1. M. K. GARBA, W. B. Yahya and B. A. Oyejola Department of Statistics,
A COMPARATIVE STUDY OF SOME ESTIMATORS FOR PANEL DATA WITH HETEROSCEDASTICITY AND CORRELATIONS By
M. K. GARBA, W. B. Yahya and B. A. Oyejola
Department of Statistics,
University of Ilorin, Ilorin, Nigeria.

2. ABSTRACT
Failure of the orthogonality assumption in panel data models leads to biased ordinary least squares (OLS) estimates. For OLS to be efficient at estimating such models, errors have to be independent and homoscedastic. These conditions are so atypical and mostly unrealistic while modeling real life data using OLS. Panel dataset is commonly afflicted with problems that typically plague time series and cross sectional data because it combines the characteristics of both datasets. This paper, therefore, examines four methods of estimating panel data models (Pooling (OLS), First-Differenced (FD), Between (BTW) and Feasible Generalized Least Squares (FGLS)) in which the assumptions of homoscedasticity, no autocorrelation and no multicollinearity are jointly violated.

3. Monte-Carlo studies were carried out at different sample sizes and varying degrees of heteroscedasticity and levels of autocorrelation and collinearity at different time periods. The results from this work showed that in small sample, irrespective of number of time periods, FGLS is preferable when heteroscedasticity is severe regardless of levels of autocorrelation and multicollinearity. But when heteroscedasticity is low or mild with moderate autocorrelation level, both FD and FGLS are preferred, while BTW performs better only when there is no autocorrelation and low degree of heteroscedasticity. However, in large sample with little time periods, both FD and BTW could be used when there is no autocorrelation and low degree of heteroscedasticity, while FGLS is preferred elsewhere.

4. INTRODUCTION
A panel data set is one where there are repeated observations on the same units. The units may be persons, families, firms, states or nations. It has the combination of the characteristics of both time- series and cross-sectional data. Therefore, problems that generally plague time-series data (i.e. autocorrelation) and cross-sectional data (i.e. heteroscedasticity) need to be addressed while analyzing panel data.

5. Panel data like other aspects of econometrics uses regression analysis as one of the statistical tools to formulate, describe and evaluate models. This regression analysis relies on some assumptions which if violated might result to one problem or the other. Among the assumptions of classical linear regression model (CLRM) are homoscedasticity, no serial correlation and orthogonality. If these assumptions are violated, we surmise the presence of heteroscedasticity, autocorrelation and multicollinearity respectively. See Schmidt (2005), Chatterjee (2006), Maddala (2008), Greene (2008), Gujarati & Porter (2009), Creel (2011), Wooldridge (2012).

6. Situation where the assumptions of CLRM are satisfied are rarely found
Situation where the assumptions of CLRM are satisfied are rarely found. The violation of each of the assumptions has attracted the attention of researchers. When heteroscedasticity and autocorrelation are present, the OLS estimators (naive approach) may still be linear, unbiased, and asymptotically normally distributed but not efficient relative to other linear and unbiased estimators . (See Greene, 2008; Baltagi et al., 2008; Olofin et al., 2010). The presence of multicollinearity renders the estimates of the parameters indeterminate. In the cases where the estimates are obtainable, the accompanying confidence intervals tend to be too large and the standard errors become infinitely large. For OLS to be optimal, it is necessary that the errors have the same variance (homoscedasticity) and that all of the errors are independent of each other (Stimson 1985; Hicks 1994; Beck and Katz 1995 & 1996).

7. This study examines the behaviours of four methods of estimating panel data models when the assumptions of no multicollinearity, no serial correlation and homoscedasticity are jointly violated. The best estimator, in terms of efficiency, among those considered in this study that is robust to the violations of the aforementioned basic assumptions is determined using absolute biases, variances and root mean square errors (RMSE) of parameter estimates. Results from this work would serve as useful guides to econometricians and students while modelling panel data that are characterized by the structure conceptualised here.

8. MATERIALS AND METHODS
This work considers one-way error component model with an endogenous and two exogenous variables. Heteroscedasticity was introduced into the model through the individual-specific error component, i.e. αi ~ (0, σαi2) and uit ~ IID(0, σu2) while performing Monte-Carlo experiments. This is in consonance with the earlier studies of Mazodier and Trongon (1978), Baltagi and Griffin (1988), Roy (2002), Bresson et al. (2006) to mention but few. First-order serial correlation was used to implant autocorrelation into the model as did by Lillard and Wallis (1978), Bhargava et al. (1983), Burke et al. (1990), Galbraith and Zind-Walsh (1995) and others. As earlier mentioned, the panel data model considered has two exogenous variables with the likelihood of existence of multicollinearity between the variables. We then studied the effects of this collinearity in terms of efficiency and stability of the four methods of estimating panel data models examined.

9. General Static Panel Data Model
A general static panel data model is
Yit = αi + β1X1it + β2X2it βkXkit + uit ...1a
or in matrix form as
Yit = αi + Xʹit β + uit …1b
for i = 1, 2, …., n and t = 1, 2, …, T
where Yit is the response for unit i at time t, αi is the
individual-specific intercept, vector Xʹit contains k
regressors for unit i at time t, vector β contains k
regression coefficients to be estimated and uit is the
error component for unit i at time t.

