In this situation, equation should (6.1) be written as:
Te = a + B jc. + u. I It It f o r f t ^ P , (6.3)
This case is presented in Figure 6.5. In this figure, the broken-line circle is not shown and the straight broken lines represent OLS regression lines if we estimate the observations of each enterprise separately. In this case, the structure of the estimates of
3 is unstable, and the regression line of the pooled regression model is simply unpredictable. Although the intercept term can be consistently estimated, the coefficient of slope is likely to be statistically insignificant. From the economic point of view, this model is not very interesting because no clear-cut policy implications can be drawn unless technical efficiency and the reform indicators display a reasonable degree of consistency and stability.
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Type 3: Intercept and Slope Heterogeneity.
In this situation, we can use the following equation to represent the model:
Te. = a , + + for a , ot^ and p, P^ (6.4)
Intercepts and slopes are allowed to vary across units in each time period. We call this the intra-temporal unrestricted model as distinguished from the unrestricted model based on both inter-temporal and intra-temporal heterogeneity.
This type of bias can be divided into two cases, shown in Figures 6.6 and 6.7. In the first case (Figure 6.6), a straightforward poohng of all observations in the panel and the assumption of identical parameters for all enterprises leads to nonsensical results for both intercept and slope. The OLS estimates, the average of the coefficients, are also difficult to predict. However, the intercepts and slopes are likely to be statistically significant in this case, and so the results are inconclusive.
The second case (Figure 6.7) is dangerously misleading. While the underlying relationship between technical efficiency and reform indicators is inconsistent, the OLS estimates predict a positive correlation. If observations of enterprises scattered around the OLS regression line are sufficiendy large, the OLS estimate may turn out to be statistically "significant". Inference based on those results would lead to misleading poUcy implications. In this case, type II error occurs, if we hypothesise that technical efficiency is positively related to economic reform.
As stated earlier, heterogenous bias may also occur with observations along the longitudinal dimension of the panel data or inter-temporally. Although the three types of heterogeneity could be presented in graphic form (analogous to Figures 6.1-6.7), it is enough to note here that the subscript i of a and (3 in the models (equations 6.2-6.4) should be replaced by t in the inter-temporal cases, where t is the index of time periods. Functions with both intra- and inter-temporal heterogeneity are called the two-way fixed effect model in the literature.
Model Selection
According to the analysis so far, it is clearly necessary to test whether parameters characterising the random outcome variable Te.^ remain constant across all enterprises and over time. Hsiao (1989) suggested using an analysis-of-covariance test to identify variation in intercepts and slopes. This is the method used most widely in panel data analysis. We will again illustrate the test procedure for intra-temporal cases, assuming as before inter-temporal homogeneity.
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Step 1: simultaneously testing the homogeneity of slopes and intercepts among sample enterprises;
Step 2: testing the homogeneity of slopes of sample enterprises as a whole; and
Step 3: testing the homogeneity of intercepts of sample enterprises as a whole.
From a statistical point of view, the first step can be separated from the other two. It is obvious that if we accept the hypothesis of homogeneity in both intercepts and slopes, no further tests are necessary. W e would use simply equation (6.2) and run the OLS regression straightforwardly on the panel data. If the hypothesis is rejected, we will need to consider other possibilities and to proceed with steps 2 and 3.
The tests in steps 2 and 3 are statistically independent and can be carried out in either order. However, the test in step 3 is meaningful only when the structure of the relationship between efficiency and economic reform indicators is reasonably stable. For this reason it is better to carry out the test in step 2 first.
