THE ECONOMETRICS OF MACROECONOMIC MODELLING
Evaluation of inflation models’ properties
The models above are estimated both for annual inflation (A4pt) and quarterly inflation (Apt) for all the inflation models, except for the NPCM where the forward-looking term on the right-hand makes the quarterly model the obvious choice. As with the Euro-area data, we shall seek to evaluate the different inflation models by comparing some of their statistical properties. In Table 8.17 we report p-values for 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 have reported F-versions of all tests.
None of the models reported in the upper part of Table 8.17 fails on the Far(i-5) or Farch(i-5) tests, hence there seems to be no serial correlation nor ARCH in the model residuals, but we see that the MdInv and the P*-model fail either on the FheTx2 test and/or the F reset test for wrong functional form. The results for the NPCM reported at the bottom of Table 8.17 indicate strong serial correlation, but as we have seen in Chapter 7, models with forward-looking expectational terms have moving average residuals under the null hypothesis that they are correctly specified. The fit of the other models vary within the range of <r = 0.35% for the ICM and the enhanced P*-model to а = 0.46% for the P*-model.
In Table 8.18 we show p-values for the encompassing tests we employed on the Euro-area data in Section 8.7.5. Recall that the statistics FEnc,1 tests the ICM against each of the six alternatives using a joint F-test for parsimonious encompassing of each of the two models in question against their minimal nesting model. The adjacent test, FEnc,2 is based on pairs of model residuals from
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Table 8.17 Mis-specification tests
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Table 8.18 Encompassing tests with ICM as incumbent model (Mi)
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the ICM (Mi) and from each of the alternative inflation models Mj. In each case we regress e1,t against the difference between the forecast errors of model j and model 1 respectively, є jt — є 1t. Under the null hypothesis that model M1, the ICM, encompasses model Mj, the coefficient of this difference should be expected to be zero and vice versa for the opposite hypothesis that model Mj encompasses Mi.
We see from Table 8.18 that the ICM outperforms most of the alternative models on the basis of the encompassing tests. We have formed a minimal nesting model for all the models, and report p-values of FEncGum tests against the minimal nesting model in the fourth column of the table. We see that the ICM and the enhanced P*-model parsimoniously encompasses the GUM. For the MdInv and the P*-model, that is, the models where we have added a set of variables from the ‘monetary’ information set, we obtain outright rejection of the corresponding set of restrictions relative to the GUM.[87] Also for the NPCM we clearly reject these restrictions.[88] Looking to the other tests, FEnc, i and FEnc,2, we find that for the ICM and the enhanced P*-model, neither model encompasses the other. The tests show that the ICM clearly encompasses the other three models.
