The random effects (error correction) model

It has been argued that instead of capturing the individual specific effects by different intercepts for each firm (the Fixed Effect Model). The effect in the Random Effect Model appears in the error component as a random disturbance that is the same for every observation for a given sample, but that is random across samples. The individual specific disturbance is one component of the total disturbance term (Murray, 2006).

The following model explains the Random Effect Model:

Yit= ß0 + ßIX1it + ß2X2it + wit

where:

ß0: is the intercept.

Xit: is the independent variable.

wit: is the composite error term.

wit= eit +uit .

eit is the random error term.

uit is the combined time series and cross-section error.

In the Random Effect Model, instead of treating ß0 as fixed, we assume that it is a random variable with a mean value of ß0 for all the individuals (firms) (no subscript i here). The intercept value for an individual firm can be expressed as:

ßIi= ßI +ei i=1,2,...,N where ei is a random error term with a mean value of zero and variance σ2. The idea here is that the firms in the sample are drawing from a much larger universe of such firms and that they have a common mean value for the intercept (=ßI), and the individual differences in the intercept values of each firm are reflected in the error term ei (Gujarati, 2003).

The underlying assumptions of the Random Effects Model are:

I. E (wit) =0.

11. The two sorts of disturbances, the uit and ei, have means of zero and are homoscedastic.

Ill. The uit are uncorrelated over time and across individuals.

IV. The ei, are uncorrelated across individuals.

VI. Var (wit) = σ2 ε + σ2 u (homoscedastic).

It is worth noting that in estimating the Random Effects Model, the Generalised Least Square (GLS) is used. This is because the GLS technique takes into account the different correlation structure of the error term in the REM (Gujarati, 2003). Gujarati argues that if we do not take this correlation structure into account, and we estimate by Ordinary Least Square (OLS), the result estimators will be inefficient.

Fixed Effects Model versus Random Effects Model

The difference between fixed and error components is that in the fixed effect each cross- sectional unit has its own (fixed) intercept value, while in the error correction model, the intercept ß0 represents the mean value of all the (cross-sectional) intercepts and the error component ei represents the (random) deviation of individual intercepts from this mean value. The error term ei is not directly observable; it is an unobservable or latent variable (Gujarati, 2003).

It has been argued that the answer to whether to use fixed effects or random effects depends on the assumption that we make about the likely correlation between the individual, or cross-section specific, error component ei and the X regressors. If it is assumed that ei and the X‘s are uncorrelated, the ECM may be appropriate, whereas if ei and the X‘s are correlated, the FEM may be appropriate (Gujarati, 2003).

Greene (2003) argues that the crucial distinction between the FEM and the REM is whether the unobserved individual effect embodies elements that are correlated with the regressors in the model, not whether these effects are stochastic or not. What makes a fixed effect appropriate for one and an error component appropriate for the other is the persistence or variability of the individual specific effect across samples (Murray,

2006). In order to select between the Fixed Effect and Error Correction Models, the Hausman test is employed. The null hypothesis is that the FEM and ECM estimators do not differ substantially. The test has asymptotic X2 distribution. If the null is rejected, the conclusion is that the ECM is not appropriate and we may be better off using FEM, in which case statistical inferences will be conditional on the ei in the sample.

All the models that are employed in the study are estimated using OLS regression and then are re-estimated with pooled sample analysis. The pooled analysis uses the full data set with cross-sectional observations in each period in the series (basic details are available in Gujarati, 2003). Fixed and random effects estimators have also used for re-estimating the models, allowing for the firm panel structure and it can be noted here that the Hausman‘s test reported later in this thesis show conclusively that fixed effects are sufficient and that a random effects model is not appropriate. Therefore, generalised least squares is not been applied in this thesis given that the variances of the observations are generally not shown to be heteroscedastic, suggesting that ordinary least squares is statistically efficient and will not give misleading inferences.

The study employs pooled sample analysis because of the advantages of pooling the sample. It has been argued that pooled samples have many advantages. Pooling data generates more informative data, more variability, less collinearity among variables, more degrees of freedom and more efficiency. Furthermore, aggregating data of many observations minimizes the bias that might result if we aggregate individuals or firms into broad aggregates (Gujarati, 2003).