We proceed to the econometric estimation in four steps: (1) we perform tests of pooling; (2) we briefly discuss the convenience of a fixed effects model versus a random effect specification; (3) we present the regression output corresponding to the spread level equations (as shown in equation 20) as well as a battery of tests intended to help discern which is the best regression; these results refer to one particular kind of estimator (i.e. Pooled-OLS, fixed effects (LSDV) and random effects (RE-FGLS); (4) we show the econometric results of the first difference model (as shown in equation 19) and compare them to those obtained in the best level equation.
5.3.1. Tests of Pooling
Following traditional panel econometric modelling (e.g. Hsiao (1986)), we proceed to test the existence of specific corporate bond spread effects in the model, not accounted by our explanatory variable set. We perform two standard Wald F–tests to check the null of: a) homogeneous slopes and intercepts in equation (20) – Pooled OLS is the right model – (against an alternative of homogeneous slopes and different intercepts – fixed effects is the proper model); and b) homogeneous slopes and different intercepts (against an alternative of different slopes in ssov? and different intercepts). The procedure of these tests as well as the full description of their results are shown in tables A2.1 and A2.2 in appendix A2.2.
The results of the first test indicate we reject the null that the model has homogeneous intercepts and slopes, at a 5% level of significance (and even at a 1% level). A robustness check, implying a change in asset returns volatility (sv24m or sv12m instead of sv1000d) yields the same conclusion.
Provided the conclusion of the first Wald-test, the results of the second test suggest we have to reject the null that the model has different intercepts but homogeneous slopes associated to SSOV? at a 5% level of significance (and even at a 1% level). A robustness check, implying a change in asset returns volatility (sv24m or sv12m instead of sv1000d) yields the same conclusion.
In conclusion, we would expect to observe different responses from SCOR? to SSOV? across different corporate bonds or issuers. This is very relevant in terms of a policy-oriented analysis. However, we wonder whether the different slopes are not the product of typical small-sample biasedness, very likely to occur in our case. Notwithstanding this result, in what follows we will assume away the heterogeneity in the slope of SSOV? in order to be able to compare RE-FGLS to other estimators.38
5.3.2. Fixed or Random Effects?
The result of the foregoing tests, notwithstanding our finding on the heterogeneity of the slopes associated to SSOV? (subject to potential biasedness), leads us to ask whether one may really consider these corporate bond-specific effects as fixed or random.
Some authors (e.g. Mundlak (1978)) have considered that this distinction is an erroneous interpretation and we should always treat these effects as random. According to him, the fixed effects model is simply analysed conditionally on the effects present in the observed sample (Greene, 2000). Moreover, this model might be viewed as applying only to the cross-sectional units in the study, not to the ones left out of the sample. The only exception would be the case when the cross-sectional units exhaust the sample. Indeed, this is our case given that all “true” corporate and banking bonds are included in our sample. Our T, on the other hand, is not long enough to assume away both effects are indistinguishable (i.e. we would know all observations). This is so because the shortness of the sample for some explanatory variables (for instance liquidity, i.e. tovc?) constrains the full utilisation of corporate spreads in the time dimension. Should we exclude these variables, we would be nearly exhausting the maximum number of available periods that correspond to BESA history.
Therefore, if no substantial differences between both estimators are found, that is, if their properties are as good and their specification errors behave similarly, we will suppose that no real differences regarding the treatment of the nature of the group effects exist in our model.
5.3.3. Model Selection
5.3.3.1. Regression Output
38 In any case, RE-FGLS would not be estimable because we could not assign weights when theβ'sare heterogeneous across units. This is another reason why we keep on working with the assumption of equal slopes.
Based on the specification suggested by equation (20) we run different regressions, each one regarding one particular kind of estimator (i.e. Pooled-OLS, fixed effects (LSDV) and random effects (RE-FGLS)). Then, we evaluate the quality and properties of these estimators in comparative perspective so as to finally be able to choose the best regression.
The first three regressions are correspondingly: Pooled-OLS (a single intercept in (20), i.e. β0), LSDV (fixed effects or within estimator, i.e. considering ui as a parametric shift in (20), additive to β0) and RE-FGLS (error components model, as it stands in equation 20). The results are reported in table 5 (appendix A3)39. All three estimators are obtained using the White heteroskedasticity-consistent standard errors & covariance. Please note that RE-FGLS estimators are both robust to heteroskedasticity (E(eit)2 =σ2i and/or E(ut)2 =σ2ui and serial correlation over time (E(eitejs)=0∀t≠s. However, as it is not necessarily the case of the LSDV estimation, we also corrected for both sources of inefficiency by using a proper weighing matrix.
