Dear Statalisters,

Linear dynamic panel data models are commonly estimated by GMM (for example with my command xtdpdgmm). Yet, when all regressors (besides the lagged dependent variable) are strictly exogenous, more efficient alternatives are available. Besides maximum likelihood estimation (for example with my command xtdpdqml), an estimator that directly corrects the dynamic panel data bias (a.k.a. Nickell bias) of the conventional fixed-effects (FE) estimator can be quite attractive because it typically retains the small variance of the FE estimator compared to GMM estimators.

My new command, xtdpdbc, implements the bias-corrected method of moments estimator described by Breitung, Kripfganz, and Hayakawa (2021). It analytically corrects the first-order condition of the FE estimator, which leads to a set of nonlinear moment conditions that can be solved with conventional numerical methods (Gauss-Newton). Another advantage of this procedure is that a formula of the asymptotic variance-covariance matrix for the calculation of standard errors is readily available, unlike the bias-corrected estimator by Kiviet (1995) that is implemented in the community-contributed xtlsdvc command by Bruno (2005).

Yet another advantage is that the estimator can accommodate higher-order lags of the dependent variable. Moreover, the moment conditions can be adjusted to create a random-effects (RE) version of the estimator, assuming that all (or some) of the exogenous regressors are uncorrelated with the unobserved group-specific effects. This RE version is not yet implemented in xtdpdbc, but will be added in due course.

In turns out that under the FE assumption the bias-corrected method of moments estimator is equivalent to the Dhaene and Jochmans (2016) adjusted profile likelihood estimator. Furthermore, if there is only a single lag of the dependent variable, it is also equivalent to the bias-corrected estimator of Bun and Carree (2005).

It should be noted that due to the nonlinearity of the bias-corrected moment functions, the estimator in general has multiple solutions and the numerical algorithm may not always converge to the correct one. The correct solution is characterized by a negativity condition on the gradient, i.e. all eigenvalues of the gradient should be negative. In the current version of xtdpdbc, the command will display a note if the gradient has positive eigenvalues. In that case, the estimation should be repeated with different starting values (using the from() option) until the correct solution is found. Starting values for the coefficient of the lagged dependent variable should typically be varied over the interval [0, 1]. Starting values for the exogenous regressors do not matter much.

In some cases, the numerical algorithm might not converge due to an almost flat criterion function. In such a case, it might help to simplify the optimization problem by concentrating out the coefficients of the exogenous regressors with option concentration. If this does not help, formal convergence could possibly be achieved by declaring the option nonrtolerance. However, the results in that case might not be very robust.

Last but not least, the command also supports unbalanced panel data.

To install the command, type the following in Stata's command window:

Please see the help file for the fairly standard command syntax and the available options:

Here is an example with second-order autoregressive dynamics:

Any comments and suggestions are welcome.

References:

Linear dynamic panel data models are commonly estimated by GMM (for example with my command xtdpdgmm). Yet, when all regressors (besides the lagged dependent variable) are strictly exogenous, more efficient alternatives are available. Besides maximum likelihood estimation (for example with my command xtdpdqml), an estimator that directly corrects the dynamic panel data bias (a.k.a. Nickell bias) of the conventional fixed-effects (FE) estimator can be quite attractive because it typically retains the small variance of the FE estimator compared to GMM estimators.

My new command, xtdpdbc, implements the bias-corrected method of moments estimator described by Breitung, Kripfganz, and Hayakawa (2021). It analytically corrects the first-order condition of the FE estimator, which leads to a set of nonlinear moment conditions that can be solved with conventional numerical methods (Gauss-Newton). Another advantage of this procedure is that a formula of the asymptotic variance-covariance matrix for the calculation of standard errors is readily available, unlike the bias-corrected estimator by Kiviet (1995) that is implemented in the community-contributed xtlsdvc command by Bruno (2005).

Yet another advantage is that the estimator can accommodate higher-order lags of the dependent variable. Moreover, the moment conditions can be adjusted to create a random-effects (RE) version of the estimator, assuming that all (or some) of the exogenous regressors are uncorrelated with the unobserved group-specific effects. This RE version is not yet implemented in xtdpdbc, but will be added in due course.