10. The model considered in this study has two exogenous
and one endogenous variables as shown below;
Yit = αi + β1X1it + β2X2it + uit … 2
where αi = α + εi
The model becomes
Yit = α + β1X1it + β2X2it + εi + uit … 3
where εi is the individual-specific error component and uit
is the combined time-series and cross-section error
component with variances σε2 and σu2 respectively.

11. Structure of the Panel Data Used for this Work
C/S units
Time Period
Yit
X1it
X2it 1 . 2 3 T
y11
y12
y13
y1T
x111
x112
x113
x11T
X211
x212
x213
x21T
y21
y22
y23
y2T
x121
x122
x123
x12T
x221
x222
x223
x22T n
yn1
yn2
yn3
ynt
x1n1
x1n2
x1n3
x1nt
x2n11
x2n 2
x2n 3
x2n T

12. Source of Data
Several Monte Carlo experiments were carried out to generate the datasets used for this work in the environment of R statistical package ( Two sizes of cross-sectional units (50 and 250), three time periods (10, 40, and 100), five levels of autocorrelation (ρ = ±0.9, ±0.5, 0), five levels of collinearity (r = ±0.9, ±0.5, 0) and three degrees of heteroscedasticity (low, mild & severe) were used for simulation (i.e 2 x 3 x 5 x 5 x 3 combinations). Each of the combinations was iterated 1000 times.

13. Brief Overview of Panel Data Estimation Methods Considered

14. 1. Pooled Estimator (OLS)
This estimator ignores the individual specific effects, stacks the data and treats them as if there is only a single index. In this case, all coefficients are constant across individuals and time periods. The estimator assumes that the intercept values are the same for all individuals and that the slope coefficients of all explanatory variables are identical.

15. The Pooled Estimator is given by This is the simplest estimator for panel data models. In most cases, it is unlikely to be adequate. In such cases, it provides a baseline for comparison with more complex estimators.

16. 2. Between Estimator (BTW)
The between estimator regresses the group means
of Y on the group means of X’s in a regression of n
observations.
Basically, it uses the cross-sectional variation by
averaging the observations over period t and converts
all the observations into individual-specific averages
then performs OLS on the transformed data.
See Baltagi 2005; Cameron and Trivedi 2005; Greene 2008;
Wooldridge, 2012

17. The Between Estimator (BTW) is given as

18. It should be noted that if OLS estimation on the pooled sample is consistent, the BTW estimator is also consistent but not as efficient as OLS when the assumptions are valid. (See Heineck, 2004). However, if the homoscedasticity assumption which says the error variances among individuals are the same is not valid, BTW estimator is expected to perform better than OLS because it takes into consideration the variation between the individuals.

19. 3. First-differenced Estimator (FD)
The first-differenced estimator exploits the special features of panel data models in that it measures the association between individual- specific one-period changes in regressors and individual-specific one-period changes in the regressand. See Wooldridge (2012) for details.

20. ΔYit = β1 ΔX1it + β2 ΔX2it + Δuit ;
Considering the model (1a), the one-period lag gives
Yi, t-1 = αi + β1X1i, t-1 + β2X2i, t-1 + ui, t-1 … 6
Subtracting the model 6 (lagged model) from model
(1a), we have
Yit - Yi, t-1 = β1 (X1it - X1i, t-1) + β2 (X2it - X2i, t-1) + (uit - ui, t-1),
… 8
Such that
ΔYit = β1 ΔX1it + β2 ΔX2it + Δuit ;
for i = 1, 2, …n and t = 2, 3, …, T

21. 4. Feasible Generalized Least Squares Estimator (FGLS)
The feasible generalized least squares estimator can be obtained from the OLS estimation of the transformed model shown below Yit* = αi* + Xʹit* β + uit* ; i = 1, 2, …., n and t = 1, 2, …, T …9 where Yit* = Yit – λi. ; Xʹit* = Xʹit – λi. ; uit* = λi and αi* = 1 – λ The term λ gives a measure of the relative sizes of the within and between unit variances. Note that if all the cross-sectional units have the same intercept, so that σα2 = 0, λ becomes 0 and the FGLS estimator reduces to OLS.

22. In compact form, GLS = (XʹΩ-1X)-1 XʹΩ-1Y … 10

23. When Ω is known GLS based on the true
variance component is BLUE and all the
feasible GLS estimators considered are
asymptotically efficient as n or T approaches
Infinity. Since Ω is often unknown, FGLS is
more frequently used rather than GLS.
See Hsiao (2002), Arellano (2003), Wooldridge (2003 & 2012),
Baltagi (2001 & 2005), Greene (2008), Creel (2011).

