This paper compares the performance of three recently proposed estimators for dynamic panel data models (LSDV bias-corrected, MLE and MDE) along with GMM. Using Monte-Carlo, we find that MLE and biascorrected estimators have the smallest bias and are good alternatives for the GMM. System-GMM outperforms the rest in ‘difficult’ designs. Unfortunately, bias-corrected estimator is not reliable in these designs which may limit its applicability.econometrics;

Lokshin, Boris

English

Research Papers in Economics

Boris Lokshin 
 
Monte-Carlo comparison of alternative estimators 
for dynamic panel data models 
 
RM/06/014 
 
 
JEL code: C23 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Maastricht research school of Economics 
of TEchnology and ORganizations 
 
Universiteit Maastricht 
Faculty of Economics and Business Administration 
P.O. Box 616 
NL - 6200 MD Maastricht 
 
phone  : ++31 43 388 3830 
fax  : ++31 43 388 4873  
 
 
 
 1
 Monte-Carlo comparison of alternative estimators for
dynamic panel data models
Boris Lokshin
a
aFaculty of Economics and Business Administration, University of Maastricht,
and UNU-MERIT
PO Box 616, 6200 MD Maastricht, The Netherlands
E-mail: B.Lokshin@os.unimaas.nl
Abstract
This paper compares the performance of three recently proposed estimators for dynamic panel data models
(LSDV bias-corrected, MLE and MDE) along with GMM. Using Monte-Carlo, we find that MLE and bias-
corrected estimators have the smallest bias and are good alternatives for the GMM. System-GMM outperforms
the rest in ‘ difficult’  designs. Unfortunately, bias-corrected estimator is not reliable in these designs which may
limit its applicability.
Keywords: Bias correction, Dynamic panel data, GMM, MLE
JEL classification: C232
1. Introduction.
The purpose of this note is to evaluate several recently proposed techniques for estimating dynamic
panel data models when T is small. Such panels are characteristic of many microeconomic panel data
sets used in applied research. Three of the recently proposed estimators for such samples, LSDV-bias-
corrected (Bun and Carree, 2005), transformed fixed-effects MLE and MDE (Hsiao et al., 2002) are
compared to each other and to the two, difference and system, GMM estimators (Blundell and Bond,
1998). When it comes to the estimation of a dynamic panel data model, GMM is often used, not in the
least because the routine is now available in such popular packages as Stata, EViews, Gauss, PcGive
and Limdep
1. One drawback of the GMM is that it requires additional decisions by the researcher on
the instruments to be used, and the estimator’ s growing finite-sample imprecision with an increase in
instruments used.
We conduct Monte-Carlo simulations in which our goal is to provide a recommendation for an
applied researcher on an appropriate technique. In particular, we focus on two issues: the accuracy of
the previous studies’  assessments that the new estimators are more precise than an established GMM;
and second, the practicality of the new estimation techniques.  The first issue is relevant because the
previous comparisons were incomplete (only difference GMM was considered in previous
simulations) or presented selectively. For example, the present only those results where g  MLE
converged to a value less than unity and compare MLE to difference GMM only.
The second issue, practicality of the techniques, is especially relevant for applied researchers.
We show for example, that the bias-corrected estimator may not have a solution altogether, while the
MLE has been reported to have convergence problems in certain designs
2. We investigate how severe
these problems are, which in fact may affect the attractiveness of these estimators for an applied
researcher.
The rest of the paper is organized as follows. Section two presents the empirical model and
describes the Monte-Carlo simulation designs. Section three discusses results and section four presents
an empirical example. Section five concludes.
2. The problem and methodology
We consider the dynamic fixed effects model
