RDP 9702: The Implementation of Monetary Policy in Australia Appendix A: Monte Carlo Procedure
April 1997
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This appendix outlines the Monte Carlo procedure used to generate confidence intervals for the OLS, IV and recursive regressions.
A.1 Ordinary Least Squares Regressions
The non-farm output equation, rewritten here for convenience, is
which may be simplified to
where Nt is the vector of explanatory variables excluding yt−1.
A sustained one per cent rise in the real interest rate leads to an effect on the level of output after j quarters (mj) of:
Estimating Equation (A2) by OLS over the 63 quarters 1980:Q3 to 1996:Q1 leads to
parameter estimates and
,
and an estimate of the standard deviation of the errors, σε
= 0.56, for both the underlying and headline models. The Monte Carlo distribution
is then generated by running 1,000 trials with each trial, i, proceeding
as follows:
-
draw a sequence of observations
from a normal distribution with mean 0 and variance
;
-
generate sequences of synthetic data
using
and
, where
and
are from the OLS estimation using the original data;
-
use the synthetic data to estimate the equation
, by OLS and hence generate parameter estimates
and
; and
-
with these parameter estimates, use Equation (A3) to calculate, for this ith
iteration, the effect of a one per cent rise in the real interest rate on
the level of output (
, j = 1,...,12, ∞) and the year-ended growth rate of output
after j quarters.
The figures in the text show the 5th, 50th and 95th percentile values for
the effect on the level of output, , and on the year-ended
growth rates,
.
A.2 Instrumental Variable Regressions
The policy reaction function, rewritten for convenience, is
Estimating the underlying CPI version of Equation (A4) by OLS over the 63 quarters
1980:Q3 to 1996:Q1 leads to fitted values ,
and an estimate of the standard deviation of the errors, σu
= 1.32. Diagnostic tests on the sample errors reveal strong signs of first-order
autocorrelation, with an estimated autocorrelation coefficient,
.
Estimating Equation (A2) by IV, using as an instrument for rt over the period 1980:Q3 to 1996:Q1 leads
to parameter estimates
and
, and an estimate
of the variance-covariance matrix of the errors from Equations (A2) and (A4),
.
The Monte Carlo distribution is then generated by running 1,000 trials with
each trial, i, proceeding as follows:
-
draw two sequences of observations
and
from a bivariate normal distribution with mean 0 and covariance matrix
, such that
, where
are independent and identically distributed;
-
generate sequences of synthetic data
using
and
, where
and
are from the IV estimation using the original data;
-
generate a sequence of synthetic data
according to
. Re-estimate Equation (A4) by OLS using
instead of rt and obtain a new set of fitted values,
;
-
estimate the equation
by IV, using
as an instrument for rt, and hence generate parameter estimates
and
; and
-
with the parameter estimates
and
, use Equation (A3) to calculate, for this ith iteration, the effect of a one per cent rise in the real interest rate on the year-ended growth rate of output,
, after j quarters.
The figures in the text show the 5th, 50th and 95th percentile values for the year-ended growth rates.
A.3 Recursive Regressions
For the recursive regressions, a new Monte Carlo distribution is estimated from 1,000 trials after each new quarter of data is added.