Panel data characterized by groupwise heteroscedasticity, cross-sectional correlation, and AR(1) serial correlation pose problems for econometric analyses. It is well known that the asymptotically efficient, FGLS estimator (Parks) sometimes performs poorly in finite samples. In a widely cited paper, Beck and Katz (1995) claim that their estimator (PCSE) is able to produce more accurate coefficient standard errors without any loss in efficiency in ¡°practical research situations.¡± This study disputes that claim. We find that the PCSE estimator is usually less efficient than Parks -- and substantially so -- except when the number of time periods is close to the number of cross-sections.Panel data estimation; Monte Carlo analysis; FGLS; Parks; PCSE; finite sample

Shu Lin

W. Robert Reed

Xiujian Chen

English

Research Papers in Economics

DEPARTMENT OF ECONOMICS 
 
COLLEGE OF BUSINESS AND ECONOMICS 
 
UNIVERSITY OF CANTERBURY 
 
CHRISTCHURCH, NEW ZEALAND 
 
 
 
A MONTE CARLO EVALUATION OF THE EFFICIENCY 
 
OF THE PCSE ESTIMATOR  
 
 
by 
 
 
Xiujian Chen 
Department of Economics 
California State University, 
Fullerton 
Shu Lin 
Department of 
Economics 
Florida Atlantic 
University 
W. Robert Reed* 
Department of 
Economics 
University of 
Canterbury 
 
 
  
 
 
 
 
 
WORKING PAPER 
 
 
No. 14/2006 
 
 
 
Department of Economics, College of Business and Economics 
University of Canterbury, Private Bag 4800, Christchurch 
New ZealandWORKING PAPER No. 14/2006 
 
 
 
A MONTE CARLO EVALUATION OF THE EFFICIENCY 
 
OF THE PCSE ESTIMATOR 
 
by  
 
Xiujian Chen 
Department of Economics 
California State University, Fullerton 
Shu Lin 
Department of Economics 
Florida Atlantic University 
W. Robert Reed* 
Department of Economics 
University of Canterbury 
 
 
November 3, 2006 
 
 
Abstract 
 
Panel data characterized by groupwise heteroscedasticity, cross-sectional correlation, and 
AR(1) serial correlation pose problems for econometric analyses.  It is well known that 
the asymptotically efficient, FGLS estimator (Parks) sometimes performs poorly in finite 
samples.  In a widely cited paper, Beck and Katz (1995) claim that their estimator 
(PCSE) is able to produce more accurate coefficient standard errors without any loss in 
efficiency in “practical research situations.”  This study disputes that claim.  We find that 
the PCSE estimator is usually less efficient than Parks -- and substantially so -- except 
when the number of time periods is close to the number of cross-sections.   
 
 
JEL Categories: C23, C15 
 
Keywords:  Panel data estimation, Monte Carlo analysis, FGLS, Parks, PCSE, finite 
sample 
 
 
*The corresponding author is W. Robert Reed, Professor of Economics, University of 
Canterbury, Private Bag 4800, Christchurch, New Zealand.  Email: 
bobreednz@yahoo.com.  Phone: +64 3 364 2846.  Fax: +64 3 364 2635. 
 
 
Acknowledgments:  An earlier draft of this paper was presented at the University of 
Oklahoma, and the 11
th International Conference on Panel Data.  We acknowledge 
helpful comments from Kevin Grier, Cynthia Rogers, and Aaron Smallwood. 
  11.   INTRODUCTION 
 
