### Abstract

First difference maximum likelihood (FDML) seems an attractive estimation methodology in dynamic panel data modeling because differencing eliminates fixed effects and, in the case of a unit root, differencing transforms the data to stationarity, thereby addressing both incidental parameter problems and the possible effects of nonstationarity. This paper draws attention to certain pathologies that arise in the use of FDML that have gone unnoticed in the literature and that affect both finite sample performance and asymptotics. FDML uses the Gaussian likelihood function for first differenced data and parameter estimation is based on the whole domain over which the log-likelihood is defined. However, extending the domain of the likelihood beyond the stationary region has certain consequences that have a major effect on finite sample and asymptotic performance. First, the extended likelihood is not the true likelihood even in the Gaussian case and it has a finite upper bound of definition. Second, it is often bimodal, and one of its peaks can be so peculiar that numerical maximization of the extended likelihood frequently fails to locate the global maximum. As a result of these pathologies, the FDML estimator is a restricted estimator, numerical implementation is not straightforward and asymptotics are hard to derive in cases where the peculiarity occurs with non-negligible probabilities. The peculiarities in the likelihood are found to be particularly marked in time series with a unit root. In this case, the asymptotic distribution of the FDMLE has bounded support and its density is infinite at the upper bound when the time series sample size T→∞. As the panel width n→∞ the pathology is removed and the limit theory is normal. This result applies even for T fixed and we present an expression for the asymptotic distribution which does not depend on the time dimension. We also show how this limit theory depends on the form of the extended likelihood.

Original language | English |
---|---|

Pages (from-to) | 35-45 |

Number of pages | 11 |

Journal | Journal of Econometrics |

Volume | 175 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2013 Jul 1 |

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### Keywords

- Asymptote
- Bounded support
- Dynamic panel
- Efficiency
- First difference MLE
- Likelihood
- Quartic equation
- Restricted extremum estimator

### ASJC Scopus subject areas

- Economics and Econometrics
- Applied Mathematics
- History and Philosophy of Science

### Cite this

*Journal of Econometrics*,

*175*(1), 35-45. https://doi.org/10.1016/j.jeconom.2013.03.003

**First difference maximum likelihood and dynamic panel estimation.** / Han, Chirok; Phillips, Peter C B.

Research output: Contribution to journal › Article

*Journal of Econometrics*, vol. 175, no. 1, pp. 35-45. https://doi.org/10.1016/j.jeconom.2013.03.003

}

TY - JOUR

T1 - First difference maximum likelihood and dynamic panel estimation

AU - Han, Chirok

AU - Phillips, Peter C B

PY - 2013/7/1

Y1 - 2013/7/1

N2 - First difference maximum likelihood (FDML) seems an attractive estimation methodology in dynamic panel data modeling because differencing eliminates fixed effects and, in the case of a unit root, differencing transforms the data to stationarity, thereby addressing both incidental parameter problems and the possible effects of nonstationarity. This paper draws attention to certain pathologies that arise in the use of FDML that have gone unnoticed in the literature and that affect both finite sample performance and asymptotics. FDML uses the Gaussian likelihood function for first differenced data and parameter estimation is based on the whole domain over which the log-likelihood is defined. However, extending the domain of the likelihood beyond the stationary region has certain consequences that have a major effect on finite sample and asymptotic performance. First, the extended likelihood is not the true likelihood even in the Gaussian case and it has a finite upper bound of definition. Second, it is often bimodal, and one of its peaks can be so peculiar that numerical maximization of the extended likelihood frequently fails to locate the global maximum. As a result of these pathologies, the FDML estimator is a restricted estimator, numerical implementation is not straightforward and asymptotics are hard to derive in cases where the peculiarity occurs with non-negligible probabilities. The peculiarities in the likelihood are found to be particularly marked in time series with a unit root. In this case, the asymptotic distribution of the FDMLE has bounded support and its density is infinite at the upper bound when the time series sample size T→∞. As the panel width n→∞ the pathology is removed and the limit theory is normal. This result applies even for T fixed and we present an expression for the asymptotic distribution which does not depend on the time dimension. We also show how this limit theory depends on the form of the extended likelihood.

AB - First difference maximum likelihood (FDML) seems an attractive estimation methodology in dynamic panel data modeling because differencing eliminates fixed effects and, in the case of a unit root, differencing transforms the data to stationarity, thereby addressing both incidental parameter problems and the possible effects of nonstationarity. This paper draws attention to certain pathologies that arise in the use of FDML that have gone unnoticed in the literature and that affect both finite sample performance and asymptotics. FDML uses the Gaussian likelihood function for first differenced data and parameter estimation is based on the whole domain over which the log-likelihood is defined. However, extending the domain of the likelihood beyond the stationary region has certain consequences that have a major effect on finite sample and asymptotic performance. First, the extended likelihood is not the true likelihood even in the Gaussian case and it has a finite upper bound of definition. Second, it is often bimodal, and one of its peaks can be so peculiar that numerical maximization of the extended likelihood frequently fails to locate the global maximum. As a result of these pathologies, the FDML estimator is a restricted estimator, numerical implementation is not straightforward and asymptotics are hard to derive in cases where the peculiarity occurs with non-negligible probabilities. The peculiarities in the likelihood are found to be particularly marked in time series with a unit root. In this case, the asymptotic distribution of the FDMLE has bounded support and its density is infinite at the upper bound when the time series sample size T→∞. As the panel width n→∞ the pathology is removed and the limit theory is normal. This result applies even for T fixed and we present an expression for the asymptotic distribution which does not depend on the time dimension. We also show how this limit theory depends on the form of the extended likelihood.

KW - Asymptote

KW - Bounded support

KW - Dynamic panel

KW - Efficiency

KW - First difference MLE

KW - Likelihood

KW - Quartic equation

KW - Restricted extremum estimator

UR - http://www.scopus.com/inward/record.url?scp=84876943110&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84876943110&partnerID=8YFLogxK

U2 - 10.1016/j.jeconom.2013.03.003

DO - 10.1016/j.jeconom.2013.03.003

M3 - Article

AN - SCOPUS:84876943110

VL - 175

SP - 35

EP - 45

JO - Journal of Econometrics

JF - Journal of Econometrics

SN - 0304-4076

IS - 1

ER -