The estimation of the P*-model in Section 8.5.4 requires additional data relative to the AWM data set. We have used a data series for broad money (M3) obtained from Gerlach and Svensson (2003) and Coenen and Vega (2001), which is shown in Figure 8.8. It also requires transforms of the original data: Figures 8.9 and 8.10 show the price gap (p — p*)t and the real money gap
Figure 8.8. The M3 data series plotted against the shorter M3 series obtained from Gerlach and Svensson (2003), which in turn is based on data from Coenen and Vega (2001). Quarterly growth rate
Figure 8.9. The upper graphs show the GDP deflator and the equilibrium price level (p*), whereas the lower graph is their difference, that is, the price gap, used in the P*-model
(rm — rm*)t along with the corresponding level series. As noted in Section 8.5.4 we have applied HP-filters to derive measures for y* and v* . Then p* can be calculated from (8.14), as well as the price and real money gaps.
The reference path for inflation is trend inflation from a smoothed HP filter, as described in Section 8.5.4. In Figure 8.11 we have plotted trend inflation together with an alternative which is the same series with the reference path
for the price (target) variable of Gerlach and Svensson (2003) substituted in for the period 1985(1)-2000(2). It is seen that the alternative reference path series share a common pattern with the series we have used. Figure 8.12 shows the corresponding graphs for the reference path of money growth.
The P*-model is estimated in two versions: one version is related to the standard formulation of the P*-model as discussed in Section 8.5.4, in which inflation is explained by the real money gap (rm — rm*) and the differences between actual price and money growth from their reference (target) paths, A4pgapt and A4mgapt.14
In order to retain comparability across the inflation models, we differ from previous studies by using the private consumption deflator rather than, for example, the GDP deflator of Trecroci and Vega (2002) or a consumer prices index like the one constructed by Gerlach and Svensson (2003). We also include four lags of inflation, two lags of output growth, Ayt, and an interest rate spread gap sgapt (defined as the deviations of the actual spread from a HP trend spread). The other version, P* enhanced, is modelled general to specific, where the general specification is based on the information set of AWM with (rm — rm*)t, A4pgapt, A4mgapt, and sgapt substituted for the equilibrium - correction terms ecmpAWM and ecmwAWM. 
After we have imposed valid restrictions, the first version based on the narrower information set becomes:
Apt = - 0.0015 + 0.60Apt_і + 0.24Apt_2 + 0.19Apt_4 (0.0005) (0.08) (0.09) (0.07)
+ 0.18Ayt_ і — 0.05A4pgapt_ 1 — 0.04A4mgapt_ 1
(0.04) t (0.02) t і (0.03) t і
+ 0.09 (rm — rm*)t-1 — 0.0006sgapt_ 1 + dummies (0.03) (0.0003)
a = 0.00211
Far(i-5)(5, 95) = 0.52[0.76] FArch(i-4)(4, 92) = 0.68[0.61]
x2normality(2) = 0.42[0.81] Fhetx?(21, 78) = 0.81[0.70]
Freset(1, 99) = 7.27[0.008**]
We find that money growth deviation from target A4mgapt_ 1 is insignificant which is in line with results reported in Gerlach and Svensson (2003). The other explanatory variables specific to the P*-model comes out significant and with expected signs. The model shows signs of mis-specification through the significant RESET-test.
The enhanced P*-model—based on the broader information set—is given by:
a = 0.00190
Far(i-5)(5, 93) = 0.65[0.66] Farch(i-4)(4, 90) = 0.74[0.56] X2normaiity(2) = 3.83[0.15] Fhetx^ (25, 72) = 0.76[0.77]
Freset(1, 97) = 0.01[0.93]
The model reduction is supported by the data, and the enhanced P* is well specified according to the standard diagnostics reported. We find the P*-model based on the broader information set variance encompasses the P*-model derived from the narrower set of variables, with a reduction of the estimated a of equation (8.21) of 10% compared with the estimated a of equation (8.20).
A striking feature of the enhanced P*-model is that the short-run explanatory variables in the first two lines are nearly identical to its counterpart in the AWM reduced form inflation equation (Aat-1 substituting for Apit-1) with coefficients of the same order of magnitude. The real money gap (rm — rm* )t-1 is highly significant, whereas sgapt drops out. Also, the P*-specific explanatory
1990 2000 1990 2000 1990 2000
1990 2000 1990 2000 1990 2000
Figure 8.13. Recursive coefficient estimates of the P*-model based
on the broad information set
variables, A4pgapt_1 and A4mgapt_4—the deviations from target—are
insignificant at the 5% level, but are retained to represent the P* mechanisms.
Figure 8.13 shows that the coefficient estimates of the enhanced P*-model are recursively stable.