RDP 9702: The Implementation of Monetary Policy in Australia Appendix A: Monte Carlo Procedure

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 Inline Equation and Inline Equation, 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:

  1. draw a sequence of observations Inline Equation from a normal distribution with mean 0 and variance Inline Equation;
  2. generate sequences of synthetic data Inline Equation using Inline Equation and Inline Equation, where Inline Equation and Inline Equation are from the OLS estimation using the original data;
  3. use the synthetic data to estimate the equation Inline Equation, by OLS and hence generate parameter estimates Inline Equation and Inline Equation; and
  4. 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 (Inline Equation, j = 1,...,12, ∞) and the year-ended growth rate of output Inline Equation after j quarters.

The figures in the text show the 5th, 50th and 95th percentile values for the effect on the level of output, Inline Equation, and on the year-ended growth rates, Inline Equation.

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 Inline Equation, 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, Inline Equation.

Estimating Equation (A2) by IV, using Inline Equation as an instrument for rt over the period 1980:Q3 to 1996:Q1 leads to parameter estimates Inline Equation and Inline Equation, and an estimate of the variance-covariance matrix of the errors from Equations (A2) and (A4), Inline Equation. The Monte Carlo distribution is then generated by running 1,000 trials with each trial, i, proceeding as follows:

  1. draw two sequences of observations Inline Equation and Inline Equation from a bivariate normal distribution with mean 0 and covariance matrix Inline Equation, such that Inline Equation, where Inline Equation are independent and identically distributed;
  2. generate sequences of synthetic data Inline Equation using Inline Equation and Inline Equation, where Inline Equation and Inline Equation are from the IV estimation using the original data;
  3. generate a sequence of synthetic data Inline Equation according to Inline Equation. Re-estimate Equation (A4) by OLS using Inline Equation instead of rt and obtain a new set of fitted values, Inline Equation;
  4. estimate the equation Inline Equation by IV, using Inline Equation as an instrument for rt, and hence generate parameter estimates Inline Equation and Inline Equation; and
  5. with the parameter estimates Inline Equation and Inline Equation, 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, Inline Equation, 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.