Forecasting and Policy Making (Paper) - Center for Financial Studies

Table 8: State Space Representation and Model Equations
structural parameters θ = (β,τ,ρg,ρz,γ,r ∗ ,π ∗ ,κ,ρR,ψ1,ψ2,σR,σg,σz)
observable variables y obs
t = [ Y GRt INFLt INTt ] ′
steady state y(θ) = [ 0 ln π ∗ ln r ∗ + ln π ∗ ] ′
deterministic trend λ = [ ln γ 0 0 ] ′
subset of endogenous
variables
y s t = [ ∆xt + zt πt Rt ] ′
endogenous variables yt = [ xt Rt πt gt zt ] ′
shocks ut = [ ǫR,t ǫz,t ǫg,t ] ′
rules. Its parameters are given by the two solution matrices gy and gu which are nonlinear functions
of the structural parameters θ. Thus, the transition equations relate the endogenous variables yt to
lags of themselves and the vector of exogenous shocks ut. Since, the measurement equations include
the deterministic growth path that is driven by labor-augmenting technological progress no separate
de-trending of the data is necessary.
Model Estimation
DSGE models are often estimated using Bayesian estimation. Bayesian estimation combines
the likelihood with prior beliefs obtained from economic theory, microeconomic data or previous
macro studies. An extensive survey of the methodology is presented in An and Schorfheide
(2007). Because the reduced-form coefficients of the state-space representations are nonlinear
functions of the structural parameters, θ, the calculation of the likelihood is not straightforward.
The Kalman filter is applied to the state space representation to set up the likelihood function
(see e.g. Hamilton, 1994, chapter 13.4). Combining the likelihood with the priors yields the log
posterior kernel lnL(θ|Y T ) + lnp(θ) that is maximized over θ using numerical methods so as to
obtain the posterior mode. In principle the posterior mode can be used to generate point forecasts
directly. However, most authors simulate the whole posterior distribution of the parameters using the
Metropolis-Hastings-Algorithm, compute density density forecasts based on draws from the poste-
rior distribution of the parameters. The mean of the density forecast is usually used as a point forecast.
Forecasting
To compute forecasts one can either estimate the model using the most recent available data vin-
tage or combine earlier parameter estimates with the most recent data. In both cases the Kalman-Filter
is applied to estimate the current state of the economy including all unobservable variables and shock
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