Panel data characterized by heteroscedasticity, serial correlation, and cross-sectional 
correlation raise serious issues for econometric analyses.  An oft-employed procedure for 
data of this sort is the FGLS estimator proposed by Parks (1968).  When the data 
generating process (DGP) is characterized by groupwise heteroscedasticity, time-
invariant cross-sectional correlation, and first-order (AR[1]) serial correlation, the Parks 
estimator is asymptotically efficient.
1  It is well-known, however, that the Parks estimator 
can sometimes perform poorly in finite samples, particularly with respect to estimating 
coefficient standard errors. 
A recent paper by Beck and Katz (1995) -- henceforth BK -- proposes an 
alternative, two-step estimator.  In the first step, the data are transformed to eliminate 
serial correlation.
2  In the second step, OLS is applied to the transformed data, and the 
standard errors are corrected for cross-sectional correlation.  Based on their Monte Carlo 
analyses, BK conclude that their “panel-corrected standard error” (PCSE) estimator 
produces more accurate standard error estimates, without any loss in efficiency.  It is 
noteworthy that the PCSE procedure assumes the same error variance-covariance matrix 
(and estimates the same parameters) as the Parks estimator.
3  
If BK’s results were generalizable to actual panel data, it would be a very useful 
finding.  It promises an important benefit without any cost.  Indeed, the paper and 
corresponding PCSE estimator have been highly influential.  A recent count identifies 
over 500 citations of BK on Web of Science (e.g. Ferguson and Schularick, 2006; 
                                                 
1 For example, in STATA, the “xtgls” options (i) “panels(hteroscedastic)”, (ii) “panels(correlated)” and (iii) 
“corr(ar1/psar1)” correspond to these three types of nonspherical error variance-covariance behaviors. 
2 The “xtpcse” procedure in STATA uses a Prais-Winsten transformation in this first stage. 
3 As a result, the better performance of the PCSE estimator cannot be attributed to the “shrinkage principle” 
(Diebold, 2004, page 45). 
  2Yermack, 2006; Dejuan and Luengo-Prado, 2006; and Lapré and Tsikriktsis, 2006).   
Further, the PCSE estimator is now a standard procedure in many statistical software 
packages, including STATA, GAUSS, RATS, and Shazam.   
Unfortunately, our analysis is unable to confirm BK’s efficiency results.  Using a 
different set of Monte Carlo parameters patterned after actual panel data, we find that the 
PCSE estimator is almost always less efficient than Parks, often substantially so.   
2.   THE DATA GENERATING PROCESS AND A MEASURE FOR RELATIVE  
      EFFICIENCY 
 
Following BK, we assume that the DGP is given by:   
(1)   , or  
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
+
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
+
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
N
2
1
N
2
1
N
2
1
ε
ε
ε
X
X
X
i
i
i
y
y
y
M M M M
x 0 β β ε X i y NT + + = x 0 β β , 
where N and T are the number of cross-sectional units and time periods, respectively;   
is a   vector of observations of the dependent variable for the i
i y
1 T ×
th cross-sectional unit;  
 is a  i 1 T ×  vector of ones; Xi  is a  1 T ×   vector of observations of the explanatory 
variable;  0 β  and  x β  are scalars; and  i ε  is a  1 T ×  vector of error terms, where ε  ~ N(0, 
NT Ω ).   
  The error structure,  NT Ω , is based on the Parks model (Parks, 1967).  It assumes 
(i) groupwise heteroscedasticity, (ii) first-order serial correlation, and (iii) time-invariant 
cross-sectional correlation, imposing the following specification for  NT Ω : 
(2)  Π Σ Ω ⊗ = NT ,  
  3where  , 
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
NN N2 N1 ε,
,2N 22 21
1N 12 11
, ,
, ,
, , ,
ε ε
ε ε ε
ε ε ε
σ σ σ
σ σ σ
σ σ σ
L
M O M M
L
L
Σ
  , 
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
− − −
−
−
−
1 ρ ρ ρ
ρ 1 ρ ρ
ρ ρ 1 ρ
ρ ρ ρ 1
3 T 2 T 1 T
3 T 2
2 T
1 T 2
L
M O M M M
L
L
L
Π it 1 t i it u + = − , ρε ε , and 
2
ij u,
ij ε, 1
σ
σ
ρ −
= . 
There are a total of 
2
2 N N
2 + +
 unique parameters in  NT Ω  (the  ’s and  ij , σε ρ ).  
Each of these must be given a value in order to generate simulated data.  BK emphasize 
that their Monte Carlo analyses were designed to provide guidance to researchers using 
panel data sets that are likely to be encountered in “practical research situations.”   
However, even a moderately sized panel data set of 10 cross-sectional units requires 
setting more than 50 unique parameters.  How is one to know whether the parameter 
values chosen by the researcher for the Monte Carlo simulations are those that typify 
practical research situations?   
Our approach is to take estimated values of these parameters from previous 
research we have done.  In particular, we use (i) real, per capita personal income data 
from U.S. states; and (ii) international real, per capita GDP data.  Further, we work with 
studies in which the dependent variable is in (i) level and (ii) difference (growth) form.  
And, since these studies derived their estimates of the  ’s and  ij , σε ρ  using residuals from 
regression equations, we reference two different regression specifications.
4   
                                                 
