Skip to main content

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


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.

This is a preview of subscription content, access via your institution.


  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 \(\delta \).


  1. Alvarez J, Arellano M (2003) The time series and cross-section asymptotics of dynamic panel data estimators. Econometrica 71(4):1121–1159

    Article  Google Scholar 

  2. Anderson TW, Hsiao C (1981) Estimation of dynamic models with error components. J Am Stat Assoc 76(375):598–606

    Article  Google Scholar 

  3. Anderson TW, Hsiao C (1982) Formulation and estimation of dynamic models using panel data. J Econ 18(1):47–82

    Article  Google Scholar 

  4. Andrews DWK (1993) Exactly median-unbiased estimation of first order autoregressive/unit root models. Econometrica 61(1):139–165

    Article  Google Scholar 

  5. Arellano M, Bover O (1995) Another look at the instrumental variable estimation of error-components models. J Econ 68(1):29–51

    Article  Google Scholar 

  6. Arellano M, Bond S (1991) Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations. Rev Econ Stud 58(2):277–297

    Article  Google Scholar 

  7. Baltagi BH (2013) Econometric analysis of panel data. Wiley, Hoboken

    Google Scholar 

  8. Baltagi BH, Kao C, Liu L (2008) Asymptotic properties of estimators for the linear panel regression model with random individual effects and serially correlated errors: the case of stationary and non-stationary regressors and residuals. Econ J 11(3):554–572

    Google Scholar 

  9. Baltagi BH, Kao C, Liu L (2012) On the estimation and testing of fixed effects panel data models with weak instruments. In: 30th anniversary edition. Emerald Group Publishing Limited, pp 199–235

  10. Baltagi BH, Kao C, Liu L (2014) Test of hypotheses in a time trend panel data model with serially correlated error component disturbances. In: Essays in Honor of Peter C. B. Phillips, vol 33 of advances in econometrics. Emerald Publishing Ltd, pp 347–394

  11. Baltagi BH, Kao C, Liu L (2017) Estimation and identification of change points in panel models with nonstationary or stationary regressors and error term. Econ Rev 36(1–3):85–102

    Article  Google Scholar 

  12. Baltagi BH, Kao C, Liu L (2019) Testing for shifts in a time trend panel data model with serially correlated error component disturbance. Econ Rev 39:745–762

    Article  Google Scholar 

  13. Blundell R, Bond S (1998) Initial conditions and moment restrictions in dynamic panel data models. J Econ 87(1):115–143

    Article  Google Scholar 

  14. Bruno GSF (2005) Approximating the bias of the LSDV estimator for dynamic unbalanced panel data models. Econ Lett 87(3):361–366

    Article  Google Scholar 

  15. Bun MJG (2003) Bias correction in the dynamic panel data model with a nonscalar disturbance covariance matrix. Econ Rev 22(1):29–58

    Article  Google Scholar 

  16. Bun MJG, Carree MA (2005) Bias-corrected estimation in dynamic panel data models. J Bus Econ Stat 23(2):200–210

    Article  Google Scholar 

  17. Bun MJG, Carree MA (2006) Bias-corrected estimation in dynamic panel data models with heteroscedasticity. Econ Lett 92(2):220–227

    Article  Google Scholar 

  18. Chao J, Kim M, Sul D (2014) Mean average estimation of dynamic panel models with nonstationary initial condition. In: Essays in Honor of Peter C. B. Phillips (advances in econometrics, vol 33), Chapter 8. Emerald Group Publishing Limited, pp 241–279

  19. Chudik A, Hashem Pesaran M, Yang J-C (2018) Half-panel jackknife fixed-effects estimation of linear panels with weakly exogenous regressors. J Appl Econ 33(6):816–836

    Article  Google Scholar 

  20. Gonçalves S, Kaffo M (2015) Bootstrap inference for linear dynamic panel data models with individual fixed effects. J Econ 186(2):407–426 (High dimensional problems in econometrics)

    Article  Google Scholar 

  21. Hahn J, Kuersteiner G (2002) Asymptotically unbiased inference for a dynamic panel model with fixed effects when both \(n\) and \(T\) are large. Econometrica 70(4):1639–1657

    Article  Google Scholar 

  22. Hahn J, Moon HR (2006) Reducing bias of MLE in a dynamic panel model. Econ Theory 22(3):499–512

    Article  Google Scholar 

  23. Han C, Phillips PCB (2010) GMM estimation for dynamic panels with fixed effects and strong instruments at unity. Econ Theory 26(1):119–151

    Article  Google Scholar 

  24. Han C, Phillips PCB, Sul D (2014) X-differencing and dynamic panel model estimation. Econ Theory 30(1):201–251

    Article  Google Scholar 

  25. Hansen BE (1999) The grid bootstrap and the autoregressive model. Rev Econ Stat 81(4):594–607

    Article  Google Scholar 

  26. Harris RDF, Tzavalis E (1999) Inference for unit roots in dynamic panels where the time dimension is fixed. J Econ 91(2):201–226

    Article  Google Scholar 

  27. Hsiao C, Zhou Q (2018) Incidental parameters, initial conditions and sample size in statistical inference for dynamic panel data models. J Econ 207(1):114–128

    Article  Google Scholar 

  28. Kao C (1999) Spurious regression and residual-based tests for cointegration in panel data. J Econ 90(1):1–44

    Article  Google Scholar 

  29. Kiviet JF (2007) Judging contending estimators by simulation: tournaments in dynamic panel data models. In: The refinement of econometric estimation and test procedures. Cambridge University Press, Cambridge, pp 282–318

  30. Mikusheva A (2007) Uniform inference in autoregressive models. Econometrica 75(5):1411–1452

    Article  Google Scholar 

  31. Mikusheva A (2012) One-dimensional inference in autoregressive models with the potential presence of a unit root. Econometrica 80(1):173–212

    Article  Google Scholar 

  32. Neyman J, Scott EL (1948) Consistent estimates based on partially consistent observations. Econometrica 16(1):1–32

    Article  Google Scholar 

  33. Nickell S (1981) Biases in dynamic models with fixed effects. Econometrica 49(6):1417–1426

    Article  Google Scholar 

  34. Pierre P, Yabu T (2009a) Estimating deterministic trends with an integrated or stationary noise component. J Econ 151(1):56–69

    Article  Google Scholar 

  35. Pierre P, Yabu T (2009b) Testing for shifts in trend with an integrated or stationary noise component. J Bus Econ Stat 27(3):369–396

    Article  Google Scholar 

  36. Phillips PCB (2014) On confidence intervals for autoregressive roots and predictive regression. Econometrica 82(3):1177–1195

    Article  Google Scholar 

  37. Robertson D, Sarafidis V, Westerlund J (2018) Unit root inference in generally trending and cross-correlated fixed-T panels. J Bus Econ Stat 36(3):493–504

    Article  Google Scholar 

Download references


This study was not funded by any grants.

Author information



Corresponding author

Correspondence to Chihwa Kao.

Ethics declarations

Conflict of interest

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.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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


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 \(\left( n,T\right) \rightarrow \infty \). \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kao, C., Liu, L. & Sun, R. A bias-corrected fixed effects estimator in the dynamic panel data model. Empir Econ 60, 205–225 (2021).

Download citation


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