Top

Empirical Economics

Published in:

05-03-2023

Likelihood-based inference for dynamic panel data models

Authors: Seung C. Ahn, Gareth M. Thomas

Published in: Empirical Economics | Issue 6/2023

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

This paper considers maximum likelihood (ML)-based inferences for dynamic panel data models. We focus on the analysis of the panel data with a large number (N) of cross-sectional units and a small number (T) of repeated time series observations for each cross-sectional unit. We examine several different ML estimators and their asymptotic and finite-sample properties. Our major finding is that when data follow unit-root processes without or with drifts, the ML estimators have singular information matrices. This is a case of Sargan (Econometrica 51:1605–1634, 1983) in which the first-order condition for identification fails, but parameters are identified. The ML estimators are consistent, but they have non-standard asymptotic distributions, and their convergence rates are lower than N^1/2. In addition, the sizes of usual Wald statistics based on the estimators are distorted even asymptotically, and they reject the unit-root hypothesis too often. However, following Rotnitzky et al. (Bernoulli 6:243–284, 2000) we show that likelihood ratio (LR) tests for unit root follow mixtures of chi-square distributions. Our Monte Carlo experiments show that the LR tests with the p-values from the mixed distributions are much better sized than the Wald tests, although they tend to slightly over-reject the unit-root hypothesis in small samples. It is also shown that the LR tests for unit roots have good finite-sample power properties.

previous article An alternative corrected ordinary least squares estimator for the stochastic frontier model

next article Approximating long-memory processes with low-order autoregressions: Implications for modeling realized volatility

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

Hahn and Kuersteiner (2002) also consider the ML estimator for the FE dynamic model with large N and T. They find the ML estimator, which is the within estimator, is consistent, but it is asymptotically biased. Hahn and Kuersteiner provide a bias-corrected ML estimator. Hayakawa and Pesaran (2015) derives the asymptotic distribution of the HPT estimator under more general conditions (i.e., non-normal and cross-sectionally heteroskedastic data).

Kruiniger (2013) also deals with general identification issues related to the GMM and quasi-maximum likelihood (QML) versions of the RE ML estimator, and QML version of the HPT ML estimator when data are not normally distributed.

See also a published paper of Alvarez and Arellano (2022), which is an updated version of Alvarez and Arellano (2004).

Alvarez and Arellano (2022) independently found a similar result for a more general model, although they do not consider the asymptotic distribution of the RE ML estimator when data follow unit-root processes.

This result was obtained in the 2004 version of this paper. Extending this result, Kruiniger (2013) shows that the convergence rates of the RE and HPT ML estimators are still N^1/4 even if data are not normally distributed.

See HPT for the derivation of $\det [\Omega_{T} (\omega )]$. The parameter $\omega$ in their paper is equivalent to $(\omega + 1)$ in this paper.

The $f(u_{i}^{*} |y_{i0} ,(\nu ,\omega )^{\prime})$ function can be easily obtained if

$$\left( {\begin{array}{*{20}c} {\alpha_{i} } \\ {\varepsilon_{i} } \\ \end{array} } \right)\sim N\left( {\left( {\begin{array}{*{20}c} {\pi_{0} y_{i0} } \\ {0_{T \times 1} } \\ \end{array} } \right),\left( {\begin{array}{*{20}c} {v\omega } & {0_{1 \times t} } \\ {0_{t \times 1} } & {\nu I_{T} } \\ \end{array} } \right)} \right),$$

conditionally on $y_{i0}$. Under this assumption, $\psi = (\delta - 1) + \pi_{0}$. The $y_{i0}$ needs not be normal. This result can be easily extended to the cases in which the model (1) includes some strictly exogenous regressors,${\mathbf{x}}_{i} = (x^{\prime}_{i0} ,x^{\prime}_{i1} ,...,x^{\prime}_{iT} )^{\prime}$:$y_{it} = \delta y_{i,t - 1} + x^{\prime}_{it} \beta + u_{it}$. For these cases, following Wooldridge (2000), we may assume that conditionally on $(y_{i0} ,{\mathbf{x}^{\prime}}_{i} )^{\prime}$

$$\left( {\begin{array}{*{20}c} {\alpha_{i} } \\ {\varepsilon_{i} } \\ \end{array} } \right)\sim N\left( {\left( {\begin{array}{*{20}c} {\pi_{0} y_{i0} + {\mathbf{x^{\prime}}}_{i} \pi_{x} } \\ 0 \\ \end{array} } \right),\left( {\begin{array}{*{20}c} {v\omega } & {0_{1 \times T} } \\ {0_{1 \times T} } & {\nu I_{T} } \\ \end{array} } \right)} \right).$$

