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A bias-corrected fixed effects estimator in the dynamic panel data model

Abstract

In this paper, we propose a biased-corrected FE estimator for the dynamic panel data model that works for the autoregressive coefficient \(\rho \in (-1,1]\). We further derive the asymptotic result of the suggested bias-corrected FE estimator. We show that when \(\rho =1\), the suggested estimator is super-consistent and is more efficient than the existing estimators that also work for \(\rho \in (-1,1]\). In addition, when the initial condition is nonstationary, many of the existing dynamic estimators become inconsistent; however, the consistency of the bias-corrected FE estimator we propose does not depend on the stationarity of the initial condition. We also compare the finite sample performances of these estimators using Monte Carlo simulations.

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Notes

  1. 1.

    This result for the case \(\rho =1\) has also been derive in Theorem 2 in Kao (1999), which discusses the spurious panel data model.

  2. 2.

    We would thank the editor for pointing this out.

  3. 3.

    We would thank a referee for pointing this out.

  4. 4.

    It is easy to show that \({\hat{\rho }}_\mathrm{HP}=2{\hat{\rho }}_\mathrm{FD}+1\) and \({\hat{\rho }}_\mathrm{FD} -\rho \overset{p}{\rightarrow }-\frac{1+\rho }{2}\) as \((n,T)\rightarrow \infty \). Hence \({\hat{\rho }}_\mathrm{HP}\overset{p}{\rightarrow }\rho \) as \((n,T)\rightarrow \infty \).

  5. 5.

    We also checked performance of FEBC using \(\delta =\left( 0.05,0.1,0.15,0.2\right) \). These Monte Carlo simulation results are available upon request. These values of \(\delta \) only affect the cases when true value of \(\rho =0.9\) or 1. When T is 10 or 20, using a large \(\delta \), such as 0.2, reduces RMSE of the FEBC estimator to almost 0 when \(\rho =1\), while ends up with a relatively large RMSE when \(\rho =0.9\). On the other hand, using a small \(\delta \), such as 0.05, ends up with a relative larger RMSE of the FEBC estimator when \(\rho =1\), while a relatively small RMSE when \(\rho =0.9\). This finding is consistent with the discussion in Sect. 2. When T is large, the results of FEBC become robust to these different values of \(\delta \).

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Correspondence to Chihwa Kao.

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Author Chihwa Kao declares that he has no conflict of interest. Author Long Liu declares that he has no conflict of interest. Author Rui Sun declares that he has no conflict of interest.

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This paper is dedicated in honour of Badi H. Baltagi’s many contributions to econometrics and in particular panel data analysis.

Proof of Theorem 1

Proof of Theorem 1

Proof

Let \(S=\left\{ T^{-\delta }\left( T+1\right) \left( 1-{\hat{\rho }} _\mathrm{FE}\right) >3\right\} \) and \({\bar{S}}=\left\{ T^{-\delta }\left( T+1\right) \left( 1-{\hat{\rho }}_\mathrm{FE}\right) \le 3\right\} \). Consider (1). When \(\left| \rho \right| <1\), it suffices to show that

$$\begin{aligned}&\sqrt{nT}\left( {\hat{\rho }}_\mathrm{FEBC}-\rho \right) -\sqrt{nT}\left( \hat{\rho }_\mathrm{FE}-\rho +\frac{1+{\hat{\rho }}_\mathrm{FE}}{T}\right) \\&\quad = \sqrt{nT}\left( {\hat{\rho }}_\mathrm{FEBC}-{\hat{\rho }}_\mathrm{FE}-\frac{1+\hat{\rho }_\mathrm{FE}}{T}\right) \overset{p}{\rightarrow }0. \end{aligned}$$

We have

$$\begin{aligned}&\lim _{\left( n,T\right) \rightarrow \infty }\Pr \left( \left| \sqrt{nT}\left( {\hat{\rho }}_\mathrm{FEBC}-{\hat{\rho }}_\mathrm{FE}-\frac{1+{\hat{\rho }}_\mathrm{FE}}{T}\right) \right|>\epsilon \right) \\&\quad =\lim _{\left( n,T\right) \rightarrow \infty }\Pr \left( \left| \sqrt{nT}\left( {\hat{\rho }}_\mathrm{FEBC}-{\hat{\rho }}_\mathrm{FE}-\frac{1+{\hat{\rho }}_\mathrm{FE}}{T}\right) \right|>\epsilon |S\right) \Pr \left( S\right) \\&\qquad +\lim _{\left( n,T\right) \rightarrow \infty }\Pr \left( \left| \sqrt{nT}\left( {\hat{\rho }}_\mathrm{FEBC}-{\hat{\rho }}_\mathrm{FE}-\frac{1+{\hat{\rho }}_\mathrm{FE}}{T}\right) \right| >\epsilon |{\bar{S}}\right) \Pr \left( {\bar{S}}\right) . \end{aligned}$$

