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

Подпись: Table 8.6 Mis-specification tests Ap model к aAp% p values FAR(1-5) FARCH(1-5) ^normality F HETx2 F RESET AWM 13 0.19 0.84 0.78 0.60 0.17 0.80 ICM 11 0.21 0.68 0.95 0.92 0.87 0.09 p* 12 0.21 0.76 0.61 0.81 0.70 0.008** P*_enh 14 0.19 0.66 0.56 0.15 0.77 0.93 NPCM 7 0.23 0.00** 0.48 0.08 О о 1—1 *

Подпись: 19

image157

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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