NPCM as a system

Equation (7.1) is incomplete as a model for inflation, since the status of xt is left unspecified. On the one hand, the use of the term forcing variable, suggests exogeneity, whereas the custom of instrumenting the variable in estimation is germane to endogeneity. In order to make progress, we therefore consider the following completing system of stochastic linear difference equations3

Apt = 6piApt+i + bP2xt + £pt - bpmt+1, (7.2)


xt = bx1Apt-1 + bx2xt—1 + £xt: 0 E |bx2 I ^ E (7.3)

Подпись: p(A) = A2 Подпись: 7 + bx2 bp1 Подпись: A +7—[bp2bx1 + bx2] • bp1 Подпись: (7.5)

and substitute xt with the right-hand side of equation (7.3). The characteristic polynomial for the system (7.3) and (7.4) is

If neither of the two roots is on the unit circle, unique asymptotically stationary solutions exist. They may be either causal solutions (functions of past values of the disturbances and of initial conditions) or future dependent solutions (functions of future values of the disturbances and of terminal conditions), see Brockwell and Davies (1991: ch. 3) and Gourieroux and Monfort (1997: ch. 12).

The future dependent solution is a hallmark of the NPC. Consider for example the case of bx1 = 0, so that xt is a strongly exogenous forcing variable in the NPCM. This restriction gives the two roots A1 = b— and A2 = bx2.

Подпись: 3Constant terms are omitted for ease of exposition.

Given the restriction on bx2 in (7.3), the second root is always less than one, meaning that xt is a causal process that can be determined from the backward solution. However, since Ai = b— there are three possibilities for Apt: (1) No stationary solution: bp1 = 1; (2) A causal solution: bp1 > 1; (3) A future dependent solution: bp1 < 1. If bx1 = 0, a stationary solution may exist even in the case of bp1 = 1. This is due to the multiplicative term bp2bx1 in (7.5). The economic interpretation of the term is the possibility of stabilising inter­action between price-setting and product (or labour) markets—as in the case of a conventional Phillips curve.

As a numerical example, consider the set of coefficient values: bp1 = 1, bp2 = 0.05, bx2 = 0.7, and bx1 = 0.2, corresponding to xt (interpreted as the output-gap) influencing Apt positively, and the lagged rate of inflation having a positive coefficient in the equation for xt. The roots of (7.5) are in this case {0.96, 0.74}, so there is a causal solution. However, if bx1 < 0, there is a future dependent solution since then the largest root is greater than one.

Finding that the existence and nature of a stationary solution is a system property is of course trivial. Nevertheless, many empirical studies only model the Phillips curve, leaving the xt part of the system implicit. This is unfor­tunate, since the same studies often invoke a solution of the well-known form[56]

Clearly, (7.6) hinges on bp1bx2 < 1 which involves the coefficient bx2 of the xt process.

If we consider the rate of inflation to be a jump variable, there may be a saddle-path equilibrium as suggested by the phase diagram in Figure 7.1. The drawing is based on bp2 < 0, so we now interpret xt as the rate of unem­ployment. The line representing combinations of Apt and xt consistent with A2pt = 0 is downward sloping. The set of pairs {Apt, xt} consistent with Axt = 0 are represented by the thick vertical line (this is due to bx1 = 0 as above). Point a is a stationary situation, but it is not asymptotically stable. Suppose that there is a rise in x represented by a rightward shift in the vertical curve, which is drawn with a thinner line. The arrows show a potential unstable trajectory towards the north-east away from the initial equilibrium. However, if we consider Apt to be a jump variable and xt as state variable, the rate of inflation may jump to a point such as b and thereafter move gradually along the saddle path connecting b and the new stationary state c.

