Tests of model specifications - Empirical results, various types of crime

LISREL 7 minimizes the function F without imposing any constraints on the admissible values of the parameter vector O. Thus the LISREL estimate of a parameter which we

6. Empirical results, various types of crime

6.3. Tests of model specifications

In order to test various model specifications we present in Table 13 the 2(²-values for various models and in Table 14 the corresponding significance probabilities. Our model framework consists now of a hierarchy of models with W3L3 on the top and W1L1 at the bottom. We have dropped all models with either MO, LO or WO in our analysis of different types of crimes in order to save time and space, cf Tables 2-4. Testing of models within this framework by likelihood ratio tests can be performed in various ways. The simplest procedure is to test each model against the general model W3L3. The significance probabilities corresponding to these tests are presented in the first five columns in Table 14. Models rejected by these tests, at a signficance level of 0.01 and 0.05, are marked in Table 15 by R⁰^•⁰¹ and ^R0^.^05,respectively.

Another procedure is to start at the top and test each model against the one immediately higher in the hierachy, and if a model is rejected, then all models below in the hierachy are also rejected. Significance probabilities corresponding to such tests can also be found in Table 14. The test results in Table 15 will in our case be the same for both procedures.

Other procedures, and other levels of significance, can be applied based on the information in Table 13, but in the following we stick to the test results presented in Table 15. In order to limit our analysis we will give no further consideration of models rejected at a significance level of 0.05.

Since our class of models is not fully identified without restrictions in the parameter space, we also reject models which do not satisfy the restrictions in (16), see section 3. It turns out that restrictions (16a) on the distribution of measurement errors, crime tendencies, and clear-up tendencies, are not binding for any of the models not rejected by the

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likelihood ratio test. The models with point estimates outside the interval hypotheses in (16b) and (16c) are marked in Table 15 with the names of the hypotheses they contradict, cf Table 1.

Observe first that for each type of crime, one of the solutions of W3L3 is rejected by at least one of our criteria, whereas the other is not. This is a remarkable result, since we could well imagine that both solutions were rejected, or none of them. Thus, in our situation, the identification problem does not seem to be empirically relevant. The solution that is not rejected is denoted solution I, the other solution II. The parameter estimates of both solutions are presented in Tables 17-27. It is seen that the estimates of solution II almost without exceptions are unprecise and rather unreasonable. Our choice of restrictions has clearly separated the appropriate solutions from the non-appropriate ones.

Considering now the less general models in Table 15 we find that quite a few of them satisfy all our criteria. The estimates of these non-rejected models are given in Tables 17-27, and the variances of the crime and clear-up tendencies are given in Table 16. In these tables we find the estimates demonstrating our observation above that the non-rejected models satify (16).

In order not to burden our presentation we have not included the estimates of the rejected models. Our inspection of these have shown, however, that almost all of them perform badly as far as precision and plausibility of estimates are concerned. Thus, our rejection criteria (16) seem to perform quite well. We have further observed that the estimates of these models to a large extent ressemble those of W3L3¹¹.

The models that meet all our criteria will on the other hand have estimates rather similar to those of W3L3I. For each type of crime parsimony of parameters may guide our choice of model within the group of these reasonable ones.

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In document Criminometrics, latent variables, panel data, and different types of crime (sider 38-45)