T t N i x y y it i it t i it ,...., 1 ; ,...., 1 , 1 , = = + + + = - e h b g (2.1)
The dependent variable, yit, is determined by the one-period own lag, an exogenous regressor, xit , an
unobserved individual effect i h  , and a random disturbance 0 ), , 0 ( ~
2 2 > e e s s e N it . We assume that xit
is not correlated with the general disturbance term.
In our Monte-Carlo experiments we follow Kiviet (1995).  Data for yit are generated by model (2.1)
and the data for xit are generated by
1 Stata routine -xtabond2- written by David Roodman, Center for Global Development, Washington, DC is
capable of estimating both system and difference GMM. E-views 5.0 release has a difference GMM available.
Gauss and PcGive have an original DPD. Limdep 8.0 can estimate a form of difference GMM.
2 In particular, Bun and Carree (2005) state that the BC estimator has convergence problems when the signal-to-
noise ratio is low for relatively high values of g  (p. 207). Hsiao et al. (2002) state that “ MLE sometimes breaks
down completely…  [when T and N are small] ”  (p. 132).  See the footnote 3 below for our approach to tracking
non-convergence cases.3
T t N i N x x it it t i it ,..., 1 and ,..., 1 ), , 0 ( ~ ,
2
1 , = = + = - x s x x r (3.1)
The individual effects i h  are generated by assuming ) , 0 ( ~
2
h s h N i  while e s is normalized to unity.
In addition to r s b x , and
2 also determines the correlation between yit and xit and is set at values 0.8
and 0.2.  In Kiviet (1995) it is argued that the relative bias of the estimators is significantly influenced
by
2
s s , the signal-to-noise ratio of the regression. In our experiments we use a combination of
relatively high
2
s s =8 with high and low correlation and relatively low
2
s s =2 with high and lowr .
Two different sample sizes are considered (N, T) = (100, 6) and (N, T) = (100, 3). Table 1 summarizes
the parameters for each of the 16 panel designs.  In generating yit and xit we set yi,-50, xi, -50 =0 and
discarded the first 50 observations, using the observations t through T for estimation. We performed
1000 replications with a fixed seed for the random number generation.
3. Monte-Carlo results
Th e  re sul ts fo r the  16 design s are  sum m ari zed in  Tabl e s 2, an d  3. U nfo rtun ately , whi le the  bi as-
corrected estimator may produce superior results in terms of bias, it is not always practical. When the
signal-to-noise ratio
2
s s  i s l ow  (de sign s 3, 4 , 11 , 12)  th e re  i s a l arg e  pe rcen tage o f case s w i th no
solution
3. The problem is exacerbated when T is relatively low, and can lead to almost 45% of such
cases. We note that even with high
2
s s , the combination of relatively high g  and r  can lead to high
share of no-solution scenario when T is relatively small (design 15).  MDE and MLE estimation
involves estimation of KT auxiliary parameters (formula B.1 in Hsiao et al., 2002), which makes
matrix i W
~
D  relatively large even for problems of moderate size. When N is not much larger than KT,
some authors suggest projecting 1 i y D  on i x D   rather than on i x D . We computed the version of MLE
and MDE where i x D  is used in place of i x D  . However, this change leads to larger bias for b  in MLE
and MDE, while bias for g  is actually somewhat smaller.  These additional results are not included in
the tables and are available upon request.
3 To avoid over- or under-reporting of non-convergence of the BC estimator, in our version of the program we
rely on the exact solution, rather than on the iterative procedure proposed in the original paper. The system of
equations (19) and (20) (see original paper) can be transformed into a polynomial of power T. We solve this
polynomial with respect tog . Then we use (20) to solve for trueb .  Note, that when T is odd the polynomial
always has at least one real root, when T is even, it may have zero real roots and T complex roots. In our Monte-
Carlo experiments, for the designs when T=6 we count the number of cases when the polynomial has no real
roots, this is reported in Table 2; in designs when T=3, when there is at least one real root, we count as non-