The jump behaviour implied by models with forward expected inflation is at odds with observed behaviour of inflation. This has led several authors to sug­gest a ‘hybrid’ model, by heuristically assuming the existence of both forward - and backward-looking agents; see, for example, Fuhrer and Moore (1995). Also Chadha et al. (1992) suggest a form of wage-setting behaviour that would

Подпись: Figure 7.1. Phase diagram for the system for the case of bpi < 1, bp2 < 0, and bxi = 0

lead to some inflation stickiness and to inflation being a weighted average of both past inflation and expected future inflation. Fuhrer (1997) examines such a model empirically and finds that future prices are empirically unimportant in explaining price and inflation behaviour compared to past prices.

In the same spirit as these authors, and with particular reference to the empirical assessment in Fuhrer (1997), GG also derive a hybrid Phillips curve that allows a subset of firms to have a backward-looking rule to set prices. The hybrid model contains the wage share as the driving variable and thus nests their version of the NPCM as a special case. This amounts to the specification

Apt = bpiEt Apt+i + bpi Apt-i + bp2xt + £pt - (7.7)

Gall and Gertler (1999) estimate (7.7) for the United States in several vari­ants—using different inflation measures, different normalisation rules for the GMM estimation, including additional lags of inflations in the equation and splitting the sample. Their results are robust—marginal costs have a signifi­cant impact on short-run inflation dynamics and forward-looking behaviour is always found to be important.

Подпись: Apt+i image108 Подпись: (7.8)

In the same manner as above, equation (8.13) can be written as

Подпись: p(A) = A3 - Подпись: —f + bx2 bpi Подпись: 1 bb a2 + f [bpi + bp2bxi + bx2] A - f bx2- bpi bpi Подпись: (7.9)

and combined with (7.3). The characteristic polynomial of the hybrid system is

Using the typical results for the expectation and backward-looking parameters, bpi = 0.25, bpi = 0.75, together with the assumption of an exogenous xt
process with autoregressive parameter 0.7, we obtain the roots {3.0, 1.0, 0.7}.[57] Thus, there is no asymptotically stable stationary solution for the rate of inflation in this case.

This seems to be a common result for the hybrid model as several authors choose to impose the restriction

Подпись: (7.10)bPi + bpi — 1,

image115 Подпись: bpi 2 bp2 1 Apt + f A pt — f xt — f Spt + nt+b bpi bpi bpi

which forces a unit root upon the system. To see this, note first that a 1-1 reparameterisation of (7.8) gives

so that if (7.10) holds, (7.8) reduces to

Подпись: (7.11)л2„ _ (1 bpi^ 2„ bp2 1 ,

A Pt+i — —j----------- A Pt — f xt — f &pt + nt+i.

bpi bpi bpi

Hence, the homogeneity restriction (7.10) turns the hybrid model into a model of the change in inflation. Equation (7.11) is an example of a model that is cast in the difference of the original variable, a so-called differenced autoregressive model (dVAR), only modified by the driving variable xt. Consequently, it rep­resents a generalisation of the random walk model of inflation that was implied by setting bpi — 1 in the original NPCM. The result in (7.11) will prove import­ant in understanding the behaviour of the NPCM in terms of goodness of fit, see later.

If the process xt is strongly exogenous, the NPCM in (7.11) can be con­sidered on its own. In that case (7.11) has no stationary solution for the rate of inflation. A necessary requirement is that there are equilibrating mech­anisms elsewhere in the system, specifically in the process governing xt (e. g. the wage share). This requirement parallels the case of dynamic homogeneity in the backward-looking Phillips curve (i. e. a vertical long-run Phillips curve). In the present context, the message is that statements about the stationarity of the rate of inflation, and the nature of the solution (backward or forward) requires an analysis of the system.

The empirical results of GG and GGL differ from other studies in two respects. First, bpi is estimated in the region (0.65, 0.85) whereas bpi is one third of bfi or less. Second, GG and GGL succeed in estimating the hybrid model without imposing (7.10). GGL (their table 2) report the estimates {0.69,0.27} and {0.88, 0.025} for two different estimation techniques. The corresponding roots are {1.09, 0.70, 0.37} and {1.11,0.70,0.03}, illustrating that as long as the sum of the weights is less than one the future dependent solution prevails.

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