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Chapter 15

ECON 6002 Econometrics Memorial University of Newfoundland. Panel Data Models. Chapter 15. Adapted from Vera Tabakova’s notes. Chapter 15: Panel Data Models. 15.1 Grunfeld’s Investment Data 15.2 Sets of Regression Equations 15.3 Seemingly Unrelated Regressions

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Chapter 15

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  1. ECON 6002 Econometrics Memorial University of Newfoundland Panel Data Models Chapter 15 Adapted from Vera Tabakova’s notes

  2. Chapter 15: Panel Data Models • 15.1 Grunfeld’s Investment Data • 15.2 Sets of Regression Equations • 15.3 Seemingly Unrelated Regressions • 15.4 The Fixed Effects Model • 15.4 The Random Effects Model • Extensions RCM, dealing with endogeneity when we have static variables Principles of Econometrics, 3rd Edition

  3. Chapter 15: Panel Data Models The different types of panel data sets can be described as: • “long and narrow,” with “long” time dimension and “narrow”, few cross sectional units; • “short and wide,” many units observed over a short period of time; • “long and wide,” indicating that both N and T are relatively large. Principles of Econometrics, 3rd Edition

  4. 15.1 Grunfeld’s Investment Data The data consist of T = 20 years of data (1935-1954) for N = 10 large firms. Let yit = INVit and x2it = Vit and x3it = Kit Value of stock, proxy for expected profits Capital stock, proxy for desired permanent Capital stock Notice the subindices!  Principles of Econometrics, 3rd Edition

  5. 15.2 Sets of Regression Equations For simplicity we focus on only two firms keep if (i==3 | i==8) in STATA Principles of Econometrics, 3rd Edition

  6. 15.2 Sets of Regression Equations Principles of Econometrics, 3rd Edition

  7. 15.2 Sets of Regression Equations Assumption (15.5) says that the errors in both investment functions (i) have zero mean, (ii) are homoskedastic with constant variance, and (iii) are not correlated over time; autocorrelation does not exist. The two equations do have different error variances Principles of Econometrics, 3rd Edition

  8. 15.2 Sets of Regression Equations reg inv v k if i==3 scalar sse_ge = e(rss) reg inv v k if i==8 scalar sse_we = e(rss) Principles of Econometrics, 3rd Edition

  9. 15.2 Sets of Regression Equations • Let Di be a dummy variable equal to 1 for the Westinghouse observations and 0 for the General Electric observations. If the variances are the same for both firms then we can run: * Create dummy variable gen d = (i == 8) gen dv = d*v gen dk = d*k * Estimate dummy variable model reg inv d v dv k dk test d dv dk Principles of Econometrics, 3rd Edition

  10. 15.2 Sets of Regression Equations Principles of Econometrics, 3rd Edition

  11. 15.2 Sets of Regression Equations * Goldfeld-Quandt test scalar GQ = sse_ge/sse_we scalar fc95 = invFtail(17,17,.05) di "Goldfeld-Quandt Test statistic = " GQ di "F(17,17,.95) = " fc95 Goldfeld-Quandt Test statistic = 7.45338 F(17,17,.95) = 2.2718929 So we reject equality at the 5% level…=> we cannot really merge the two equations for now… Principles of Econometrics, 3rd Edition

  12. 15.3 Seemingly Unrelated Regressions This assumption says that the error terms in the two equations, at the same point in time, are correlated. This kind of correlation is called a contemporaneous correlation. Under this assumption, the joint regression would be better than the separate simple OLS regressions Principles of Econometrics, 3rd Edition

  13. 15.3 Seemingly Unrelated Regressions Econometric software includes commands for SUR (or SURE) that carry out the following steps: • Estimate the equations separately using least squares; • Use the least squares residuals from step (i) to estimate ; • Use the estimates from step (ii) to estimate the two equations jointly within a generalized least squares framework. Principles of Econometrics, 3rd Edition