convergence those solutions that are smaller than g -LSDV. The unique solution to the polynomial when T=3
and the root is less than g -LSDV is a negative number between -3 and -4. Convergence for MLE has been
always achieved. The starting values for MLE were the MDE estimates. We used NLPDD Double Dogleg
optimization routine in SAS for MLE and used the outer product of gradient to compute variance. Estimates for
MDE are obtained from B.5 based on initial
2
u s andg , b from Anderson-Hsiao IV estimator, using
2 , - t i y  and
) ( 1 , , - - t i t i x x  as instruments.4
4. Application to US unemployment dynamics
We applied the various methods for estimating model (2.1) to the data set used in Bun and Carree
(2005) for an analysis of unemployment dynamics at the US state level observed over 1991-2000. In
the model the current unemployment rate is a function of the previous period’ s unemployment and
economic growth rate. The equation also includes the time and individual fixed effects.
Table 4 presents the recomputed results (original paper) for the BC estimator along with the MLE,
MDE and GMM, estimators that we test in a Monte Carlo.  For the GMM results the lagged growth
rate is treated as exogenous and an IV-style instrument for it is used. MLE and BC estim ates are
virtually identical, implying an adjustment rate of about 40% a year. The coefficient on the lagged
dependent variable is higher for both GMM estimators, while the effect of economic growth is smaller
compared to the bias-corrected estimates. The former result would suggest a lower adjustment rate
between 23% and 30% a year.
5. Conclusions
We have three main conclusions. MLE has the lowest average bias for g  and has the highest average
ranking among all the estimators in all designs considered
4. System-, but not difference-GMM
performs better than the other estimators in “ difficult”  designs, with low signal-to-noise ratio coupled
with high values ofg ; it has also the lowest RMSE for g  and b  among considered estimators. Bias-
corrected estimator can match MLE in terms of bias, and System-GMM in terms of RMSE, but
unfortunately has a very high percentage of non-convergence in “ difficult”  designs, which makes its
use in such cases problematic for applied researchers.
Acknowledgement: I thank Martin Carree for useful comments
4 The ranking is computed by giving 5 points for the first place, 4 for the second and so on in each of the designs,
and is based on average cumulative score for biases on g and b .5
References
Bun, M. and Carree M. (2005) Bias-corrected estimation in dynamic panel data models. Journal of
Business and Economic Statistics, 23(2), 200-10.
Blundell, R. and Bond S. (1998) Initial conditions and moment restrictions in dynamic panel data
models, Journal of Econometrics, 87, 115-143.
Hsiao, C., Pesaran, H. and Tahmiscioglu A. (2002) Maximum likelihood estimation of fixed effects
dynamic panel data models covering short time periods. Journal of Econometrics 109, 107-150.
Kiviet, J., (1995) On bias, inconsistency and efficiency of various estimators in dynamic panel data
models. Journal of Econometrics 68, 53-78.6
Table 1 Summary of Monte-Carlo simulation parameters, N=100, e s =1.
Design
number
T g b 2
s s r
x s h s % break-
downs
BC
1 6 0.2 0.8 2 0.8 0.88 0.8 0
2 2 0.2 1.61 0.8 0
3 6 0.8 0.2 2 0.8 0.40 0.2 26.8
4 2 0.2 1.18 0.2 23.2
5 6 0.2 0.8 8 0.8 1.76 0.8 0
6 8 0.2 3.25 0.8 0
7 6 0.8 0.2 8 0.8 2.10 0.2 0.6
8 8 0.2 6.24 0.2 0
9 3 0.2 0.8 2 0.8 0.88 0.8 0
10 2 0.2 1.61 0.8 0
11 3 0.8 0.2 2 0.8 0.40 0.2 43.0
12 2 0.2 1.18 0.2 41.0
13 3 0.2 0.8 8 0.8 1.76 0.8 0
14 8 0.2 3.25 0.8 0
15 3 0.8 0.2 8 0.8 2.10 0.2 17.3
16 8 0.2 6.24 0.2 0.27
Table 2 Bias estimates and RMSE for g  using various estimators*
Des. g  bias g  RMSE
GMM1 GMM2 MLE MDE BC
* GMM1 GMM2 MLE MDE BC
1 -0.0199   0.0228 -0.0015 -0.0089 -0.0015 0.0044  0.0054 0.0016  0.0020  0.0016
2 -0.0073   0.0108 -0.0004 -0.0049 -0.0004 0.0016  0.0016 0.0008  0.0009  0.0008
3 -0.0611 -0.0028   0.0250 -0.0379 -0.0357 0.0143  0.0029 0.0151  0.0106  0.0060
