THE ECONOMETRICS OF MACROECONOMIC MODELLING

# Comparing the forecasting properties of the models

Figure 8.15 shows graphs of 20 quarters of one-step ahead forecasts with +/- two forecast errors to indicate the forecast uncertainty for the five models we have estimated. It is difficult to tell from the diagrams by means of ‘eyeball’ econometrics whether there are any differences between them. So there is a need for formal tests: Table 8.9 provides a summary of the forecasting prop­erties of the different inflation models as it reports root mean squared forecast errors (RMSFEs) along with their decomposition into forecast error bias and standard errors. The models are re-estimated on a sample up to the start of the forecasting horizon, and then used to forecast quarterly inflation until 2000(3). Two horizons are considered: a 36-period horizon starting in 1991(4), and a 20-period horizon starting in 1995(4). The first three lines of Table 8.9 show the RMSFE of inflation from the AWM, and its decomposition into mean fore­casting bias and standard deviation sdev. The other rows of the table shows the same three components of the RMSFE-decomposition for each of the other inflation models, measured relative to the results for the AWM, such that, for example, a number greater than one indicates that the model has a larger RMSFE than the AWM. For one-step forecasts 20 quarters ahead, we find that all competing models beat the AWM on the RMSFE—and bias—criteria, whereas AWM is superior according to sdev.

Tables 8.10 and 8.11 show the results from forecast encompassing tests, regressing the forecast errors of model 1, £1t, against the difference between the forecast errors of model j and model 1 respectively, єjt — £1t[83] Under the null that there is no explanatory power in model j beyond what is already reflected in model 1, the expected regression coefficient is zero. In the tables we report p-values when we run the forecast encompassing test in both directions. The AWM is used as benchmark (model 1) in Table 8.10 and the table contains evidence that AWM forecast encompasses three out of four competitors over 20 quarters (and the fourth—the P*-model enhanced—comes close to being encompassed at the 5% level), while the reverse is not true. Over 36 quarters there is clear evidence that the AWM forecast encompasses the NPCM, but is

 AWM — 1-step Forecasts AWM — Dp

 ICM — 1-step Forecasts (ICM) — Dp

 Pstar model — Pstar 1-step Forecasts — Dp

 Enhanced Pstar model — 1-step Forecasts — Dp

 NPCM model — NPC 1 step Forecasts — Dp Figure 8.15. Forecasts of quarterly inflation in the Euro area with five different models: over the period 1995(4)-2000(3). The models are: (a) the AWM; (b) the ICM; (c) the P*-model; (d) the enhanced P*-model; and (e) the NPCM. The bars show 2x forecast errors

Table 8.9

Forecasting the quarterly rate of inflation. RMSFE and its decomposition: bias, standard deviations, and RMSFE of different inflation models, relative to the AWM

 Д4р model Forecasting Др 1991(4)-2000(3) 1995(4)-2000(3) AWM RMSFE 0.0022 0.0021 Bias 0.0011 0.0016 sdev 0.0019 0.0014 ICM Rel. RMSFE 1.08 0.82 Rel. bias 1.28 0.42 Rel. sdev 1.01 1.14 p* Rel. RMSFE 0.92 0.88 Rel. bias 0.55 0.38 Rel. sdev 1.02 1.26 PA enh Rel. RMSFE 0.76 0.73 Rel. bias 0.09 0.13 Rel. sdev 0.88 1.10 NPCM Rel. RMSFE 1.11 0.73 Rel. bias 0.20 0.06 Rel. sdev 1.29 1.12

Table 8.10

Forecast encompassing tests over 36 and 20 periods, ending in 2000(3)

 Model k F Enc gum(j, 63) Forecast encompassing tests: p-values p-value 1991(4)-2000(3) 1995(4)-2000(3) Mi vs. Mj Mj vs. Mi Mi vs. Mj Mj vs. Mi AWM 13 0.19 16 0.08 ICM 11 0.21 18 0.00** 0.08 0.06 0.96 0.03* P* 12 0.12 17 0.00** 0.04* 0.02* 0.38 0.003** PAenh 14 0.19 15 0.04* 0.002** 0.42 0.88 0.067 NPCM 7 0.23 22 0.00** 0.21 0.00** 0.35 0.03*

The AWM is used as benchmark.

itself overwhelmingly forecast encompassed by the enhanced P*-model (based on the same broad information set).

In Table 8.11 the ICM is used as benchmark (model 1). The ICM is not fore­cast encompassing any competitor over 20 quarters, but is, as noted above, itself forecast encompassed by the AWM. Over 36 quarters ICM forecast encompasses

Table 8.11

Forecast encompassing tests over 36 and 20 periods, ending in 2000(3)

 Model k F Enc gumC), 63) Forecast encompassing tests: p-values p-value 1991(4)-2000(3) 1995(4)-2000(3) Mi vs. Mj Mj vs. Mi Mi vs. Mj Mj vs. Mi ICM 11 0.21 18 0.00** AWM 13 0.19 16 0.08 0.06 0.08 0.03* 0.96 P* 12 0.12 17 0.00** 0.11 0.06 0.87 0.06 PAenh 14 0.19 15 0.04* 0.001** 0.18 0.09 0.22 NPCM 7 0.23 22 0.00** 0.64 0.00** 0.10 0.17

The ICM is used as benchmark.

the NPCM, and—like the AWM—it is forecast encompassed by the enhanced version of the P*-model.

An important caveat applies to the results in this section. In interpreting the favourable results for the P*-model it should be borne in mind that the forecasts made for the P*-specifications are greatly helped by the two-sided filters used to define the equilibrium values for, say rm*, as described in Section 8.5.4.[84]

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## THE ECONOMETRICS OF MACROECONOMIC MODELLING

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