Under this assumption, the log-density of $\Delta y_{i}^{*}$ conditional on $(y_{i0} ,{\mathbf{x^{\prime}}}_{i} )^{\prime}$ is obtained by replacing $(\Delta y_{i1} - \psi y_{i0})$ and $(\Delta y_{i} - \delta \Delta y_{i, - 1})$ in $\ell_{RE,i} (\theta)$ by $(\Delta y_{i1} - \psi y_{i0} - {\mathbf{x^{\prime}}}_{i} \pi_{0,x})$ and $(\Delta y_{i} - \delta \Delta y_{i, - 1} - \Delta {\mathbf{X}}_{i} \beta)$, respectively, where

$${\mathbf{x}}_{i}^{{\prime }} \pi_{0,x} = \Delta x_{i1}^{{\prime }} \beta + {\mathbf{x}}^{{\prime }}_{i} \pi_{x} ;\,\Delta {\mathbf{X}}_{i} = (\Delta x_{i2} ,\Delta x_{i3} ,...,\Delta x_{iT} )^{{\prime }}.$$

This conditional ML method is in contrast to the ML estimation method of Bhargava and Sargan (BS, 1983). They consider two cases: one in which $(y_{i1} ,y_{i2} ,...,y_{iT} )^{\prime}$ is normal conditionally on $(y_{i0} ,\mathbf{x}^{\prime}_{i} )^{\prime}$; and the other in which $(y_{i0} ,y_{i1} ,...,y_{iT} )^{\prime}$ is normal conditionally on ${\mathbf{x}}_{i}$ and ${\text{E}}(y_{i0} |{\mathbf{x}}_{i} ) = \mathbf{x}^{\prime}_{i} \pi_{x}$. Their second case is related to the RE model we consider here, except that BS additionally assumes that $\alpha_{i}$ and ${\mathbf{x}}_{i}$ are independent.

We can estimate $\xi_{T}$ instead of ω. In simulations, we found that the ML algorithms converge faster when $\xi_{T}$ is estimated. Accordingly, we have estimated $\xi_{T}$ for our simulations.

HPT in fact include an intercept term for the Δy_i1 equation. But the interceptor term equals zero under our zero-mean assumption.

For the cases in which $y_{it} = \delta y_{i,t - 1} + x^{\prime}_{it} \beta + u_{it}$ and the $x_{it}$ are strictly exogenous to $\varepsilon_{it}$, HPT assumes that conditionally on ${\mathbf{x}}_{i}$,

$$\left( {\begin{array}{*{20}c} {p_{i} } \\ {\varepsilon_{i} } \\ \end{array} } \right)\sim N\left( {\left( {\begin{array}{*{20}c} {\Delta {\mathbf{x^{\prime}}}_{i} \pi_{x} } \\ 0 \\ \end{array} } \right),\left( {\begin{array}{*{20}c} {\nu \omega_{HPT} } & {0_{1 \times T} } \\ {0_{1 \times T} } & {\nu I_{T} } \\ \end{array} } \right)} \right).$$

Then, the log-density of $(\Delta y_{i1} ,...,\Delta y_{iT} )^{\prime}$ conditional on ${\mathbf{x}}_{i}$ equals $\ell_{HPT,i} (\theta_{HPT})$ with $\Delta y_{i1}$ and $(\Delta y_{i} - \delta \Delta y_{i, - 1})$, respectively, replaced by $(\Delta y_{i1} - \Delta {\mathbf{x^{\prime}}}_{i} \pi_{x})$ and $(\Delta y_{i} - \delta \Delta y_{i, - 1} - \Delta {\mathbf{X}}_{i} \beta)$, where $\Delta {\mathbf{x}}_{i} = (\Delta x^{\prime}_{i1} ,...,\Delta x^{\prime}_{iT} )^{\prime}$ and $\Delta {\mathbf{X}}_{i} = (\Delta x_{i1} ,...,\Delta x_{iT} )^{\prime}$. See HPT for the stationary conditions under which ${\text{E}}(p_{i} |{\mathbf{x}}_{i})$ is linear in $\Delta {\mathbf{x}}_{i}$.

