πt+1= Xtβt+1 + et+1 et+1 ~N(0,ht)
and Xt = [1 πt, ..., πt-k, ut ...,ut-m] (1)
ht=h+aet2-1+λht-1 (2)
βt+1=βt+Vt+1 where Vt+1~N(0,Q) (3)
where πt+1 denotes the rate of inflation between t and t+1; Xt is a vector of
explanatory variables known at time t containing current and lagged values of
inflation and the unemployment rate ut; et+1 describes the shocks to the inflation
process that cannot be forecast with information known at time t, and is assumed to be
normally distributed with a time-varying conditional variance ht. The conditional
variance is specified as a GARCH(p,q) process, that is, as a linear function of past
squared forecast errors, e2t-i, and past variances, ht-j.
Further,βt+1 = [β0π,t+1,β1π,t+1,...,βkπ,t+1,β1u,t+1,...,βmu,t+1] denotes the time-varying parameter
vector, and Vt+1 is a vector of shocks to βt+1, assumed to be normally distributed with
a homoscedastic covariance matrix Q. The updating equations for the Kalman filter
are:
πt+1= XtEtβt+1 +εt+1 (4)
Ht = XtΩt+itXt' + ht (5)
Et+1βt+2 = Et βt+ι + [Ωt+itXtHt-1>t+1 (6)
Ωt+2∣ t+1 = [I - Ωt+ιkXt'Ht-ιXt ]Ωt+ψ + Q (7)
where Ωt+1∣t is the conditional covariance matrix of βt+1 given the information set at
time t, representing uncertainty about the structure of the inflation process.
As Equation (5) indicates, the conditional variance of inflation (short-run uncertainty),
Ht, can be decomposed into: (i) the uncertainty due to randomness in the inflation
shocks et+1, measured by their conditional volatility ht (impulse uncertainty); (ii) the
uncertainty due to unanticipated changes in the structure of inflation Vt+1, measured
by the conditional variance of Xtβt+1, which is XtΩt+1∣tXt' = St (structural
uncertainty). The standard GARCH model can be obtained as a special case of this
model if there is no uncertainty about βt+1, so that Ωt+1∣t = 0.
In this case, the conditional variance of inflation depends solely on impulse
uncertainty. Equations (6) and (7) capture the updating of the conditional distribution
of βt+1 over time in response to new information about realised inflation. As indicated
by Equation (6), inflation innovations, defined as εt+1 in Equation (4), are used to
update the estimates of βt+1 . These estimates are then used to forecast future inflation.
If there are no inflation shocks and parameter shocks, so that πt+1 =πt=... = πt-k for
all t, we can calculate the steady-state rate of inflation, πt*+1 , as:
**
πt+1 = β0,t+1
[1-(∑ik=1βiπ,t+1)]
(8)