  14. 15.3 Seemingly Unrelated Regressions Principles of Econometrics, 3rd Edition

  15. 15.3 Seemingly Unrelated Regressions * Open and summarize data (which is already in wide format!!!) use grunfeld2, clear summarize * SUR sureg ( inv_ge v_ge k_ge) ( inv_we v_we k_we), corr test ([inv_ge]_cons = [inv_we]_cons) ([inv_ge]_b[v_ge] = [inv_we]_b[v_we]) ([inv_ge]_b[k_ge] = [inv_we]_b[k_we]) Principles of Econometrics, 3rd Edition Slide 15-15

  16. 15.3.1 Separate or Joint Estimation? There are two situations where separate least squares estimation is just as good as the SUR technique : • when the equation errors are not contemporaneously correlated; • when the same (the “very same”) explanatory variables appear in each equation. If the explanatory variables in each equation are different, then a test to see if the correlation between the errors is significantly different from zero is of interest. Principles of Econometrics, 3rd Edition

  17. 15.3.1 Separate or Joint Estimation? In this case we have 3 parameters in each equation so: Principles of Econometrics, 3rd Edition

  18. 15.3.1 Separate or Joint Estimation? Testing for correlated errors for two equations: LM = 10.628 > 3.84 (Breusch-Pagan test of independence: chi2(1)) Hence we reject the null hypothesis of no correlation between the errors and conclude that there are potential efficiency gains from estimating the two investment equations jointly using SUR. Principles of Econometrics, 3rd Edition

  19. 15.3.1 Separate or Joint Estimation? Testing for correlated errors for three equations: Principles of Econometrics, 3rd Edition

  20. 15.3.1 Separate or Joint Estimation? Testing for correlated errors for M equations: Under the null hypothesis that there are no contemporaneous correlations, this LM statistic has a χ2-distribution with M(M–1)/2 degrees of freedom, in large samples. Principles of Econometrics, 3rd Edition

  21. 15.3.2 Testing Cross-Equation Hypotheses Most econometric software will perform an F-test and/or a Wald χ2–test; in the context of SUR equations both tests are large sample approximate tests. The F-statistic has J numerator degrees of freedom and (MTK) denominator degrees of freedom, where J is the number of hypotheses, M is the number of equations, and K is the total number of coefficients in the whole system, and T is the number of time series observations per equation. The χ2-statistic has J degrees of freedom. Principles of Econometrics, 3rd Edition

  22. 15.4 The Fixed Effects Model • SUR is OK when the panel is long and narrow, not when it is short and wide. Consider instead… We cannot consistently estimate the 3×N×T parameters in (15.9) with only NT total observations. But we can impose some more structure… We consider only one-way effects and assume common slope parameters across cross-sectional units Principles of Econometrics, 3rd Edition

  23. 15.4 The Fixed Effects Model All behavioral differences between individual firms and over time are captured by the intercept. Individual intercepts are included to “control” for these firm specific differences. Principles of Econometrics, 3rd Edition

  24. 15.4.1 A Dummy Variable Model This specification is sometimes called the least squares dummy variable model, or the fixed effects model. Principles of Econometrics, 3rd Edition

  25. 15.4.1 A Dummy Variable Model Principles of Econometrics, 3rd Edition

  26. 15.4.1 A Dummy Variable Model These N–1= 9 joint null hypotheses are tested using the usual F-test statistic. In the restricted model all the intercept parameters are equal. If we call their common value β1, then the restricted model is: So this is just OLS, the pooled model Principles of Econometrics, 3rd Edition

  27. 15.4.1 A Dummy Variable Model reg inv v k Principles of Econometrics, 3rd Edition

  28. 15.4.1 A Dummy Variable Model We reject the null hypothesis that the intercept parameters for all firms are equal. We conclude that there are differences in firm intercepts, and that the data should not be pooled into a single model with a common intercept parameter. Principles of Econometrics, 3rd Edition

