, need to treat as multivariate time series. • Ranks of matrices. X must have full column rank. (Xi may not, if Ti < K.) • Strict exogeneity and dynamics. If xit contains yi,t-1 then xit cannot be strictly exogenous. Xit will be correlated with the unobservables in period t-1. (To be revisited later.) • Empirical characteristics of microeconomic data

  10. Estimating β • β is the partial effect of interest • Can it be estimated (consistently) in the presence of (unmeasured) ci? • Does pooled least squares “work?” • Strategies for “controlling for ci” using the sample data • Using a proxy variable.

  11. The Pooled Regression • Presence of omitted effects • Potential bias/inconsistency of OLS – depends on ‘fixed’ or ‘random’

  12. OLS with Individual Effects

  13. Mundlak’s Estimator Mundlak, Y., “On the Pooling of Time Series and Cross Section Data, Econometrica, 46, 1978, pp. 69-85.

  14. Chamberlain’s (1982) Approach Use a linear projection, not necessarily the conditional mean. This “regression” can be computed T times, using one year at a time. How would we reconcile the multiple estimators of each parameter?.

  15. Chamberlain’s (1982) Approach

  16. Proxy Variables • Proxies for unobserved effects: e.g., Test score for unobserved ability • Interest is in δ(xit,ci)=E[yit|xit,ci]/xit • Since ci is unobserved, we seek APE = Ec[δ(xit,ci)] • Proxy has two characteristics • Ignorable in the model: E[yit|xit,zi,ci] = E[yit|xit,ci] • ‘Explains’ ci in that E[ci|zi,xit] = E[ci|zi]. In the presence of zi, xit does not further ‘explain ci.’ • Then, Ec[δ(xit,ci)] = Ez{E[yit|xit,zi]/xit} • Proof: See Wooldridge, pp. 23-24. • Loose ends: • Where do you get the proxy? • What is E[yit|xit,zi]? Use the linear projection and hope for the best.

  17. Estimating the Sampling Variance of b • s2(X́X)-1? • Correlation across observations • Heteroscedasticity • A “robust” covariance matrix • Robust estimation (in general) • The White estimator • A Robust estimator for OLS.

  18. A ‘Cluster’ Estimator

  19. Cluster Estimator (cont.)

  20. Cornwell and Rupert Data Cornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 YearsVariables in the file are EXP = work experienceWKS = weeks workedOCC = occupation, 1 if blue collar, IND = 1 if manufacturing industrySOUTH = 1 if resides in southSMSA = 1 if resides in a city (SMSA)MS = 1 if marriedFEM = 1 if femaleUNION = 1 if wage set by union contractED = years of educationLWAGE = log of wage = dependent variable in regressions These data were analyzed in Cornwell, C. and Rupert, P., "Efficient Estimation with Panel Data: An Empirical Comparison of Instrumental Variable Estimators," Journal of Applied Econometrics, 3, 1988, pp. 149-155.  See Baltagi, page 122 for further analysis.  The data were downloaded from the website for Baltagi's text.

  21. Application: Cornell and Rupert

  22. Bootstrapping Some assumptions that underlie it - the sampling mechanism Method: 1. Estimate using full sample: --> b 2. Repeat R times: Draw n observations from the n, with replacement Estimate  with b(r). 3. Estimate variance with V = (1/R)r [b(r) - b][b(r) - b]’

  23. Bootstrap Application matr;bboot=init(7,21,0.)$ Store results here name;x=one,occ,…,exp$ Define X regr;lhs=lwage;rhs=x$ Compute b calc;i=0$ Counter Proc Define procedure regr;lhs=lwage;rhs=x;quietly$ … Regression matr;{i=i+1};bboot(*,i)=b$... Store b(r) Endproc Ends procedure exec;n=20;bootstrap=b$ 20 bootstrap reps matr;list;bboot' $ Display results

  24. Results of Bootstrap Procedure

  25. Bootstrap Replications Full sample result Bootstrapped sample results

  26. Bootstrap variance for a panel data estimator • Panel Bootstrap = Block Bootstrap • Data set is N groups of size Ti • Bootstrap sample is N groups of size Ti drawn with replacement.

