Determination of different types of fixed effects in three-dimensional panels*

Abstract

In this paper, we propose a jackknife method to determine the type of fixed effects in three-dimensional panel data models. We show that with probability approaching 1, the method can select the correct type of fixed effects in the presence of only weak serial or cross-sectional dependence among the error terms. In the presence of strong serial correlation, we propose a modified jackknife method and justify its selection consistency. Monte Carlo simulations demonstrate the excellent finite sample performance of our method. Applications to two datasets in macroeconomics and international trade reveal the usefulness of our method.

Keywords:

• Consistency
• cross-validation
• fixed effect
• individual effect
• jackknife
• three-dimensional panel

JEL CLASSIFICATION:

• C23
• C33
• C51
• C52
• F17
• F47

Notes

1 If we further decompose MSE into bias and variance terms in the simulations, we can also find that for M3, M4 and M7, both the bias and variance terms are important no matter whether bias-correction is corrected. For M1, M2, M5 and M6, the bias terms are relatively small, and therefore, the variance terms play a dominant role regardless of whether one corrects the bias or not.

2 The jackknife method was originally proposed by Quenouille (1956) and Tukey (1958). It can be used for different purposes, such as bias-correction, inference and model selection. The theoretical work on jackknife for model specification includes Allen (1974), Stone (1974), Geisser (1975), Wahba and Wold (1975), Li (1987), Efron (1983, 1986), Picard and Cook (1984), Andrews (1991), Shao (1993), Hansen and Racine (2012), and Lu and Su (2015), among others.

3 If any one of Models 1, 2, 5, and 6 is selected, then there is no problem to identify ϑ along with β. However, we cannot identify $\vartheta$ directly in Models 3, 4 and 7 from the usual fixed-effects estimation procedure when Model $3^{\prime },4^{\prime }$ or $7^{\prime }$ is selected. In this case, we could consider the following two-step post-selection procedure to estimate ϑ if needed: 1) In the first step, we obtain the consistent estimator ${\widehat{\gamma }}_{ij}$ of ${\tilde{\gamma }}_{ij}$ based on Model $3^{\prime },4^{\prime }$ or $7^{\prime },$ whichever is selected; 2) In the second step, we run a linear regression of ${\widehat{\gamma }}_{ij}$ on d_ij to estimate ϑ under the additional identification restriction that d_ij and γ_ij are uncorrelated. Of course, one must take into account the estimation error from the first stage when making inference on $\vartheta .$ We leave the systematic study of this issue to future research.

4 To the best of our knowledge, there is no theoretical justification for AIC and BIC in the context of determining fixed effects in 3-D panels. In fact, we are not aware of any systematic study of alternative approaches in our context.

5 For Model 1, noting that $D_1=\varnothing ,$ we implicitly define $d_{\mathit{ijt},1}=0,$ $M_{D_1}=I_{\mathit{NMT}},$ and $X_{D_m}^{\ast }={(X^{\prime }X)}^{-1}.$ In this case, $A_{\mathit{ijt},1}=u_{\mathit{ijt}},$ $B_{\mathit{ijt},1}=d_{\mathit{ijt}}^{\ast \prime }{\pi }^{\ast }-x_{\mathit{ijt}}^{\prime }{(X^{\prime }X)}^{-1}{X}^{\prime }{D}^{\ast }{\pi }^{\ast },$ and $C_{\mathit{ijt},1}=-x_{\mathit{ijt}}^{\prime }{(X^{\prime }X)}^{-1}{X}^{\prime }U.$

6 Admittedly, this rate requirement does not appear very restrictive and looks quite reasonable in many applications.