In the first procedure to estimate our model we treat the panel data as a cross section. The procedure to estimate a cross-section model follows Blundell and Meghir (1987) approach. The aim of this estimation is to be able to compare the estimates obtained with cross-sectional data to the ones obtained with panel data.
In appendix A we derive the asymptotic variance following the approaches by Ham (1982) and Wooldridge (1995).
As highlighted by Blundell and Meghir (1987) it would be interesting to test for heteroscedasticity and for independence, in the context of this model. In order to do this, we follow their approach.
An interesting test to be carried out is to compare the cross-sectional model of IPM and the time aggregated model. The motivation for this test is that under the assumption of strict exogeneity, the null hypothesis states that both the maximum likehhood estimator in the IPM (the true model), and the time aggregated model estimator, p^^, are consistent, although the aggregated model estimates are not efficient. However under the alternative hypothesis, and still assuming strict exogeneity, the time aggregated estimator remains consistent but the IPM estimator does not.^®
As pointed out by Mroz (1987), given that the estimator under the null hypothesis may not be efficient, we can not use Hausman’s (1978) formula for the covariance matrix of the differences between the two sets of estimates. Therefore we follow Mroz’s (1987) approach to construct the covariance matrix of
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C 5 ’ P T’A ]
Therefore under Ho we have the chi-squared test based on the Wald criterion (1.20) W = = [P^^ - PrA]’^ “'[P c5 “ PtaI
where S = Vhr^P^^ - p^.^ . Under the null hypothesis, W is asymptotically distributed as a chi-squared with K degrees of freedom. In order to carry out the test we need to calculate the asymptotic variance of P from the infrequency of
A test for normality is also implemented for the error term in the frequency of purchase equation for the cross-sectional model.
The characteristics of this test are that if we accept the Ho, we accept the IPM as the true model against the aggregated model, but if we reject the null hypothesis we are not sure if we are rejecting the specification o f the IPM or the strict exogeneity assumption. In the second case, we should test for strict exogeneity.
See Mroz (1987), for a detailed description of the formula to calculate the covariance matrix of , 0 ^ j - All his formulae are based on White (1982a). In our case we specially follow case 3 of the appendix, where he reports the formula when one of the estimators is a multi-stage estimator and the other is a maximum likelihood estimator.
purchase model. In appendix A there is a description of the asymptotic variance for the estimator.
The second set of tests are based on the panel data estimators. After estimating the model using Wooldridge (1995) approach, it is possible to test for the exogeneity of the individual fixed effects (i.e., E (a . |z. ) = 0
Another diagnostic that has been carried out is the efficiency gains when estimating the IPM using the information provided in the frequency of purchase equation. Equation (1.4) above holds because of the assumption that E(y*|z.) = E (y.Jz.). It follows that regressing y., on z. also yields a consistent estimate.^^ Comparing both estimators it is possible to assess the efficiency gains obtained given that the estimator using equation (1.4) should be more efficient because it uses more structure.
Finally, we can compare the IPM with cross-sectional data estimates, , with the estimates obtained from the IPM with panel data, The motivation of this test is based on the idea that under the hypothesis of no individual effects, both models are consistently estimated but the estimates from the IPM with panel data are inefficient (there is no need of estimating the model with panel data if we do not have fixed effects). However under the alternative hypothesis of correlated individual effects, only the IPM with panel data estimates remain consistent.
Under Ho we have the chi-squared test based on the Wald criterion
(1.21) W =
There is another way to test for the exogenenity of the fixed effects. Under the maintained assumption that E (a .|z,.) is a linear function of z . , a Hausman test based on the comparison of the time aggregated estimator and Wooldridge’s (1995) type estimator for the IPM, provides also a test for the exogenenity of the individual effects.
A similar estimator has been proposed by Keen (1986) in a cross sectional context. This point has been already made by Meghir and Robin (1992).
Both estimators used to estimate equation (1.4) can be taken to carry out this test.
where Ê =