Evaluation of the inflation models’ properties
In this section, we summarise the statistical properties of the different inflation models, in order to make more formal comparisons. In Table 8.6 we have collected the pvalues for the misspecification tests for residual autocorrelation, autoregressive conditional heteroskedasticity, nonnormality, and wrong functional form. With the exception of the normality tests which are x2(2), we report Fversions of all tests, as in the previous sections. We also report k, the number of estimated coefficients, and aAp%, the estimated standard error.
One way of condensing this information is to perform encompassing tests.[80] In Table 8.7 we consider AWM as the incumbent model, the one we want to
Table 8.7 Encompassing tests with AWM as incumbent model

Mj encompasses M is tested by running the regression of the residuals from model Mj, £j, t, on the same difference (with changed sign). The simple Ftest of the hypothesis that the difference has no (linear) effect is reported in the table. Following Mizon and Richard (1986) and Hendry and Richard (1989), a congruent encompassing model can account for the results obtained by rival models, and hence encompassing tests form a richer basis for model comparison than ordinary goodnessoffit measures.
Tables 8.7 and 8.8 show results from the two encompassing tests explained above, and in addition we report a test for parsimonious encompassing. We have embraced all five models in forming their minimal nesting model, and report pvalues of FEncGum tests in the fourth column of the two tables.[81] We see that only the AWM parsimoniously encompasses the general unrestricted model (GUM[82]). For all the other models we reject the corresponding set of restrictions relative to the GUM (at the 5% level). In some cases, neither of the pair of models encompasses the other. When both tests lead to rejection this is prima facie evidence that both models are misspecified; see Ericsson (1992).
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