The RE-FGLS estimates show how inefficient are the weights assigned by Pooled-OLS to the between-units variation in relation to the LSDV estimator. Indeed, an overwhelmingly majority of the point estimates (βs) yielded by the RE-FGLS regression are much closer to the LSDV than to the pooled-OLS ones (the exception being the signs and values of sigspotm? , sv1000d? and d1? in the regression where autocorrelation and
heteroskedascity are remedied –though sigspotm? is not statistically significant). The closeness of RE-FGLS to LSDV estimators results from the fact that RE-FGLS attaches less weight to the between variations than Pooled-OLS.
RE-FGLS may converge to Pooled-OLS or to LSDV estimators as a special case, as we said before. The degree of convergence hinges on the variancesσu, σe and T, the time dimension. Moreover, these three parameters determine the magnitude of the weights used by RE-FGLS. This is explained in appendix A2.3.
Once Pooled-OLS estimates are discarded as a result of their relative inefficiency, we have to decide which estimator between RE-FGLS and LSDV yields the most reliable and robust results. In order to do this we perform several tests. A first step is to test the null that σu=0 or see what the value of the estimate forσeis. This comes next.
39
In all regressions, d1 and sigspotm? (i.e. leverage and interest rate volatility) yielded the more robust results.
5.3.3.2. Test for Existence of Random Effects
This is typically a Breusch-Pagan test to check whether the null that σu =0 ((iii) from the Gauss-Markov assumptions laid out above) can be rejected or not. The results reported in appendix A2.4 confirm the rejection of the null that no random effects on the cross-sectional units are present. This is evidence in favour of the RE-FGLS estimator.
While the result of this test tells us we should not discard the RE-FGLS model, it does not conclude, on the other hand, that we should rule out the (corrected) LSDV model. For instance, it may happen that given the specificities of the sample, LSDV turned out to be close to RE-FGLS, as it seems to be the case in our exercise. Put differently, were all cross sections and almost the entire T dimension known, there would not be a “real” random distribution because the population would then be known. Therefore, the distinction made between both kinds of effects should ultimately result irrelevant, as Mundlak (1978) pointed out.
Suppose, notwithstanding, we concluded the other way around (accepting the null of no random effects, or "the variance of the u's is equal to zero"). This conclusion would favour the adoption of the LSDV model in any case. Again, we should take into account Mundlak’s argument: irrespective of what this test concludes, we might end up with both effects being undistinguishable.
Recapitulating, the value and statistical significance of σu can have a bearing on determining whether the RE-FGLS model is close to Pooled-OLS or LSDV; or, in other words, on determining the weights implied by the RE-FGLS estimator –as appendix A2.3 demonstrates. However, as it is also shown in appendix A2.3, these weights also depend on the size of T and the magnitude of the variance of regressionσ2e: for small T, when
e 2
σ tends to zero, irrespective of the value of σ2u, the RE-FGLS estimator will converge to the LSDV model. This seems to be our case: indeed, σ2e ≈0. Put differently, when
0 2 ≈
e
σ all variation across units is due to the different uis, which, because they are constant across time, would be equivalent to the LSDV estimator used in the fixed effects model.
5.3.3.3. Haussman’s Test of Endogeneity
A further way to test whether the RE-FGLS estimators are superior to those obtained through LSDV is to perform a Haussman’s test. Briefly, under the null ofE(Xk,itui)=0, which implies exogenous regressors, both LSDV and RE-FGLS estimators are consistent but only the latter is efficient. By contrast, under the alternative, the random effects estimators are inconsistent while the within hold consistent. We do the
test by using both LSDV and LSDV corrected for serial correlation and heteroskedasticity. The results of Haussman’s test suggest the null can be convincingly accepted in both cases40
In conclusion, despite the RE-FGLS estimates tend to converge to the within or LSDV ones, we could still prefer the first in virtue of its higher relative efficiency (under the null of exogenous regressors). However, when we correct the LSDV estimators for the loss of efficiency implied by heteroskedastic variances and serially correlated residuals, we get the best fit, for equivalent properties. A final test accepts the null of no cross residual-correlation in the corrected LSDV model (see appendix A2.4). Thus, we select the corrected LSDV model as our representative regression output for corporate spreads in levels.41