An intriguing question would be whether the Lancaster and/or HPT estimators are consistent under broader circumstances than the RE estimator is. Consider the two conditions: (i)$p\lim_{N \to \infty } N^{ - 1} \Sigma_{i} p_{i}^{2} < \infty$ is finite; and (ii) $p\lim_{N \to \infty } N^{ - 1} \Sigma_{i} (y_{io} ,\alpha_{i} )^{\prime}(y_{i0} ,\alpha_{i})$ is finite. Obviously, condition (ii) is stronger than (i). Kruiniger (2002, 2013) finds that a necessary condition for consistency of the Lancaster estimator is (i). It can be shown that the same condition is required for the consistency of the HPT estimator. As long as the condition holds and the errors $\varepsilon_{it}$ are i.i.d. and uncorrelated with $p_{i}$, both the Lancaster and HPT estimators are consistent, even if data are not normal. In contrast, it can be shown that the consistency of the RE estimator requires the stronger restriction (ii). Thus, it is true that that the Lancaster and/or the HPT estimator are consistent under more general conditions. However, there are few realistic cases in which condition (i) holds, while (ii) is violated. Condition (ii) would be an acceptable assumption for most of the panel studies (at least the studies with short panels). If (ii) holds, all the three estimators are consistent, and thus the distinction between RE and FE becomes unimportant.

See Kruiniger (2002).

Lancaster (2002) also acknowledges this point.

If the means of the $y_{i0}$ and $\alpha_{i}$ are nonzero, we need to include an intercept term, say a, in the log-likelihood function replacing $(\Delta y_{i1} - \psi y_{i0} )$ by $(\Delta y_{i1} - \psi y_{i0} - a)$. Then, the unit-root hypothesis (15) implies a = 0 in addition to the restrictions in θ_*. For this case, the LR statistic for testing all the restrictions implied by (15) is a mixture of $\chi^{2} (3)$ and $\chi^{2} (4)$.

We may consider an alternative Wald-type test statistic,$N(\hat{\delta }_{RE} - 1)^{4} /\Upsilon^{11}$. Proposition 6 implies that under $H_{o}^{UND}$, this statistic follows a mixture of $\chi^{2} (1)$ and zero.

These LR tests are two-tail tests. When T is fixed, the RE likelihood function is well defined in the neighborhood of $\delta = 1$. Thus, the RE ML estimator is not subject to the boundary problem raised by Andrews (1999), and the hypothesis of $\delta_{o} = 1$ can be tested against the two-tail alternative hypothesis of $\delta_{0} \ne 1$. This justifies use of the LR tests. However, some one-tail alternatives of the LR tests would be more powerful since $\delta_{o}$ is unlikely to be greater than one, although we do not investigate them here.

Arellano and Bover propose to use for GMM the moment conditions,${\text{E}}[\Delta y_{it} (y_{is} - \delta y_{i,s - 1} )] = 0$, t < s. These moment conditions are valid under (7), but not under (1) and (2). Observe that these moment conditions can identify the true value of $\delta$ even if data follow unit-root processes without drifts. In contrast, the moment conditions by Arellano and Bond (1991),${\text{E}}(y_{it} (\Delta y_{is} - \delta \Delta y_{i,s - 1} )) = 0$, t < s, are motivated by the model given by (1) and (2). As Blundell and Bond (1998) find, these moment conditions are unable to identify $\delta$ if data follow random walk processes because the level instruments y_it are uncorrelated with differenced regressors Δy_i,s-1.

The Lancaster estimator is quite sensitive to the choice of the starting parameter values used for algorithms.

For the data generating process used for simulations, $0 \le \left| {\psi_{o} } \right| \le 0.25$ because $\psi = - (1 - \delta )\sigma_{q0}^{2} /(\sigma_{\eta }^{2} + \sigma_{q0}^{2})$. As discussed in Sect. 2, The RE and HPT ML estimators are equivalent if $\psi_{o} = 0$(the case in which $y_{i0}$ and $\delta y_{i1}$ are uncorrelated. It appears that the RE and HPT ML estimators show similar performances in Table 1 because the values used for $\psi_{o}$ are small.