24. Criteria for Assessing the Performances of the Estimators
The assessments of the estimators considered
were based on the following criteria.
Absolute bias
Variance
Root mean squared error
Each of the criteria are defined in the next slide

26. Friedman Test and LSD
After the estimators were evaluated based on the above criteria they were all being ranked according to their relative performances to one another. A test of significance was carried out to verify if the sums of ranks of other estimators are actually significantly different from that of the estimator proclaimed best. Because the ranks are on ordinal scale of measurements, a non-parametric statistical test developed by Milton Friedman (1939) was employed to perform this task.

27. RESULTS AND DISCUSSION
The four estimators were assessed using each of the criteria stated earlier. These estimators were ranked using the ranks 1, 2, 3 and 4 with rank 1 assigned to the best estimator that has the lowest value of the absolute bias, variance and root mean square error. A rank of 2 is assigned to the second best estimator and so on. For instance, the performances of the estimators using variance criterion when we have 50 cross-sectional units and 10 time periods at all levels of autocorrelation and multicollinearity and degrees of heteroscedasticity are presented in the next slide.

28. Table 1: Ranks of the Estimators Using Variance Criterion when N = 50 and T = 10

29. The ranks for each of the estimators were summed for all levels of multicollinearity to determine an estimator with lowest sum of ranks at each of the autocorrelation levels and degrees of heteroscedasticity. Here, the effects of multicollinearity are repressed to assess the upshots of autocorrelation and heteroscedasticity which are respectively the problems that are peculiar to time- series and cross-sectional data which are brass tacks of panel data.

30. Table 2: Sum of Ranks of the Estimators Using Variance Criterion when N = 50 and T = 10

31. Table 3: Preference of the Estimators for each of the Sample Sizes

32. It can be deduced from Table 3 that at N = 50; T = 10 and at high level of autocorrelation (±0.9) irrespective of degree of heteroscedasticity, the performances of FD and FGLS estimators are not significantly different from each other. Therefore, each of them is efficient for this scenario. Meanwhile, for moderate autocorrelation level (±0.5), any of FD and BTW estimators could be used at low degree of heteroscedasticity. The FD estimator is preferred when there is mild heteroscedasticity, while under severe heteroscedasticity, the FD and FGLS estimators performed better. But at no autocorrelation, BTW estimator which regresses the group means of the dependent variable on the group means of independent variables using OLS is efficient irrespective of the degree of heteroscedasticity for sample size N =50; T = 10. The efficient estimators for other sample sizes at various conditions are presented accordingly in Table 3.

33. CONCLUSION
Various results obtained in this work generally showed that the behaviours of the four estimators investigated for modeling various panel data vary as the violations are varied. Failure of the orthogonality assumption makes the OLS estimators to be biased and imprecise. For OLS to be accurately used in estimating the parameters of panel data models, errors have to be independent and homoscedastic. These conditions are so atypical and mostly unrealistic in many real life situations that would have warranted the use of OLS for modeling panel data efficiently.

34. Our findings from Monte Carlo experiments for several combinations of violations show that in small sample, irrespective of number of time periods, FGLS is preferable when heteroscedasticity is severe regardless of autocorrelation level. But when heteroscedasticity is low or mild with moderate autocorrelation level, both FD and FGLS are preferred, while BTW performs better only when there is no autocorrelation and low degree of heteroscedasticity. That is, in large sample with little time periods, both FD and BTW could be used when there is no

35. Also when the degree of heteroscedasticity is mild and there is no autocorrelation in large sample with little time periods, any of the FD and FGLS would produce efficient results. Finally, when severe degree of heteroscedasticity is present and autocorrelation is apparent in large sample regardless of time periods, FGLS is superior. Meanwhile, both FD and FGLS are suitable when there is low heteroscedasticity despite the existence of autocorrelation and multicollinearity.

36. Generally, we observe based on the various results obtained in this study that, it is always necessary to assess the degree of heteroscedasticity and level of autocorrelation while modeling panel data in order to ensure efficient results. In case there is more than one predictor in the model, the strength of relationship between or among the predictors needs to be examined for possible presence of multicollinearity in the data as to avoid erroneous inferences that may arise from the use wrong method for estimating the model.

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39. ABRIDGE CODES FOR SIMULATION
set.seed(123) id = rep(1:samp, each = period) time = rep(1:period, samp) sigma <- matrix(c(var1, r*sqrt(var1*var2), r*sqrt(var1*var2), var2), 2) mv <- mvrnorm(samp*period, c(mu1,mu2), sigma) pData$x1 <- mv[,1] pData$x2 <- mv[,2] U_it <- rnorm(samp*period, 0, sigma^2_u) u_it <- rho*U_itlag + e_it e_it <- rnorm(samp*period, 0, sigma^2_e) hetero_i <- rep(rnorm(samp, sd = round(seq(2, het, length = samp))), each = period) alpha_i <- alpha + hetero_i pData$y <- alpha_i + beta_1*pData$x1 + beta_2*pData$x2 + u_it pData <- data.frame(id, time, pData$y, pData$x1, pData$x2, u_it) pdat = function(samp, period, mu1, mu2, var1, var2, r, rho, alpha, beta_1, beta_2, sigma^2_u, sigma^2_e, het)

40. THANK YOU