THE ECONOMETRICS OF MACROECONOMIC MODELLING

# 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 p-values for the mis-specification tests for residual autocorrela­tion, autoregressive conditional heteroskedasticity, non-normality, and wrong functional form. With the exception of the normality tests which are x2(2), we report F-versions 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

 Ap k u c Ш LL gum(j> 83) p-values for two types of encompassing tests model p-value FEnc,1 FEnc,2 Mi vs. Mj Mj vs. Mi M1 vs. Mj Mj vs. M1 AWM 13 0.19 16 0.08 ICM 11 0.21 18 0.00** 0.75 0.006** 0.24 0.00** P* 12 0.21 17 0.00** 0.06 0.00** 0.03* 0.00** P*_enh 14 0.19 15 0.04* 0.11 0.04* 0.009** 0.005** NPCM 7 0.23 22 0.00** Table 8.8 Encompassing tests with ICM as incumbent model Ap k F Enc gum(j> 83) p-values for two types of encompassing tests model p-value FEnc,1 FEnc,2 Mi vs. Mj Mj vs. Mi M1 vs. Mj Mj vs. M1 ICM 11 0.21 18 0.00** AWM 13 0.19 16 0.08 0.006** 0.75 0.00** 0.24 P* 12 0.21 17 0.00** 0.002** 0.000** 0.017* 0.001** P*_enh 14 0.19 15 0.04* 0.003** 0.26 0.000** 0.013* NPCM 7 0.23 22 0.00**

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 F-test 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 goodness-of-fit 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 p-values 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 mis-specified; see Ericsson (1992).

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## THE ECONOMETRICS OF MACROECONOMIC MODELLING

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