Forecast errors of stylised inflation models

We formulate a simple DGP to investigate the theoretical forecasting capabilities of the ICM and the PCM, thus providing a background for the interpretation of the actual forecast errors in Section 11.3.3. The variable symbols take the same meaning as in the earlier chapters on wage-price mod­elling (see Chapter 6), hence (in logs) wt is the wage rate, pt is the consumer price index, pit denotes import prices, and ut is the rate of unemployment.

In order to obtain an analytically tractable distillation of the models, we introduce simplifying assumptions. For example, we retain only one cointegrating relationship, the ‘wage-curve’, and we also abstract from productivity.[112] Thus (11.44) is a simplified version of the dynamic wage equation of Chapter 6:

A(w - p)t = к - nw [(w - p)t-i + Xut-i - p]+ew, t, nw > 0, X > 0.


The wage-curve is the term in square brackets. The parameter p denotes the mean of the long-run relationship for real wages, that is, E[(w - p)t-1 - Xut-1 - p] = 0. Since we abstract from the cointegration rela­tionship for consumer prices, the simultaneous equation representation of the inflation equation is simply that Apt is a linear function of Apit and Awt, and the reduced form equation for Apt is:

Apt = Фр + у pi Apit - np[(w - p)t-i + Xut-i - p]+ ep, t,

VPi > 0, Пр > 0. (11.45)

Multi-step (dynamic) forecasts of the rate of inflation require that also import price growth and the rate of unemployment are forecasted. In order to simplify as much as possible, we let Apit and ut follow exogenous stationary processes:

Apit = Фрі + tpi, t, (11.46)

Aut = фи - nuut-i + euit, Пи > 0. (11.47)

IT denotes the information set available in period T. The four disturbances (ew, t, ep, t, epi, t, eu, t) are innovations relative to IT, with contemporaneous covariance matrix П. Thus, the system (11.44)-(11.47) represents a simple DGP for inflation, the real wage, import price growth, and the rate of unemployment. The forecasting rule

Apt+h = E[ApT+h It] = a0 + aiSpi + a2E[(w - p)T+h-i | It]

+ a3E[uT+h-i It], h =1, 2,...,H


with coefficients

a0 фp + Пpp,

ai (уpi,

a2 = Пp,

a3 — npX

is the minimum MSFE predictor of ApT+h, by virtue of being the conditional expectation.

First, we abstract from estimation uncertainty and assume that the param­eters are known. The dynamic ICM forecast errors have the following means and variances:

Подпись: Е[Арт+hПодпись: Var[ApT+hApT+h, icm 11 ] — 0, (11.49)


APT+h, ICM I IT ] — Vp + Vpi + «2 Yj (1 — )2{h-1-t)ai


+ a, l(nw Л)2 ^(1 - 'nwf{-h-1-i)


X ^(1 - nu)2{t-f>VU j=1

+ «3 ^(1 - nu)2{h-1-t)vl. (11.50)


The first two terms on the right-hand side of (11.50) are due to ep, T+h and £pi, T+h. The other terms on the right-hand side of (11.50) are only relevant for h — 2, 3, 4,.. .,H. The third and fourth terms stem from (w — p)T+h-1—it is a composite of both wage and unemployment innovation variances. The last line contains the direct effect of Var[uT+h-1] on the variance of the inflation forecast. In addition, off-diagonal terms in H might enter.

We next consider the case where a forecaster imposes the PCM restriction nw — 0 (implying np — 0 as well). The ‘Phillips curve’ inflation equation is then given by:

Apt — «о + a,1Apit + a3ut-1 + «p, t, (11.51)


ao — «о + a^E[ut-1] + «2Р and ip, t — Cp, t + «2[(w — p)t-1 — Лщ-1 — р].

This definition ensures a zero-mean disturbance E[ePjt | IT] — 0. Note also that Var[ep t | It-1] — <rp, that is, the same innovation variance as in the ICM-case. The PCM forecast rule becomes

Apt+h, PCM — E[Apt+h, PCM 1 IT ] — «0 + «1 dpi + «4uT+h-1.

The mean and variance of the 1-step forecast error are

E[ApT+1 — Apt +1,pcm I It] — («1 — «1)dpb + ut(«3 — «з)ит

+ «2 {(w — p)t — ЛE[ut] — р}, Var[ApT+1 — ApT +1,pcm 1 It ] — Vp + Vpi.

The 1-step ahead prediction error variance conditional on It is identical to the ICM-case. However, there is a bias in the 1-step PCM forecast arising from two sources: first, omitted variables bias implies that «1 — «1 and/or «3 — a3,

in general. Second,

(w - р)т - AE[wt] - p = 0

unless (w - p)T = E[(w — p)t], that is, the initial real wage is equal to the long-run mean of the real-wage process.