In turns out that under the FE assumption the bias-corrected method of moments estimator is equivalent to the Dhaene and Jochmans (2016) adjusted profile likelihood estimator. Furthermore, if there is only a single lag of the dependent variable, it is also equivalent to the bias-corrected estimator of Bun and Carree (2005).

It should be noted that due to the nonlinearity of the bias-corrected moment functions, the estimator in general has multiple solutions and the numerical algorithm may not always converge to the correct one. The correct solution is characterized by a negativity condition on the gradient, i.e. all eigenvalues of the gradient should be negative. In the current version of xtdpdbc, the command will display a note if the gradient has positive eigenvalues. In that case, the estimation should be repeated with different starting values (using the from() option) until the correct solution is found. Starting values for the coefficient of the lagged dependent variable should typically be varied over the interval [0, 1]. Starting values for the exogenous regressors do not matter much.

In some cases, the numerical algorithm might not converge due to an almost flat criterion function. In such a case, it might help to simplify the optimization problem by concentrating out the coefficients of the exogenous regressors with option concentration. If this does not help, formal convergence could possibly be achieved by declaring the option nonrtolerance. However, the results in that case might not be very robust.

Last but not least, the command also supports unbalanced panel data.

To install the command, type the following in Stata's command window:

Code:

net install xtdpdbc, from(http://www.kripfganz.de/stata/)

Code:

help xtdpdbc

Code:

. webuse psidextract . xtdpdbc lwage wks south smsa ms exp exp2 occ ind union, lags(2) Bias-corrected estimation Iteration 0: f(b) = .00415219 Iteration 1: f(b) = 7.766e-06 Iteration 2: f(b) = 2.040e-09 Iteration 3: f(b) = 2.132e-16 Group variable: id Number of obs = 2975 Time variable: t Number of groups = 595 Obs per group: min = 5 avg = 5 max = 5 ------------------------------------------------------------------------------ lwage | Coefficient Std. err. z P>|z| [95% conf. interval] -------------+---------------------------------------------------------------- lwage | L1. | .2777891 .0713708 3.89 0.000 .1379049 .4176733 L2. | .0777857 .0411693 1.89 0.059 -.0029045 .158476 | wks | -.0000815 .0014887 -0.05 0.956 -.0029992 .0028363 south | .0828634 .0950579 0.87 0.383 -.1034466 .2691735 smsa | -.0304335 .0293295 -1.04 0.299 -.0879182 .0270513 ms | -.0096381 .0294365 -0.33 0.743 -.0673326 .0480565 exp | .06042 .012486 4.84 0.000 .0359478 .0848921 exp2 | -.0002095 .0001089 -1.92 0.054 -.0004229 3.86e-06 occ | -.029654 .0222952 -1.33 0.183 -.0733517 .0140437 ind | .0189437 .025248 0.75 0.453 -.0305414 .0684289 union | -.0044655 .030205 -0.15 0.882 -.0636661 .0547351 _cons | 3.283092 .5078034 6.47 0.000 2.287815 4.278368 ------------------------------------------------------------------------------

References:

- Breitung, J., S. Kripfganz, and K. Hayakawa (2021). Bias-corrected method of moments estimators for dynamic panel data models.
*Econometrics and Statistics*, forthcoming. - Bruno, G. S. F. (2005). Estimation and inference in dynamic unbalanced panel-data models with a small number of individuals.
*Stata Journal 5*(4), 473-500. - Bun, M. J. G., and M. A. Carree (2005). Bias-corrected estimation in dynamic panel data models.
*Journal of Business & Economic Statistics 23*(2), 200-210. - Dhaene, G., and K. Jochmans (2016). Likelihood inference in an autoregression with fixed effects.
*Econometric Theory 32*(5), 1178-1215. - Kiviet, J. F. (1995). On bias, inconsistency, and efficiency of various estimators in dynamic panel data models.
*Journal of Econometrics 68*(1), 53-78.

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