  29. 15.4.2 The Fixed Effects Estimator Principles of Econometrics, 3rd Edition

  30. 15.4.2 The Fixed Effects Estimator Principles of Econometrics, 3rd Edition

  31. 15.4.2 The Fixed Effects Estimator Principles of Econometrics, 3rd Edition

  32. 15.4.2 The Fixed Effects Estimator Principles of Econometrics, 3rd Edition

  33. 15.4.2 The Fixed Effects Estimator Principles of Econometrics, 3rd Edition

  34. 15.4.2 The Fixed Effects Estimator Principles of Econometrics, 3rd Edition

  35. 15.4.2 The Fixed Effects Estimator ONE PROBLEM: Even with the trick of using the within estimator, we still implicitly (even if no longer explicitly) include N-1 dummy variables in our model (not N, since we remove the intercept), so we use up N-1 degrees of freedom. It might not be then the most efficient way to estimate the common slope ANOTHER ONE. By using deviations from the means, the procedure wipes out all the static variables, whose effects might be of interest In order to overcome this problem, we can consider the random effects/or error components model Principles of Econometrics, 3rd Edition Slide 15-35

  36. 15.5 The Random Effects Model Average intercept Usual error Randomness of the intercept Principles of Econometrics, 3rd Edition

  37. 15.5 The Random Effects Model Because the random effects regression error has two components, one for the individual and one for the regression, the random effects model is often called an error components model. a composite error Principles of Econometrics, 3rd Edition

  38. 15.5.1 Error Term Assumptions v has zero mean v has constant variance If there is no correlation between the individual effects and the error term Principles of Econometrics, 3rd Edition

  39. 15.5.1 Error Term Assumptions But now there are several correlations that can be considered. • The correlation between two individuals, i and j, at the same point in time, t. The covariance for this case is given by Principles of Econometrics, 3rd Edition

  40. 15.5.1 Error Term Assumptions • The correlation between errors on the same individual (i) at different points in time, t and s. The covariance for this case is given by Principles of Econometrics, 3rd Edition

  41. 15.5.1 Error Term Assumptions • The correlation between errors for different individuals in different time periods. The covariance for this case is Principles of Econometrics, 3rd Edition

  42. 15.5.1 Error Term Assumptions The errors are correlated over time for a given individual, but are otherwise uncorrelated This correlation does not dampen over time as in the AR1 model Principles of Econometrics, 3rd Edition

  43. 15.5.2 Testing for Random Effects This is xttest0 in Stata if H0 is not rejected you can use OLS Principles of Econometrics, 3rd Edition

  44. 15.5.3 Estimation of the Random Effects Model Is the transformation parameter Principles of Econometrics, 3rd Edition

  45. 15.5.4 An Example Using the NLS Data Is the transformation parameter Principles of Econometrics, 3rd Edition

  46. Summary for now Pooled OLS vs different intercepts: test (use a Chow type, after FE or run RE and test if the variance of the intercept component of the error is zero (xttest0)) You cannot pool onto OLS? Then… FE vs RE: test (Hausman type) Different slopes too perhaps? => use SURE or RCM and test for equality of slopes across units

  47. Summary for now Note that there is within variation versus between variation The OLS is an unweighted average of the between estimator and the within estimator The RE is a weighted average of the between estimator and the within estimator The FE is also a weighted average of the between estimator and the within estimator with zero as the weight for the between part

  48. Summary for now The RE is a weighted average of the between estimator and the within estimator The FE is also a weighted average of the between estimator and the within estimator with zero as the weight for the between part So now you see where the extra efficiency of RE comes from!...

  49. Summary for now The RE uses information from both the cross-sectional variation in the panel and the time series variation, so it mixes LR and SR effects The FE uses only information from the time series variation, so it estimates SR* effects

  50. Summary for now With a panel, we can learn about dynamic effects from a short panel, while we need a long time series on a single cross-sectional unit, to learn about dynamics from a time series data set

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