Inflation and Inflation Uncertainty in the Euro Area

πt+1= Xtβt+1 + et+1     et+1 ~N(0,ht)

and X_t = [1 π_t, ..., π_t-k, u_t ...,u_t-m] (1)

h_t=h+ae_t²_-1+λh_t-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,β_t+1 = [β₀^π_,t+1,β₁^π_,t+1,...,β_k^π_,t+1,β₁^u_,t+1,...,β_m^u_,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)

H_t = XtΩt+itXt' + h_t                                                              (5)

E_t+1βt+2 = E_t βt+ι + [Ωt+itXtHt-₁>t+1                                                    (6)

Ωt+2∣ t+1 = [I - Ωt+ι_kXt'Ht-ιXt ]Ω_t+ψ + Q                                              (7)
where Ω_t+1∣_t is the conditional covariance matrix of β_t+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 X_tβ_t+1, which is X_tΩ_t+1∣_tX_t' = S_t (structural
uncertainty). The standard GARCH model can be obtained as a special case of this
model if there is no uncertainty about β_t+1, so that Ω_t+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 β_t+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 β_t+1 . These estimates are then used to forecast future inflation.
If there are no inflation shocks and parameter shocks, so that π_t+1 =π_t=... = π_t-k for
all t, we can calculate the steady-state rate of inflation, π_t^*₊₁ , as:

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