Two-way fixed effects versus panel factor-augmented estimators: asymptotic comparison among pretesting procedures

Abstract

Empirical researchers may wonder whether or not a two-way fixed effects estimator (with individual and period fixed effects) is sufficiently sophisticated to isolate the influence of common shocks on the estimation of slope coefficients. If it is not, practitioners need to run the so-called panel factor augmented regression instead. There are two pretesting procedures available in the literature: the use of the estimated number of factors and the direct test of estimated factor loading coefficients. This article compares the two pretesting methods asymptotically. Under the presence of the heterogeneous factor loadings, both pretesting procedures suggest using the common correlated effects (CCE) estimator. Meanwhile, when factor loadings are homogeneous, the pretesting method utilizing the estimated number of factors always suggests more efficient estimation methods. By comparing asymptotic variances, this article finds that when the slope coefficients are homogeneous with homogeneous factor loadings, the two-way fixed effects estimation is more efficient than the CCE estimation. However, when the slope coefficients are heterogeneous with homogeneous factor loadings, the CCE estimation is, surprisingly, more efficient than the two-way fixed effects estimation. By means of Monte Carlo simulations, we verify the asymptotic claims. We demonstrate how to use the two pretesting methods through the use of an empirical example.

Keywords:

• CCE estimator
• factor number
• maximum test
• panel factor-augmented estimator
• two-way fixed effects estimator

JEL Classification::

• C33

Acknowledgments

Helpful comments on the original version were received from Alexander Chudik, Yoonseok Lee, and participants in Asian Econometric Society Group meeting and Midwest Econometric meeting. We thank valuable comments from the Editor, the Associate Editor and two anonymous referees. Also, special thanks goes to Ryan Greenaway-McGrevy for editorial help.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Note that CRT (2015b) also propose a pre-test for $F_t=F$ for all t. The procedure is exactly identical, but here we do not consider this test jointly since in practice, the null hypothesis of ${\gamma }_i=\gamma$ becomes of interest.

2 As CRT (2015b) claim, there is no reason to test the null of homogeneous factor loadings when the number of common factors is more than one. To see this, let $u_{it}={\gamma }_{1i}{F}_{1t}+{\gamma }_{2i}{F}_{2t}+{\epsilon }_{it}.$ Suppose that ${\gamma }_{1i}={\gamma }_1\mathrm{ }\text{and}\mathrm{ }{\gamma }_{2i}={\gamma }_2$ for all $i.\mathrm{ }\text{Then}\mathrm{ }{u}_{it}$ has a single factor, or $u_{it}=F_t+{\epsilon }_{it}\mathrm{ }\text{with}\mathrm{ }{F}_t={\gamma }_1{F}_{1t}+{\gamma }_2{F}_{2t}.\mathrm{ }\text{If}\mathrm{ }{\gamma }_{1i}={\gamma }_1$ but ${\gamma }_{2i}\ne {\gamma }_2,\mathrm{ }\text{then}\mathrm{ }{u}_{it}$ has two factors.

3 The Mahalanobis distance is a well-known statistic to measure the degree of outlyingness. As ${\widehat{\gamma }}_i$ departs further from its center or central location, the outlyingness approaches infinity. There are many statistical outlyingness functions available. See Zuo and Serfling (2000) for more discussions.

4 As we mentioned earlier, if the number of common factors to w_it is more than one, the factor loadings are heterogeneous.

5 Sul (2019) reports that BN’s IC${}_2$ criterion performs best among other criteria considered by Bai and Ng (2002).

6 Note that instead of the CCE estimators, one may consider IE least squares estimators suggested by Bai (2009). However, in this article we consider only the CCE estimators to avoid any issues related to weak factors. When both $x_{it}$ and u_it in (4) have weak factors, it is well known that Bai’s estimator becomes inconsistent. Meanwhile, the CCE estimator is still consistent in this case.

7 To see this, assume that ${\gamma }_i=\gamma +{ϵ}_i,$ with ${ϵ}_i=O_p(n^{-1/2}).$ Then as $n,T\to \infty$ with $T/n\to 0,$ the following condition becomes

$\frac{T}{\hspace{0.17em}\text{ln}\hspace{0.17em}n}\left||{ϵ}_i\right|{|}^2=\frac{T}{n\hspace{0.17em}\text{ln}\hspace{0.17em}n}{O}_p(1)\to 0,$

which implies the failure of Theorem 3 in CRT (2015b).

8 See Lee and Sul (2020b) for the asymptotic comparison between the MG and the conventional pooled estimations.

9 See Appendix A in Stock and Watson (1998) for more discussions about different imputing methods of dealing with specific data irregularities.