Power Comparison of Some Tests for the Error Component Model Based on Panel Data M. Ishaq Bhatti Department of Statistics & O.R., University of Kuwait, P. O. Box 5969, Safat Kuwait City, Kuwait and School of International Business, Griffith University, Australia
[email protected] 1. Introduction In recent years, there has been an increasing use of regression analysis based on panel data in the area of applied statistics, econometrics, time series forecasting, and social and biological sciences. Often data arising from these areas are modelled by routine cross-sectional and time-series techniques without due regards being given to the complication of the data being double dimension and also contains individual, time, and temporal effects in the regression disturbances. Moreover, the data contain additional complications of repeated observations on individual households, firms, sectors, regions, and blocks over different time. The evaluation of statistical theory and methods as well as increasing and efficient means of data collections and computing facilities have made it possible to model effectively the most complicated structure of the data and then efficiently use it to make statistical inference about the population parameters. Recently, Rao et al (1993) and Baltagi (1996) used error component model to model survey and panel data, respectively. This paper uses a unified model which could be useful in modelling both survey and panel data. It derives the point optimal invariant (POI) test for testing individual (time-specific) and panel effects and computes the sizes and powers of some existing tests (DurbinWatsin (DW), one and two-sided lagranges multiplier (LM1) and LM2), the new optimal, (locally best (LB), locally mean most powerful (LMMP) and the POI) tests to assess the relative strength of the power performance. An empirical power comparison of these tests demonstrate that the power of the POI test is closest to the power envelop (PE) to be called as an approximately uniformly most powerful (UMP) invariant test for the error component model. 2. The Model and the Test Statistics Consider the linear regression model of the form p
y = ∑ β x + u , t = 1, 2, …, T, and i= 1, 2, …, N where u = µ +ν , in which N stands for the number of individuals (e.g. households) in the it
k
k =1
it
i
itk
it
it
sample and t = 1, 2, …, T stands for the length of the observed time series. Each of the µi's (I=1, .., N) are called an individual effects and vit is the usual error term (white noise). In this formulation, if we assume that i is the cluster or block identifier and t is the observation identifier in the given cluster then this model can be used to capture two stage cluster sample data (Bhatti (1995)). Further extension of this model may be done by incorporating number of individuals in households and the number of households within a region or block (J) over the length of observed time series, T. The basic outline of the model (1) has been drafted by the pioneers Balestra and Nerlove (1966) and then recently reviewed by Baltagi (1996) under the standard assumptions of µ and v
being mutually independent with
σ
2
= σ µ + σ ν and is called one way error component model. 2
2
The extension of this model to two- and multiple- way error component model is considered here. This paper tests the panel, individual and the regional effects. This study considered the testing of one effect in the presence of the others. For these testing problems various tests have been derived and their power and sizes have been studied. Particularly, we considered the DW, modified version of the DW, LM1, LM2, LB, LMMP and the POI tests. The power of the PO tests is being used as a bench mark to compute the PE for a particular testing problem and then gauge and compares its power performance with other tests to see which test has the closest power with the PE to be called as approximatel UMP test. It is of interest to note that the POI test optimise the power at a pre-determine point away from the null hypothesis in the alternative parameter space, S(.5) and/or S(.8) hereas LB optimise the power curve locally under the null while the LMMP maximise the average curvature of the power hypersurface in the neighbourhood of the null hypothesis (King and Wu (1992).. The brief summary of the power comparison of these tests is given below for the selected sample and the time periods to demonstrate its application using an example from selected panel data. 3. The Power Comparison In this comparison, it is observed that for one of our testing problems LM1 test is LB and LMMP for selected T, N and p. This study show that the POI test is superior than the existing tests in the arae of panel data modelling. Furthermore, in some selected N and T, it is observed that the POI test is UMP invariant. 7DEOHRIWKH3RZHUVRIYDULRXV7HVWVIRU7 S DQGα 8VLQJ$XVWUDOLDQ3DQHO'DWD Test Statistics ρ= N=32 PE POI LM1 LM2 DW N=64 PE POI LM1 LM2 DW
ρ = 0.1
0.3
0.5
.164 .163 .163 .118 .095
.442 .442 .427 .343 .245
.686 .684 .656 .678 .452
.297 .295 .296 .215 .147
.772 .772 .763 .689 .485
.952 .951 .944 .919 .797
REFERENCES Balestra, P. and Nerlove, M. (1966). Pooling Cross section and time-series data in the estimation of Dynamic Model the demand for natural gas, Econometrica, 34, 585-612. Baltagi, H. B. (1996). Econometric Analysis of Panel Data, John Wile, NY. Bhatti, M. I. (1995). Testing Regression models based on sample survey data, Avebury, UK. Rao, J. N. K., Sutradhar, B. C. and Yue, K. (1993). Generalised least squares F-Test in regression analysis with two stage cluster samples, Journal of the American Statistical Association, 88 (424), 1388-1391.