4 The main difference in the two regression specifications is that version 1 includes cross-sectional fixed 
effects, while version 2 includes both cross-sectional and time period fixed effects.  To give an idea of the 
  4This allows us to create eight different artificial data environments, each patterned 
after U.S. or international data (INT), income data that are either in level (L) or difference 
(D) form, and a particular type of residual-producing regression specification (1 or 2).  
We designate these US-L1, US-D1, US-L2, US-D2, INT-L1, INT-D1, INT-L2, and INT-
D2.  Our Monte Carlo experiments set values for the  ’s and  ij , σε ρ  so that they “look 
like” estimated values from actual panel sets of a particular size (N,T) for each of the 
eight data environments.  
  Like BK, we generate 1000 simulated panel data sets for each (N,T) experiment.  
For every panel data set (replication), we estimate  x β  in Equation (1) using both Parks 
and PCSE.
5  Define   as the true population value of 
*
x β x β , and   and   as the 
Parks and PCSE estimates of 
(r)
Parks β ˆ (r)
PCSE β ˆ
x β  for a given replication, r.  We compare the efficiency 
performance of the two estimators using BK’s measure of relative efficiency:   
( )
() ∑
∑
=
=
−
−
⋅ =
1000
1 r
2
x
(r)
PCSE
1000
1 r
2
x
(r)
Parks
100 Efficiency   Relative
*
*
ˆ
ˆ
β β
β β
. 
“Relative Efficiency” values less than 100 indicate that PCSE is less efficient than Parks. 
3.   RESULTS 
  Table 1 reports the results of our Monte Carlo simulations.  The presentation is 
patterned after Table 5 (page 642) in BK.  BK report that the PCSE estimator is either 
more efficient, or only slightly less efficient, than the Parks estimator except for “extreme 
                                                                                                                                                 
difference this makes for the residuals, the R
2 associated with the first specification usually ran around 0.60 
for the US-Level data, compared to R
2 values that were typically over 0.90 using the second specification. 
5 All the programs used for this analysis were written in SAS/IML.  The formulae for the Parks and PCSE 
estimators were constructed to exactly match the output from STATA’s “xtgls” and “xtpcse” procedures, 
using the (i) “panels(correlated)” and “corr(ar1)”, and (ii) “correlation(ar1)” options, respectively (we note 
that the default cross-sectional correlation option for the “xtpcse” option is groupwise heteroscedasticity 
and time-invariant cross-sectional correlation). 
  5cases” (page 645) where the “average contemporaneous correlation is at least 0.50 and 
the time sample is quite long” (page 642).  In contrast, we find that the PCSE estimator is 
almost always less efficient than Parks, sometimes substantially so.  For panel data sets of 
size N=10 and T=20, we find that “Relative Efficiency” is less than 50% in four of the 
eight artificial data environments, and never higher than 74%.   
As expected, we find that the efficiency advantage of Parks increases 
monotonically with T.  As T increases, there are more observations available to estimate 
each cross-sectional covariance term.  This increases the reliability of the FGLS 
estimates, enhancing the associated efficiency advantages.  With a few exceptions, the 
efficiency of the PCSE estimator approaches that of Parks only when T is very close to 
N.    
Why are our results different from those of BK?  Table 2 makes it clear that it is 
not because our simulated data are driven by extreme values of either serial correlation or 
cross-sectional correlation.  Each cell reports the (i) average correlation coefficient and 
(ii) average, absolute value of the cross-sectional correlation terms for the 1000 data sets 
corresponding to that experiment.  There is a wide range of serial correlation behavior 
across the different artificial data environments.  Further, most of the simulated data sets 
(cf. the last six columns) have average contemporaneous correlation values well below 
the 0.50 value that BK identify as problematic.   
Most likely, the strong performance of the PCSE estimator reported by BK is an 
artifact of the particular parameter values they selected for their simulations.  As 
discussed above, even small panel data sets with relatively few cross-sectional units have 
a large number of unique parameters in the error variance-covariance matrix.  It is 
  6difficult to know how one should set these parameters.  For example, the average 
contemporaneous correlation may be less important for determining efficiency than the 
dispersion of the cross-sectional covariance values.  Further, it is unclear which particular 
combinations represent “practical data situations.”
6  For these reasons, we think that 
simulations using error variance-covariance parameter values that are patterned after real 
panel data sets provide a better means of evaluating likely estimator performance in 
“practical research situations.” 
4.   CONCLUSION 
This paper evaluates the efficiency of the PCSE estimator.  Beck and Katz (1995) claim 
that the PCSE estimator is more efficient, or only slightly less efficient, than the Parks 
estimator except for extreme cases that researchers are unlikely to encounter in practice. 
In contrast, this study finds that the PCSE estimator is usually less efficient than 
Parks -- and substantially so -- except when the number of time periods is close to the 
number of cross-sections.  Our findings are consistent across a wide variety of Monte 
Carlo environments patterned after actual panel data.  They indicate that researchers 
should be aware that use of the PCSE estimator may come at a considerable cost in 
efficiency. 
 