Ahn SC, Schmidt P (1995) Efficient estimation of models for dynamic panel data. J Econom 68:5–27CrossRef

Ahn SC, Schmidt P (1997) Efficient estimation of dynamic panel data models: alternative assumptions and simplified assumptions. J Econom 76:309–321CrossRef

Alvarez A, Arellano M (2004) Robust likelihood estimation of dynamic panel data models, mimeo. CEMFI, Spain

Alvarez A, Arellano M (2022) Robust likelihood estimation of dynamic panel data models. J Econom 226:21–61CrossRef

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

Andrews D (1999) Estimation when a parameter is on a boundary. Econometrica 67:1341–1384CrossRef

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

Arellano M, Bover O (1995) Another look at the instrumental variables estimation of error-component models. J Econom 68:29–51CrossRef

Bhargava A, Sargan JD (1983) Estimating dynamic random effects models from panel data covering short time periods. Econometrica 51:1635–1659CrossRef

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

Bond S, Windmeijer F (2002) Finite sample inference for GMM estimators in linear panel data models, IFS, mimeo.

Breitung J, Meyer W (1994) Testing for unit roots using panel data; are wages on different bargaining levels cointegrated. Appl Econ 26:353–361CrossRef

Cox DR, Reid N (1987) Parameter orthogonally and approximate conditional inference (with discussion). J Royal Stat Soc B 49:1–39

Gouriéroux C, Monfort A, Trognon A (1984) Pseudo maximum likelihood methods: Theory. Econometrica 52:681–700CrossRef

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

Harris R, Tzavalis E (1999) Inference for unit roots in dynamic panels where the time dimension is fixed. J Econom 91:2001–2226CrossRef

Hayakawa K, Pesaran MH (2015) Robust standard errors in transformed likelihood estimation of dynamic panel data models with cross-sectional heteroskedasticity. J Econom 118:111–134CrossRef

Hsiao C (1986) Analysis of panel data. University Press, Cambridge, UK, Cambridge

Hsiao C, Pesaran MH, Tahmiscioglu AK (2002) Maximum likelihood estimation of fixed effects dynamic panel data models covering short time period. J Econom 109:107–150CrossRef

Im K, Pesaran MH, Shin Y (2003) Testing for unit roots in heterogeneous panels. J Econom 115:53–74CrossRef

Kruiniger H (2008) Maximum likelihood estimation and inference methods for the covariance stationary panel AR(1)/unit root model. J Econom 144:447–464CrossRef

Kruiniger H (2013) Quasi ML estimation of the panel AR(1) model with arbitrary initial conditions. J Econom 173:175–188CrossRef

Kruiniger H (2002) On the estimation of panel regression models with fixed effects, Queen Mary, University of London, mimeo.

Lancaster T (2002) Orthogonal parameters and panel data. Rev Econ Stud 69:647–666CrossRef

Levin A, Lin F, Chu C (2002) Unit root tests in panel data: asymptotic and finite-sample properties. J Econom 108:1–24CrossRef

Moon H, Phillips PCB (2004) GMM estimation of autoregressive roots near unity with panel data. Econometrica 72:467–522CrossRef

Moon H, Phillips P, Perron B (2007) Incidental trends and the power of panel unit root tests. J Econom 141:416–459CrossRef

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

Nickell S (1981) Biases in dynamic models with fixed effects. Econometrica 49:1399–1416CrossRef

Rao CR (1973) Linear statistical inference and its applications, 2nd edn. Wiley, New YorkCrossRef

Rotnitzky A, Cox DR, Bottai M, Robins J (2000) Likelihood-based inference with singular information matrix. Bernoulli 6:243–284CrossRef

Sargan JD (1983) Identification and lack of identification. Econometrica 51:1605–1634CrossRef

Thomas G (2005) Maximum likelihood based estimation of dynamic panel data Models, Ph.D. dissertation, Arizona State University.

Wooldridge JM (2000) A framework for estimating dynamic, unobserved effects panel data models with possible feedback to future explanatory variables. Econ Lett 68:245–250CrossRef

Title: Likelihood-based inference for dynamic panel data models
Authors: Seung C. Ahn
Gareth M. Thomas
Publication date: 05-03-2023
Publisher: Springer Berlin Heidelberg
Published in: Empirical Economics / Issue 6/2023
Print ISSN: 0377-7332
Electronic ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-023-02375-0

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft"

Springer Professional "Wirtschaft+Technik"

Other articles of this Issue 6/2023

Assessing the impacts of pandemic and the increase in minimum down payment rate on Shanghai housing prices

Internal adjustment costs of firm-specific factors and the neoclassical theory of the firm

Identification and estimation of categorical random coefficient models

Indirect inference estimation of stochastic production frontier models with skew-normal noise

Predicting binary outcomes based on the pair-copula construction

Information loss in volatility measurement with flat price trading

Premium Partner