Estimation of fixed effects dynamic panel data models: linear differencing or conditional expectation

Abstract

This note discusses the pros and cons of using the conditional mean approach of Mundlak and Chamberlain and the linear difference approach to deal with the incidental parameters issue in estimating fixed effects dynamic panel data models. The importance of the data generating process of the explanatory variables and the proper treatment of initial values for either approach to get asymptotically unbiased estimators are demonstrated both analytically and through Monte Carlo studies.

Keywords:

• Conditional mean approach
• fixed effects dynamic panel models
• incidental parameter
• initial value distribution
• linear difference approach

JEL Codes:

• C01
• C13
• C23

Acknowledgments

The author also wishes to thank two referees for helpful comments and Yimeng Xie for computational assistance.

Notes

1 Monte Carlo studies conducted by Hsiao and Zhou (2018) show that, although, using the ith individual’s time series mean, ${\overline{x}}_i=\frac{1}{T}{\sum }_{t=1}^T{x}_{it}$ is more restrictive than using ${\underset{˜}{x}}_i,$ using ${\overline{x}}_i$ in lieu of ${\underset{˜}{x}}_i$ tends to perform better if N and T are of the same magnitude.

2 The estimator (3.7)(3.7) $$\left(\begin{array}{c}\widehat{\rho }\\ \widehat{\beta }\\ \widehat{\underset{˜}{a}}\end{array}\right)={\left[{\displaystyle \sum _{i=1}^N\left(\begin{array}{c}{\underset{˜}{y}}_{i,-1}^{\prime }\\ X_i\prime \\ {(\underset{˜}{e}{\tilde{x}}_i\prime )}^{\prime }\end{array}\right)}{V}^{-1}\left(\begin{array}{c}{\underset{˜}{y}}_{i,-1}^{\prime }\\ X_i\prime \\ {(\underset{˜}{e}{\tilde{x}}_i\prime )}^{\prime }\end{array}\right)\prime \right]}^{-1}\left[{\displaystyle \sum _{i=1}^N\left(\begin{array}{c}{\underset{˜}{y}}_{i,-1}^{\prime }\\ X_i\prime \\ {(\underset{˜}{e}{\tilde{x}}_i\prime )}^{\prime }\end{array}\right)}{V}^{-1}\underset{˜}{y_i}\right].$$(3.7) is identical to the Bai (2013) factor estimator.

3 Equation (4.3)(4.3) $${\theta }_{i1}=\Delta \widetilde{\underset{˜}{x}}{\prime }_i{\widetilde{\underset{˜}{a}}}^{*}+{\tilde{\omega }}_i^{*}.$$(4.3) is derived by backcasting $\Delta x_{i,1-j}$ if $\Delta x_{it}$ is stationary. As argued in Remark 3.1, since $\Delta x_{i,1-j}$ is time varying, it is expected that using $\Delta {\underset{˜}{x}}_i$ or $\Delta \underset{˜}{x_i^{*}}=(\Delta x_{i2},\dots ,\Delta x_{iT}^{*}),T^{*}<T,$ performs better than using $\Delta {\overline{x}}_i=\frac{1}{T}{\sum }_{t=2}^T\Delta x_{it}.$ We prefer to use $\Delta \underset{˜}{x_i^{*}}$ because $\left|\rho \right|<1,$ it is expected that $(\Delta x_{i,T^{*}+1},\dots ,\Delta x_{iT})$ provides little predictive power for ${\theta }_{i1}$.

4 If there is a trend for x_it, (4.14)(4.14) $$x_{it}=d_i+gt+{(1-L)}^{d}b(L){ϵ}_{it},$$(4.14) restricts the trend coefficient to be identical over i.

Additional information

Funding

This work is supported in part by the China Natural Science Foundation (grant # 71631004).