The first term is zero given that, if S is true, we have \({\hat{\rho }} _\mathrm{FEBC}={\hat{\rho }}_\mathrm{FE}+\frac{1+{\hat{\rho }}_\mathrm{FE}}{T}\) so that \(\Pr \left( \left| \sqrt{nT}\left( {\hat{\rho }}_\mathrm{FEBC}-{\hat{\rho }}_\mathrm{FE}-\frac{1+{\hat{\rho }}_\mathrm{FE}}{T}\right) \right| >\epsilon |S\right) =0\). The second term is zero since

$$\begin{aligned}&T^{-\delta }\left( T+1\right) \left( 1-{\hat{\rho }}_\mathrm{FE}\right) -3\\&\quad =T^{-\delta }\left( T+1\right) \left( 1-\rho \right) -T^{-\delta } \sqrt{\frac{T}{n}}\frac{T+1}{T}\left[ \sqrt{nT}\left( {\hat{\rho }}_\mathrm{FE} -\rho +\frac{1+\rho }{T}\right) \right] \\&\qquad -3+T^{-\delta }\frac{T+1}{T}\left( 1+\rho \right) \\&\quad =T^{-\delta }\left( T+1\right) \left( 1-\rho \right) +O_{p}\left( T^{-\delta }\sqrt{\frac{T}{n}}\frac{T+1}{T}\right) +O_{p}\left( T^{-\delta }\frac{T+1}{T}\right) \rightarrow \infty \end{aligned}$$

using\(\sqrt{nT}\left( {\hat{\rho }}_\mathrm{FE}-\rho +\frac{1+\rho }{T}\right) =O_{p}\left( 1\right) \) and \(T^{-\delta }\left( T+1\right) \left( 1-\rho \right) \rightarrow \infty \) as \(T\rightarrow \infty \). Hence \(\Pr \left( {\bar{S}}\right) \rightarrow 0\) as \(T\rightarrow \infty \). Therefore, \(\Pr \left( \left| \sqrt{nT}\left( {\hat{\rho }}_\mathrm{FEBC}-{\hat{\rho }}_\mathrm{FE}-\frac{1+{\hat{\rho }}_\mathrm{FE}}{T}\right) \right| >\epsilon \right) \) \(\rightarrow 0\) as \(\left( n,T\right) \rightarrow \infty \).

Consider (2). When \(\rho =1\), we have

$$\begin{aligned}&\lim _{\left( n,T\right) \rightarrow \infty }\Pr \left( \left| \sqrt{n}T\left( {\hat{\rho }}_\mathrm{FEBC}-1\right) \right|>\epsilon \right) \\&\quad =\lim _{\left( n,T\right) \rightarrow \infty }\Pr \left( \left| \sqrt{n}T\left( {\hat{\rho }}_\mathrm{FEBC}-1\right) \right|>\epsilon |S\right) \Pr \left( S\right) \\&\qquad +\lim _{\left( n,T\right) \rightarrow \infty }\Pr \left( \left| \sqrt{n}T\left( {\hat{\rho }}_\mathrm{FEBC}-1\right) \right| >\epsilon |{\bar{S}}\right) \Pr \left( {\bar{S}}\right) . \end{aligned}$$

Notice that

$$\begin{aligned} \Pr \left( S\right)&=\Pr \left( T^{-\delta }\left( T+1\right) \left( 1-{\hat{\rho }}_\mathrm{FE}\right)>3\right) \\&=\Pr \left( T^{-\delta }\left[ 3+\left( T+1\right) \left( 1-\hat{\rho }_\mathrm{FE}-\frac{3}{T+1}\right) \right]>3\right) \\&=\Pr \left( 3T^{-\delta }-\frac{1}{\sqrt{n}T^{\delta }}\frac{T+1}{T}\sqrt{n}T\left( {\hat{\rho }}_\mathrm{FE}-1+\frac{3}{T+1}\right) >3\right) \\&\rightarrow 0 \end{aligned}$$

as \(\left( n,T\right) \rightarrow \infty \) since \(\sqrt{n}T\left( \hat{\rho }_\mathrm{FE}-1+\frac{3}{T+1}\right) \) \(=O_{p}\left( 1\right) \), \(T^{-\delta }\rightarrow 0\) and \(\frac{1}{\sqrt{n}T^{\delta }}\rightarrow 0\). Hence the first term is zero. For the second term, if \({\bar{S}}\) is true, \(\hat{\rho }_\mathrm{FEBC}=1\) so that \(\Pr \left( \left| \sqrt{nT}\left( {\hat{\rho }} _\mathrm{FEBC}-1\right) \right| >\epsilon |{\bar{S}}\right) =0\). Thus, \(\Pr \left( \left| \sqrt{n}T\left( {\hat{\rho }}_\mathrm{FEBC}-1\right) \right| >\epsilon \right) \) \(\rightarrow 0\) as \(\left( n,T\right) \rightarrow \infty \). \(\square \)

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Kao, C., Liu, L. & Sun, R. A bias-corrected fixed effects estimator in the dynamic panel data model. Empir Econ 60, 205–225 (2021). https://doi.org/10.1007/s00181-020-01995-0

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Keywords

  • Dynamic panel data
  • Bias-corrected estimator
  • Fixed effects
  • Nonstationarity
  • Super-efficient estimator
  • Initial condition