The P*-model of inflation

In the P*-model (Hallman et al. 1991) the long-run equilibrium price level is defined as the price level that would result with the current money stock, mt, provided that output was at its potential (equilibrium level), y*, and that velocity, vt = pt + yt — mt, was at its equilibrium level v*:

pt = mt + vt — y*t. (8.14)

The postulated inflation model is given by

Apt = E(Apt I It-i) + ap(pt-i — p—1) + @zzt + £t, (8.15)

where the main explanatory factors behind inflation are inflation expectations, E(Apt | It-1), the price gap, (pt-1 — pt_ 1), and other variables denoted zt. Note that if we replace the price gap in (8.15) with the output gap, we obtain the NPCM (8.12) discussed in the previous section with the expectations term backdated one period.

In order to calculate the price gap one needs to approximate the two equi­libria for output, yt, and velocity, vt, respectively. The price gap, (pt — pt), is obtained by subtracting pt from both sides of (8.14) and applying the identity vt = pt + yt — mt. It follows that the price gap is decomposed into the velocity gap, (vt — v*), minus the output gap, (yt — y*):

(pt — p*) = (vt — v*) — (yt — y*). (8.16)

The P*-model can alternatively be expressed in terms of the real money gap, rmt — rm*, where rm* = mt — p*. The inverse relationship holds trivially between the real money gap and price gap, that is, (rmt — rm*) = —(pt — p*), and thus the P*-model predicts that there is a direct effect on inflation from the lagged real money gap (rm — rm*)t-l. Moreover, in the P*-model, fluctuations in the price level around its equilibrium, p*, are primarily driven by fluctuations in velocity and output.

Another defining characteristic of recent studies adopting the P*-model is that inflation is assumed to be influenced by Д4pgapt, which is the change in the difference between the actual inflation Д4pt and a reference or target path Д4pt, and also by an analogous variable for money growth, Д4mgapt. The reference path for money growth Д4гпt is calculated in a similar way as suggested in Gerlach and Svensson (2003), referred to below. If we know the inflation target (or reference path for inflation in the case when no explicit target exists), we can calculate the corresponding reference path for money growth as follows (see Bofinger 2000):

Д4 m t = Д4pt + Д4У? — Д4 v*. (8.17)

In our empirical estimates of the P*-model below we have simply let the reference value for inflation, Д^, vary with the actual level of smoothed infla­tion and Д4pgapt is defined accordingly. The heuristic interpretation is that the monetary authorities changed the reference path according to the actual behaviour, adapting to the many shocks to inflation in this period and we calculate Д4pt with a Hodrick-Prescott (HP) filter[71] with a large value of the parameter which penalises non-smoothness, that is, we set A = 6400 to avoid volatility in Д^. Likewise, we apply the HP-filter to derive measures for the equilibrium paths for output, y*, and velocity, v*, and in doing so, we use A = 1600 to smooth output series y* and A = 400 to smooth velocity v*. Д4гаt follows from (8.17), as does Д4mgapt.

Gerlach and Svensson (2003) estimate a variant of the P*-model (8.15), and they find empirical support for the P*-model on aggregated data for the Euro area. In this study Gerlach and Svensson introduce and estimate a measure for the inflation target in the Euro area as a gradual adjustment to the (implicit) inflation target of the Bundesbank, and they interpret the gradual adjustment as a way of capturing a monetary policy convergence process in the Euro area throughout their estimation period (1980(1)-2001(2)).

Gerlach and Svensson (2003) find a significant effect of the energy com­ponent of consumer price index on inflation measured by the total consumer price index, and when they include the output gap in (8.15), in addition to the

real money gap, both gaps come out equally significant, indicating that each is an important determinant of future price changes. By contrast, they find that the Eurosystem’s money-growth indicator, defined as the gap between current M3 growth and its reference value, has little predictive power beyond that of the output gap and the real money gap.

Trecroci and Vega (2002) re-estimate the AWM equation for the GDP deflator at factor prices for the period 1980(4)-1997(4), and they find that (an earlier version of) the Gerlach and Svensson P* equation (without the output gap) outperforms the AWM price equation (for qt) in out of sample fore­casts for the period 1992(1)-1997(4) at horizons ranging from 1 to 8 periods ahead.[72] Likewise, Nicoletti Altimari (2001) finds support for the idea that monetary aggregates contain substantial information about future price devel­opments in the Euro area and that the forecasting performance of models with money-based indicators improves as the forecast horizon is broadened.

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