Based on the assumption that W, is a random variable independently and normally distributed with zero mean and constant variance c ] , an F-test is used in each step of the procedure. The rationale underlying this joint test is that variation in individual enterprises of intercepts and slopes is not important, but that collectively significant heterogeneity of a group of individuals is of concem and should be corrected. For the first step, we impose a set of linear restrictions on all intercepts and slopes and test statistically whether the restrictions hold. The hypothesis can be formulated as:
(6.5) s.t. Hi: a, =a, =...= a,
In this case there are (k + 1){N -1) restrictions, where N is the total number of enterprises and K is the number of socio-economic variables. Let us define the residual sum of square of the unrestricted model (equation 6.5) as E^ and that of the pooled regression model as E^. Under hypothesis H^, the explained sum of square of the model decreases by (E^- EJ due to the restrictions imposed on intercepts and slopes. The unrestricted residual sum of square (E^) divided by the variance ( o ] ) has a Chi-square distribution with NT - N ( K +I) degree of fi-eedom; the difference in the two values of the residual sum of square ( E ^ - E J divided by the variance has a Chi-square distribution with + degree of freedom. Given that the two random variables are independently distributed, an F-statistic can be calculated as follows:
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EJ[NT-N{K + \)]
where T is the number of periods covered by the data.'^ If the F value is statistically
insignificant we stop the testing procedure and use the pooled regression model. Otherwise, we go on to steps 2 and 3.
For step 2, the hypothesis can be formulated as:
r e „ = a , + p , x , , + M „
We calculate the f-statistic in the same way as shown in equation (6.6), except that the
numerator has K{N — 1) degree of freedom. In this step, we have two alternative ways
of conducting the test: one conditional on homogenous intercepts, and the other an
unconditional test. But we do not know a priori whether or not the intercepts are
homogenous and so need to do both tests. If the null hypothesis (H^) is not rejected, we proceed to test intercepts on the condition of homogenous slope; if it is rejected, we carry out the unconditional test.
Step 3 is equivalent to testing the linear restriction on the intercepts alone and can be formulated as:
sJ. Hi a, =a. =...= a. (6.8.1)
or
SJ. H": a, =...= a. (6.8.2)
depending on the results of the test in step 2. In this case the F-statistic calculated using
equation (6.6) has N degree of freedom in the numerator. As noted earlier, the test for
homogeneity of intercepts is only economically interesting when the structure of the relationship between technical efficiency and reform indicators is reasonably stable, both
i "•in the case of unbalanced panel data NT should be replaced by ^ N.^.
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qualitatively and quantitatively. Based on this observation, we will only proceed with the test in step 3 if only hypothesis {H]) is not rejected.
If H] is rejected, the decision about model selection will be made on the basis of the results of steps 2 and 3. If the null hypothesis is rejected in both steps, then the unrestricted model (equation 6.4) should be used for estimation; if the null hypothesis in step 2 {H]) is rejected but that in step 3 is not, the model represented by equation (6.3) would be used; if the null hypothesis in step 2 is not rejected but that in step 3 is rejected, then the cell-mean corrected regression model (equation 6.2) should be used to estimate the impact of economic reform on technical efficiency.
It should be noted that two uncomfortable situations may emerge. In the first, the hypothesis H^ is rejected in step 1 but, if we go on with the testing procedure, not in the other two steps. In other words, while the overall test suggests that intercepts and slopes are not collectively homogenous and that we should use the intra-temporal unrestricted model, individual tests indicate that they are not heterogeneous and therefore that the intra-temporal unrestricted model is not an appropriate specification. The opposite situation may also crop up, in which hypothesis H] passes the F-test in step 1, but fails individual tests in steps 2 and 3. The former case suggests that the pooled regression model should be used, and the latter that models with homogenous intercepts and slopes are inappropriate. These contradictory results may be due to differences in null hypotheses in overall and individual tests. If they do occur, they may make it difficult for us to come to a meaningful and useful conclusion.
The test procedure for inter-temporal homogeneity is analogous to the three-step procedure set down for intra-temporal homogeneity. The only differences lie in the degree of freedom in the formulation of the F-statistic (6.6), and some notations. In the inter-temporal case, the subscript i wiU be replaced by t, and N by T, where t is the index for time periods and T is total number of periods.
In this study, the significance of the inter-temporal test lies in our interest in whether the response of efficiency to economic reform measures is consistent over time. Inter- and intra-temporal testing procedures can be carried out separately. In the testing process, we first conduct the inter-temporal test without imposing any mtra-temporal restrictions on intercepts and slopes. Intra-temporal tests are then carried out based on the selected inter-temporal model.