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

 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

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.

The data consist of T = 20 years of data ( ) for N = 10 large firms. Let y it = INV it and x 2it = V it and x 3it = K it Notice the subindices! Value of stock, proxy for expected profits Capital stock, proxy for desired permanent Capital stock

For simplicity we focus on only two firms keep if (i==3 | i==8) in STATA

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

reg inv v k if i==3 scalar sse_ge = e(rss) reg inv v k if i==8 scalar sse_we = e(rss)

 Let D i 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

* 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 = F(17,17,.95) = So we reject equality at the 5% level…=> we cannot really merge the two equations for now…

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

Econometric software includes commands for SUR (or SURE) that carry out the following steps: (i) Estimate the equations separately using least squares; (ii) Use the least squares residuals from step (i) to estimate ; (iii) Use the estimates from step (ii) to estimate the two equations jointly within a generalized least squares framework.

* 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])

There are two situations where separate least squares estimation is just as good as the SUR technique : (i) when the equation errors are not contemporaneously correlated; (ii) 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.

In this case we have 3 parameters in each equation so:

Testing for correlated errors for two equations: LM = > 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.

Testing for correlated errors for three equations:

Testing for correlated errors for M equations: Under the null hypothesis that there are no contemporaneous correlations, this LM statistic has a χ 2 -distribution with M(M–1)/2 degrees of freedom, in large samples.

Most econometric software will perform an F-test and/or a Wald χ 2 –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 χ 2 -statistic has J degrees of freedom.

 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

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.

This specification is sometimes called the least squares dummy variable model, or the fixed effects 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 β 1, then the restricted model is: So this is just OLS, the pooled model

reg inv v k

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.

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

Randomness of the intercept Usual error Average intercept

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

v has zero mean v has constant variance If there is no correlation between the individual effects and the error term

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

 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

 The correlation between errors for different individuals in different time periods. The covariance for this case is

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

This is xttest0 in Stata if H0 is not rejected you can use OLS

Is the transformation parameter

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

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

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!...

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

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

If the random error is correlated with any of the right-hand side explanatory variables in a random effects model then the least squares and GLS estimators of the parameters are biased and inconsistent. This bias creeps in through the between variation, of course, so the FE model will avoid it

We expect to find because Hausman proved that

The test statistic to the coefficient of SOUTH is: Using the standard 5% large sample critical value of 1.96, we reject the hypothesis that the estimators yield identical results. Our conclusion is that the random effects estimator is inconsistent, and we should use the fixed effects estimator, or we should attempt to improve the model specification.

The Hausman test assumes that the RE estimator used in the comparison is fully efficient, which requires that the unobserved effect and the idiosyncratic error are both i.i.d. (Cameron & Trivedi MMA page 719)  often not the case => t he hausman command yields incorrect statistic Example: If the error terms are cluster, (e.g. due to autocorrelation across time for an individual, then the RE estimator is not efficient) Solutions:  do a panel bootstrap of the Hausman test  or use the Wooldridge (2002) robust version of Hausman test.

 Test for gamma =0 in:  To run in Stata, generate the RE differences and the mean differences Principles of Econometrics, 3rd Edition

 To run in Stata, generate the RE differences and the mean differences manually  See an example here: pages of Cameron&Trivedi’s MUS book Principles of Econometrics, 3rd Edition

If the random error is correlated with any of the right- hand side explanatory variables in a random effects model then the least squares and GLS estimators of the parameters are biased and inconsistent. Then we would have to use the FE model But with FE we lose the static variables? Solutions? HT, AM, BMS, instrumental variables models could help

We can generalise the random effects idea and allow for different slopes too: Random Coefficients Model Again, the now it is the slope parameters that differ, but as in RE model, they are drawn from a common distribution The RCM in a way is to the RE model what the SURE model is to the FE model Further issues

Unit root tests and Cointegration in panels Dynamics in panels Further issues

 Of course it is not necessary that one of the dimensions of the panel is time as suchExample: i are students and t is for each quiz they take  Of course we could have a one-way effect model on the time dimension instead  Or a two-way model  Or a three way model! But things get a bit more complicated there… Further issues

 Another way to have more fun with panel data is to consider dependent variables that are not continuous  Logit, probit, count data can be considered  STATA has commands for these  Based on maximum likelihood and other estimation techniques we have not yet considered Further issues

Another extension is to consider mixed linear models (Cameron&Trivedi MUS page 305) Stata’s xtmixed fits linear mixed models. From Stata;s help: Mixed models contain both fixed effects and random effects. The fixed effects are analogous to standard regression coefficients and are estimated directly Further issues

 The random effects are not directly estimated but are summarized according to their estimated variances and covariances  Although random effects are not directly estimated, you can form best linear unbiased predictions (BLUPs) of them (and standard errors) by using predict after xtmixed  Random effects may take the form of either random intercepts or random coefficients, and the grouping structure of the data may consist of multiple levels of nested groups.  Mixed models are also known as multilevel models and hierarchical linear models  Quite rare in the econometric literature Undergraduate Econometrics, 3 rd Edition Principles of Econometrics, 3rd Edition

 Some particular specifications of the mixed linear models result in more standard models  OLS, RE are special cases of mixed linear models  Another one is known as the Random Coefficients Model  RCM also allows groupwise heteroskedasticity rather than imposing homoskedasticity like its mixed linear model equivalent Undergraduate Econometrics, 3 rd Edition Principles of Econometrics, 3rd Edition

Example in Cameron & Trivedi MUS page 310 Undergraduate Econometrics, 3 rd Edition Principles of Econometrics, 3rd Edition

 Data (available through Cameron & Trivedi’s MUS textbook ancillary files) :  mus08psidextract.dta  (PSID wage data from Baltagi and Khanti- Akom (1990))  I cut for you the first 994 observations  mus08psidextract994 Undergraduate Econometrics, 3 rd Edition Principles of Econometrics, 3rd Edition

DO we have a name for this test? Econometrics, 3 rd Edition

 You can understand the use of the FE model as a solution to omitted variable bias  If the unmeasured variables left in the error model are not correlated with the ones in the model, we would not have a bias in OLS, so we can safely use RE  If the unmeasured variables left in the error model are correlated with the ones in the model, we would have a bias in OLS, so we cannot use RE, we should not leave them out and we should use FE, which bundles them together in each cross-sectional dummy Further issues

 Another criterion to choose between FE and RE  If the panel includes all the relevant cross-sectional units, use FE, if only a random sample from a population, RE is more appropriate (as long as it is valid) Further issues

Wooldridge’s book on panel data Baltagi’s book on panel data Greene’s coverage is also good Readings

Slide Principles of Econometrics, 3rd Edition  Balanced panel  Breusch-Pagan test  Cluster corrected standard errors  Contemporaneous correlation  Endogeneity  Error components model  Fixed effects estimator  Fixed effects model  Hausman test  Heterogeneity  Least squares dummy variable model  LM test  Panel corrected standard errors  Pooled panel data regression  Pooled regression  Random effects estimator  Random effects model  Seemingly unrelated regressions  Unbalanced panel

Slide Principles of Econometrics, 3rd Edition

Slide (15A.1) (15A.2) (15A.3)

Principles of Econometrics, 3rd Edition Slide (15A.4) (15A.5)

Principles of Econometrics, 3rd Edition Slide (15A.6) (15A.7)