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 N_{t} is the vector of explanatory variables excluding y_{t−1}.
A sustained one per cent rise in the real interest rate leads to an effect on the level of output after j quarters (m_{j}) 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, σ_{ε} = 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 i^{th} 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 5^{th}, 50^{th} and 95^{th} 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, σ_{u} = 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 r_{t} 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 r_{t} and obtain a new set of fitted values, ;
- estimate the equation by IV, using as an instrument for r_{t}, and hence generate parameter estimates and ; and
- with the parameter estimates and , use Equation (A3) to calculate, for this i^{th} 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 5^{th}, 50^{th} and 95^{th} 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.