Which panel data estimator should I use?

Abstract

This study employs Monte Carlo experiments to evaluate the performances of a number of common panel data estimators when serial correlation and cross-sectional dependence are both present. It focuses on fixed effects models with less than 100 cross-sectional units and between 10 and 25 time periods (such as are commonly employed in empirical growth studies). Estimator performance is compared on two dimensions: (i) root mean square error and (ii) accuracy of estimated confidence intervals. An innovation of our study is that our simulated panel data sets are designed to look like ‘real-world’ panel data. We find large differences in the performances of the respective estimators. Further, estimators that perform well on efficiency grounds may perform poorly when estimating confidence intervals and vice versa. Our experimental results form the basis for a set of estimator recommendations. These are applied to ‘out of sample’ simulated panel data sets and found to perform well.

Acknowledgements

This article has benefited from comments by Bill Greene, Peter Kennedy, Peter Phillips, Gareth Thomas (from EViews), David Drukker (from Stata) and seminar participants at the universities of Auckland, Canterbury, Otago, Victoria and Waikato in New Zealand. Culpability for remaining errors are ours alone.

Notes

¹ This quote is taken from Peter Kennedy's A Guide to Econometrics, 5th edn (2003, p. 405).

² As a result, the PCSE estimator has been widely adopted. A recent Web of Science search found approximately 700 citations of Beck and Katz (1995). Applications of this estimator can be found in Bitzer and Stephan (2007), Mosca (2007), Lago-Penas (2006) and Marques (2005).

³ This is consistent with the ‘shrinkage principle,’ well-known in the forecasting literature, that imposing incorrect restrictions on a model can improve forecast performance (Diebold, 2004, p. 45).

⁴ We follow other Monte Carlo studies in equating efficiency with MSE (cf. Beck and Katz, 1995), but recognize that FGLS is biased in small samples.

⁵ In its most general form, the Parks model assumes groupwise, first-order serial correlation. In contrast, our experiments model the DGP with an AR(1) parameter, ρ, that is the same across groups. We do this for two reasons. First, Beck and Katz (1995) recommend that researchers should impose a common AR(1) parameter when estimating the Parks model, and we wanted our full Parks model estimators to be correctly specified. Second, having a single AR(1) parameter facilitates characterization and comparison of serial correlation within and across the simulated data sets.

⁶ Details are provided in an Appendix available from the authors.

⁷ We do not consider dynamic panel data models (cf. Roodman, 2006) as these entail additional issues not primarily related to the structure of the error variance–covariance matrix.

⁸ We thank Peter Phillips for recommending the use of ‘partial FGLS’ to distinguish these estimators from conventional FGLS.

⁹ Specifically, Greene (2003, p. 333) recommends using a robust estimator that incorporates cross-sectional dependence.

¹⁰ Estimators 9 through 11 can be thought of as the (partial) FGLS analogues for the OLS estimators 4, 3 and 2, respectively.

¹¹ For example, Stata uses linear methods for estimating ρ, while EViews employs a nonlinear procedure.

¹² Of particular note is the way that Stata calculates confidence intervals when cluster( ) is chosen. Let be the estimated coefficient covariance matrix unadjusted for degrees of freedom. The cluster option makes the following d.f. adjustment,and NT is the total number of observations, K is the number of estimated coefficients and C is the number of ‘clusters’ (either N or T in our notation). Further, in calculating confidence intervals (and p-values), Stata uses the t critical value with C − 1 degrees of freedom. Contrast this with the conventional approach of using NT − K degrees of freedom. This difference can have a substantial impact on the width of the confidence interval. Note that EViews follows the latter when estimating its analog of cluster( ) SEs.

¹³ The main difference between the two residual-producing regression specifications is that only one included time period fixed effects (both included group fixed effects). The inclusion of time period fixed effects substantially reduces (but does not eliminate) cross-sectional dependence (cf. Roodman, 2006).

¹⁴ As these values are experiment-specific, the associated range on the level of individual data sets is even larger.

¹⁵ Driscoll and Kraay (1998) find that coverage rates for the SUR estimator decline as the degree of cross-sectional dependence increases (Table 1, p. 554).

¹⁶ When N increases by 1, the number of unique parameters in the Parks error variance–covariance matrix increases by N + 1, while the number of observations increases by T, and recall that T ≥ N.

¹⁷ The p-value for the coefficient of N when it is included as an additional variable in a specification that already includes (T/N) is 0.344. The p-value for the coefficient of T when it is added to a specification that already includes (T/N) is 0.192.

¹⁸ Asymptotic theory is not helpful in identifying key determinants in this case because the respective estimators incorrectly/incompletely model the error variance–covariance matrix.

¹⁹ While it is not distinguished in the legend, FGLS (Parks) is identifiable by its wild swings and extreme values.

²⁰ As in the previous case, we find no evidence that this threshold value is significantly affected by either N or T.