For dynamic h period ahead forecasts, the PCM prediction error becomes

Арт+fe - Арт+h, pcm = (ai - ai)5pb + (аз - йз)йт+ь-1


+ аз Ed - nu)h 1 l^u, T+i + epi, T+h + Cp, T+h i=1

+ a2(w - p)T+h-i - a2(AE[ut] - p).

Taking expectation and variance of this expression gives:

E[ApT+h - ApT+h, pcM I It] = (ai - ai)^pi + (a4 - a4)«T+h-i

+ a2{E[(w - p)t+h-i I It] - AE[ut] - p},

Var[ApT+h - ApT+h, pcm 11 ] = Var[ApT+h - ^pT+h, icm 11L

for h = 2, 3, ...,H.

Hence systematic forecast error is again due to omitted variables bias and the fact that the conditional mean of real wages h - 1 periods ahead, departs from its (unconditional) long-run mean. However, for long forecast horizons, large H, the bias expression can be simplified to become

E[ApT+h - ApT+hpcm 1 I-t] ~ (ai - ai)Spi + (a4 - a4) —


since the conditional forecast of the real wage and of the rate of unemployment approach their respective long-run means.

Thus far we have considered a constant parameter framework: the param­eters of the model in equations (11.44)-(11.47) remain constant not only in the sample period (t = 1,...,T) but also in the forecast period (t = T +1,. ..,T + h). However, as discussed, a primary source of forecast failure is structural breaks, especially shifts in the long-run means of cointegrating relationships and in parameters of steady-state trend growth. Moreover, given the occurrence of deterministic shifts, it no longer follows that the ‘best’ econometric model over the sample period also gives rise to the minimum MSFE; see, for example, Section 11.2.

That a tradeoff between close modelling and robustness in forecasting also applies to wage-price dynamics is illustrated by the following example: assume that the long-run mean p of the wage-equation changes from its initial level to a new level, that is, p ^ p*, before the forecast is made in period T, but that the change is undetected by the forecaster. There is now a bias in the (1-step) ICM real-wage forecast:

E[(w - p)t +i - (w-p)t +i, ICM IIt] = - nw[p - p*L (11.52)

which in turn produces a non-zero mean in the period 2 inflation forecast error:

E[ApT+2 - Apt+2,icm 1 IT] = - a2nw [p - p*]. (11.53)

The PCM-forecast on the other hand, is insulated from the parameter change in wage formation, since (w — p)Tdoes not enter the predictor—the fore­cast error is unchanged from the constant parameter case. Consequently, both sets of forecasts for ApT+2+h are biased, but for different reasons, and there is no logical reason why the PCM forecast could not outperform the ICM forecast on a comparison of biases. In terms of forecast properties, the PCM, despite the inclusion of the rate of unemployment, behaves as if it was a dVAR, since there is no feedback from wages and inflation to the rate of unemployment in the example DGP.

Finally, consider the consequences of using estimated parameters in the two forecasting models. This does not change the results about the forecast biases. However, the conclusion about the equality of forecast error variances of the ICM and PCM is changed. Specifically, with estimated parameters, the two models do not share the same underlying innovation errors. In order to see this, consider again the case where the ICM corresponds to the DGP. Then a user of a PCM does not know the true composition of the disturbance ep, t in (11.51), and the estimated PCM will have an estimated residual variance that is larger than its ICM counterpart, since it is influenced by the omitted wage-curve term. In turn, the PCM prediction errors will overstate the degree of uncertainty in inflation forecasting. We may write this as

Var[<=p t I It, PCM] > Var^t | It, ICM]

to make explicit that the conditioning is with respect to the two models (the DGP being unknown). From equation (11.51) it is seen that the size of the dif­ference between the two models’ residual variances depends on (1) the strength of equilibrium correction and (2) the variance of the long-run wage curve.

The main results of this section can be summarised in three points:

1. With constant parameters in the DGP, forecasting using the PCM will bias the forecasts and overstate the degree of uncertainty (i. e. if the PCM involves invalid parameter restrictions relative to the DGP).

2. PCM forecasts are however robust to changes in means of (omitted) long-run relationships.

3. Thus PCM shares some of the robustness of dVARs, but also some of their drawbacks (specifically, excess inflation uncertainty).

In sum, the outcome of a forecast comparison is not a given thing, since in practice we must allow for the possibility that both forecasting models are mis-specified relative to the generating mechanism that prevails in the period we are trying to forecast. A priori we cannot tell which of the two models will forecast best. Hence, there is a case for comparing the two models’ forecasts directly, even though the econometric evidence presented in earlier chapters has gone in favour of the ICM as the best model.

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