Revisiting the Great Moderation using the Method of Indirect Inference

model. An auxiliary time series model is then tted to each set of data and the sampling
distribution of the coecients of the auxiliary time series model is obtained from these
estimates of the auxiliary model. A Wald statistic is computed to determine whether
functions of the parameters of the time series model estimated on the actual data lie in
some condence interval implied by this sampling distribution.
Following Minford, Theodoridis and Meenagh (2009) we take a VAR(1) for the three
macro variables (interest rate, output gap and ination) as the appropriate auxiliary
model and treat as the descriptors of the data the VAR coecients and the variances
of these variables. The Wald statistic is computed from these 8 . Thus eectively we are
testing whether the observed dynamics and volatility of the chosen variables are explained
by the simulated joint distribution of these at a given condence level. The Wald statistic
is given by:
X1
( ) 0 ( ) (6)
()
the squared `Mahalanobis distance', where is the vector of VAR estimates of the chosen
descriptors yielded in each simulation, with and P ()
representing the corresponding
sample means and variance-covariance matrix of these calculated across simulations,
respectively 9 .
Figure 2 illustrates the whole testing procedure. While panel A of the gure summarises
the main steps just described, the `mountain-shaped' diagram in the second panel
gives an example of how the `reality' is compared to model predictions using the Wald
8 Note that the VAR impulse response functions, the co-variances, as well as the auto/cross correlations
of the left-hand-side variables will all be implicitly examined when the VAR coecient matrix is
considered, since the former are functions of the latter.
9 Smith (1993), for his demonstration of model estimation, originally used VAR(2) as the auxiliary
model. His VAR included the logged output and the logged investment and he tried to maximize the
model's capacity in tting the dynamic relation between these. To this end he included the ten VAR
coecients (including two constants) in his vector of data descriptors. Here, since a VAR(1) is chosen
to provide a parsimonious description of the data and the models are tested against their capacity in
tting the data's dynamic relations and size, the vector of chosen data descriptors includes nine VAR(1)
coecients and three data variances. No constant is included since the data are demeaned and detrended.
In the Supporting Annex we show our results that follow are robust to the choice of VAR: it turns
out that using a VAR of higher orders, though strengthening the test's rejection power, will not cause
change in the ranking between the models.
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