  27. Using First Differences Eliminating the heterogeneity

  28. OLS with First Differences With strict exogeneity of (Xi,ci), OLS regression of Δyit on Δxit is unbiased and consistent but inefficient. GLS is unpleasantly complicated. In order to compute a first step estimator of σε2 we would use fixed effects. We should just stop there. Or, use OLS in first differences and use Newey-West with one lag.

  29. Two Periods With two periods and strict exogeneity, This is a classical regression model. If there are no regressors,

  30. Difference-in-Differences Model With two periods and strict exogeneity of D and T, This is a linear regression model. If there are no regressors,

  31. Difference in Differences

  32. www.oft.gov.uk/shared_oft/reports/Evaluating-OFTs-work/oft1416.pdfwww.oft.gov.uk/shared_oft/reports/Evaluating-OFTs-work/oft1416.pdf

  33. Treatment Schools Control Schools Treatment is not voluntary

  34. Treatment (Intervention) Effect = 1 + 2 if SS school

  35. In order to test robustness two versions of the fixed effects model were run. The first is Ordinary Least Squares, and the second is heteroscedasticity and auto-correlation robust (HAC) standard errors in order to check for heteroscedasticity and autocorrelation.

  36. D-in-D Model: Natural Experiment With two periods and strict exogeneity, This is a classical regression model. If there are no regressors,

  37. D-i-D • Card and Krueger: “Minimum Wages and Employment: A Case Study of the Fast Food Industry in New Jersey and Pennsylvania,” AER, 84(4), 1994, 772-793. • Pennsylvania vs. New Jersey • 1991, NJ raises minimum wage • Compare change in employment PA after the change to change in employment in NJ after the change. • Differences cancel out other things specific to the state that would explain change in employment.

  38. A Tale of Two Cities • A sharp change in policy can constitute a natural experiment • The Mariel boatlift from Cuba to Miami (May-September, 1980) increased the Miami labor force by 7%. Did it reduce wages or employment of non-immigrants? • Compare Miami to Los Angeles, a comparable (assumed) city. • Card, David, “The Impact of the Mariel Boatlift on the Miami Labor Market,” Industrial and Labor Relations Review, 43, 1990, pp. 245-257.

  39. Difference in Differences

  40. Applying the Model • c = M for Miami, L for Los Angeles • Immigration occurs in Miami, not Los Angeles • T = 1979, 1981 (pre- and post-) • Sample moment equations: E[Yi|c,t,T] • E[Yi|M,79] = β79 + γM • E[Yi|M,81] = β81 + γM + δ • E[Yi|L,79] = β79 + γL • E[Yi|M,79] = β81 + γL • It is assumed that unemployment growth in the two cities would be the same if there were no immigration.

  41. Implications for Differences • Neither city exposed to migration • E[Yi,0|M,81] - E[Yi,0|M,79] = [β81 + γM ] – [β79 + γM] ( Miami) • E[Yi,0|L,81] - E[Yi,0|L,79] = [β81 + γL ] – [β79 + γL] (LA) • Both cities exposed to migration • E[Yi,1|M,81] - E[Yi,1|M,79] = [β81 + γM ] – [β79 + γM] + δ (Miami) • E[Yi,1|L,81] - E[Yi,1|L,79] = [β81 + γL ] – [β79 + γL] + δ (LA) • One city (Miami) exposed to migration: The difference in differences is. • Miami change - Los Angeles change • {E[Yi,1|M,81] - E[Yi,1|M,79]} – {E[Yi,0|L,81] - E[Yi,0|L,79]} = δ (Miami)

  42. The Tale 1979 1980 1981 1982 1983 1984 1985 In 79, Miami unemployment is 2.0% lower In 80, Miami unemployment is 7.1% lower From 79 to 80, Miami gets 5.1% better In 81, Miami unemployment is 3.0% lower In 82, Miami unemployment is 3.3% higher From 81 to 82, Miami gets 6.3% worse