We start with a recursive VAR model of the state vector
where the vector Zt is the transpose of [Z[, Zft_1, Z*_2], and U is a lower triangular matrix with ones on the diagonal and nonzero off-diagonal elements only in the first three rows, the off-diagonal elements of which are estimated so as to make the residuals in et orthogonal to one another. The first three rows of the vectors et contain the VAR residuals e^, e^t and while the other elements are zero. The number of lags included in our VAR is sufficient to eliminate nearly all evidence of serial correlation in the disturbances itat on.
The first equation in this VAR is our estimate of the monetary policy rule. This estimated rule has the same structure as (1.23), except that it also includes a white noise residual eu. Note that while the interest rate comes first in the casual ordering, the timing of the variables ensures that the interest rate in period t responds to inflation and output in period t, wrhile these variables only react to lagged interest rates. We suppose that ei,( is independent of the two “real” disturbances Yts and Gt so that it is exclusively a monetary policy disturbance. (Note that these identifying assumptions are ones that are implied by the decision lags assumed in our theoretical model.)
Under these assumptions, we can estimate not only the coefficients of the historical monetary policy rule, but also the impulse responses of output, inflation and the interest rate to a monetary policy disturbance. We can then recover most of the structural parameters of our model by minimizing the discrepancy between the estimated responses of these variables to the monetary disturbance and the responses predicted by our theoretical model when the systematic part of the monetary policy rule is given by the estimated coefficients in (1.23).12 By calibrating the remaining parameters on the basis of other evidence, we obtain numerical values for the model parameters a, /?, 7, сг, в and uj. These are, respectively, .66, .99, .63, .16, 7.88 and .47, so that к equals .024 and ф equals .53.
Armed with our parameter values and the VAR, we can reconstruct the stochastic processes for the structural disturbances as follows.13 Equation (1.11) gives Gt as the sum of Yt and a times the expected long term real rate. Given that the VAR allows us to forecast both inflation and interest rates, this expected long term real rate is a function of Zt. Similarly, solving (1.20) for Xt as a function of inflation and expected inflation, and substituting this into (1.21), we find that Yts must be given by