 
                                                 
6 For example, the simulations underlying the Relative Efficiency results of BK’s Table 5 assume no serial 
correlation. 
 
  7  8
REFERENCES 
 
Beck, Nathaniel and Jonathan N. Katz, 1995, What to do (and not to do) with time-series 
cross-section data, American Political Science Review 89, 634-647.  
 
Dejuan, Joseph P. and Maria Jose Luengo-Prado, 2006, Consumption and aggregate 
constraints: international evidence, Oxford Bulletin of Economics and Statistics 
68, 81-99. 
 
Diebold, Francis X, 2004, Elements of Forecasting, 3
rd Edition. (South-Western, Mason: 
Ohio). 
 
Ferguson, Niall and Moritz Schularick, 2006, The empire effect: the determinants of 
country risk in the first age of globalization: 1880-1913, Journal of Economic 
History 66, 283-312. 
 
Lapré, Michael A. and Nikos Tsikriktsis, 2006, Organizational learning curves for 
customer dissatisfaction: heterogeneity across airlines, Management Science 52, 
352-366. 
 
Parks, Richard, 1967, Efficient estimation of a system of regression equations when 
disturbances are both serially and contemporaneously correlated, Journal of the 
American Statistical Association 62, 500-509. 
 
Yermack, David, 2006, Flights of fancy: corporate jets, CEO perquisites, and inferior 
shareholder returns, Journal of Financial Economics 80, 211-242. 
 
 Table 1: 
Relative Efficiency of PCSE Compared to Parks (%) 
 
   Data Generating Process
c
N
a T
b US-L1 US-G1 US-L2 US-G2 INT-L1  INT-G1  INT-L2  INT-G2 
            
5  10  99.4 95.7 61.8 56.7  108.9  99.9 61.5 57.6 
  15  78.0 95.3 51.0 42.1 94.2 89.0 48.0 43.1 
  20  55.5 83.3 41.2 33.1 94.4 80.9 38.8 31.7 
  25  52.4 74.5 34.5 26.6 94.6 74.3 31.8 25.9 
            
10  10  98.4 96.5 94.1 93.5 93.9 94.7 93.9 92.1 
  15  94.0 88.0 71.5 67.9 82.3 84.8 71.2 66.6 
  20  74.3 81.5 59.3 68.0 76.0 78.2 57.6 52.5 
  25  54.4 66.1 49.6 44.7 71.3 74.0 49.9 43.9 
            
20  20  91.1 96.1 97.1 96.8 97.5 95.9 97.3 95.9 
  25  80.0 78.8 82.8 79.5 86.5 81.3 82.4 78.6 
            
 
NOTE:   ( )
() ∑
∑
=
=
−
−
⋅ =
1000
1 r
2
x
(r)
PCSE
1000
1 r
2
x
(r)
Parks
100 Efficiency   Relative
*
*
ˆ
ˆ
β β
β β
.  Values less than 100% indicate that PCSE is less efficient than Parks. 
a Number of cross-sectional units. 
b Number of time periods. 
c Indicates the type of actual panel data after which the simulated data are patterned.  See Section 2 for category definitions. 
 