4 -0.0551 -0.0014   0.0239 -0.0344 -0.0275 0.0124  0.0027 0.0136  0.0095  0.0052
5 -0.0127   0.0151 -0.0008 -0.0045 -0.0008 0.0026  0.0040 0.0007  0.0008  0.0007
6 -0.0025   0.0036 -0.0001 -0.0015 -0.0001 0.0005  0.0005 0.0003  0.0003  0.0003
7 -0.0590 -0.0075   0.0030 -0.0154   0.0017 0.0133  0.0035  0.0034  0.0073  0.0030
8 -0.0189 -0.0011   0.0004 -0.0076   0.0005 0.0032  0.0010  0.0010  0.0017  0.0010
9 -0.0226 -0.0021   0.0009 -0.0010   0.0003 0.0190  0.0141  0.0073  0.0099  0.0069
10 -0.0090   0.0008   0.0007 -0.0024   0.0005 0.0061  0.0049  0.0032  0.0035  0.0033
11 -0.0687 -0.0281   0.0223 -0.0250 -0.1451 0.0863  0.0143 0.0434  0.0496  0.0337
12 -0.0616 -0.0212   0.0323 -0.0211 -0.1246 0.0764  0.0127 0.0448  0.0457  0.0287
13 -0.0158 -0.0051 -0.0005   0.0007 -0.0005 0.0140  0.0148 0.0027  0.0039  0.0027
14 -0.0029   0.0004   0.0004 -0.0006   0.0003 0.0019  0.0017  0.0010  0.0010  0.0010
15 -0.0787 -0.0431   0.0330   0.0085 -0.0253 0.1100  0.0299  0.0310  0.0498  0.0137
16 -0.0222  -0.0087   0.0079   0.0069   0.0062 0.0223  0.0048  0.0076  0.0120  0.0065
AB  0.0324 0.0109 0.0096 0.0113 0.0232 0.0243 0.0074 0.0111 0.0130 0.0072
* The numbers for the BC estimator are based only on those cases where the exact solution was found (designs
3, 4, 10, 11, 15).
Table 3 Bias estimates and RMSE for b  using various estimators*
Des. b  bias b  RMSE
GMM1 GMM2 MLE MDE BC
GMM1 GMM2 MLE MDE BC
1   0.0050  -0.0152 0.0017   0.0040   0.0017 0.0042  0.0041  0.0025  0.0025  0.0025
2 -0.0004 -0.0010 0.0005   0.0003   0.0005 0.0011  0.0009  0.0008  0.0008  0.0008
3   0.0084   0.0002  0.0037   0.0032   0.0076 0.0209  0.0056  0.0120  0.0112  0.0110
4 -0.0032   0.0004 0.0034  -0.0014   0.0006 0.0020  0.0014  0.0016  0.0015  0.0014
5   0.0024  -0.0097 0.0008   0.0021   0.0008 0.0010  0.0019  0.0007  0.0007  0.0007
6   0.0000  -0.0003 0.0002   0.0002   0.0002 0.0002  0.0002  0.0002  0.0002  0.0002
7   0.0014   0.0024  0.0007   0.0009   0.0007 0.0007  0.0006  0.0004  0.0004  0.0004
8 -0.0012   0.0001 0.0002  -0.0003   0.0002 0.0001  0.0000  0.0001  0.0001  0.0001
9   0.0015   0.0002  0.0038   0.0033   0.0038 0.0128  0.0085  0.0098  0.0099  0.0097
10 -0.0011   0.0000  0.0021   0.0014   0.0020 0.0033  0.0019  0.0024  0.0025  0.0025
11   0.0063   0.0094  0.0081   0.0061   0.0117 0.0624  0.0134  0.0506  0.0480  0.0406
12 -0.0029   0.0012 0.0061   0.0004 -0.0056 0.0063  0.0032  0.0050  0.0047  0.0039
13   0.0005   0.0025  0.0020   0.0017   0.0020 0.0033  0.0063  0.0024  0.0025  0.0024
14 -0.0001   0.0002  0.0009   0.0007   0.0009 0.0008  0.0005  0.0006  0.0006  0.0006
15 -0.0025   0.0144  0.0028   0.0014   0.0017 0.0026  0.0042  0.0018  0.0018  0.0017
16 -0.0016   0.0002  0.0012   0.0011   0.0010 0.0004  0.0001  0.0002  0.0003  0.0002
AB  0.0024 0.0036 0.0024 0.0018 0.0026 0.0076 0.0033 0.0057 0.0055 0.0049
*The numbers for the BC estimator are based only on those cases where the exact solution was found (designs 3,
4, 10, 11, 15).8
Table 4 Empirical results for the unemployment-growth model
BC MLE
Fixed effects
a
Minimum
distance
GMM1
b GMM2
c
g  0.6152***  0.6167***  0.5926***  0.6919***  0.7746***
SE(g )  0.0463  0.0336  0.0335  0.0728  0.0631
b -0.0567*** -0.0566*** -0.0579*** -0.0433*** -0.0430***
SE(b )  0.0120  0.0119  0.0118  0.0164  0.0130
LL 1747.88
a. Standard errors are square roots of diagonals of the negative inverse of the Hessian
b. GMM1 is the difference, one-step estimator (Arellano and Bond)
c. GMM2 is the system, one-step estimator (Blundell and Bond)
*** Indicates significance at 99% level

Bias-corrected estimation in dynamic panel data models.

Initial conditions and moment restrictions in dynamic panel data models,

Maximum likelihood estimation of fixed effects dynamic panel data models covering short time periods.

On bias, inconsistency and efficiency of various estimators in dynamic panel data models.

Monte-Carlo comparison of alternative estimators for dynamic panel data models

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Monte-Carlo comparison of alternative estimators for dynamic panel data models

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