  9Table 2: 
Mean Serial Correlation and Mean Cross-Sectional Correlation of Simulated Data Sets 
 
   Data Generating Process
c
N
a T
b US-L1 US-G1 US-L2 US-G2 INT-L1  INT-G1  INT-L2  INT-G2 
            
5  10  0.35 / 0.65  0.09 / 0.59  0.42 / 0.36  0.19 / 0.34  0.41 / 0.36  -0.09 / 0.27  0.38 / 0.38  -0.08 / 0.34 
  15  0.50 / 0.77  0.16 / 0.67  0.56 / 0.35  0.26 / 0.32  0.57 / 0.34  -0.06 / 0.23  0.56 / 0.36  -0.06 / 0.31 
  20  0.59 / 0.86  0.20 / 0.70  0.66 / 0.34  0.29 / 0.31  0.66 / 0.35  -0.04 / 0.20  0.67 / 0.34  -0.02 / 0.30 
  25  0.64 / 0.89  0.21 / 0.70  0.73 / 0.34  0.30 / 0.30  0.73 / 0.35  -0.04 / 0.19  0.74 / 0.33  -0.02 / 0.29 
            
10  10  0.38 / 0.62  0.08 / 0.57  0.50 / 0.34  0.09 / 0.31  0.47 / 0.39  -0.03 / 0.30  0.49 / 0.37  -0.04 / 0.31 
  15  0.53 / 0.71  0.17 / 0.65  0.63 / 0.31  0.16 / 0.28  0.62 / 0.37  0.01 / 0.27  0.66 / 0.35  0.00 / 0.28 
  20  0.62 / 0.79  0.23 / 0.66  0.73 / 0.30  0.19 / 0.27  0.71 / 0.37  0.03 / 0.26  0.75 / 0.35  0.01 / 0.27 
  25  0.69 / 0.81  0.24 / 0.66  0.79 / 0.28  0.20 / 0.26  0.76 / 0.37  0.03 / 0.24  0.79 / 0.35  0.02 / 0.25 
            
20  20  0.63 / 0.77  0.19 / 0.65  0.71 / 0.32  0.12 / 0.29  0.72 / 0.35  0.04 / 0.25  0.72 / 0.30  0.02 / 0.25 
  25  0.70 / 0.79  0.21 / 0.65  0.77 / 0.31  0.14 / 0.28  0.78 / 0.35  0.05 / 0.24  0.78 / 0.29  0.02 / 0.23 
            
 
NOTE:  Each cell summarizes serial and cross-sectional correlations of a 1000 simulated, panel data sets of respective size (N,T).  For 
each simulated data set, we estimate a value for ρ  and 
2
N N
2 +
 values for the respective  ’s.  The first number in each cell reports 
the average value of 
ij , σε
ρ ˆ  across the respective 10000 simulated data sets.  The second number reports the average of the absolute value 
of the respective estimates of  .   ij , σε
a Number of cross-sectional units. 
b Number of time periods. 
c Indicates the type of actual panel data after which the simulated data are patterned.  See Section 2 for category definitions. 
  10

Consumption and aggregate constraints: international evidence,

Efficient estimation of a system of regression equations when disturbances are both serially and contemporaneously correlated,

Flights of fancy: corporate jets, CEO perquisites, and inferior shareholder returns,

Organizational learning curves for customer dissatisfaction: heterogeneity across airlines,

The empire effect: the determinants of country risk in the first age of globalization: 1880-1913,

What to do (and not to do) with time-series cross-section data,

A Monte Carlo Evaluation of the Efficiency of the PCSE Estimator

http://www.econ.canterbury.ac.nz/RePEc/cbt/econwp/0614.pdf

A Monte Carlo Evaluation of the Efficiency of the PCSE Estimator

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics