RDP 9702: The Implementation of Monetary Policy in Australia Appendix B: Estimating the Bias from Ordinary Least Squares

It is convenient to rewrite the model for non-farm GDP growth, Equation (1) in the text, as

where Zt−1 = yt−1χ*wt−1 (χ* is the cointegrating vector between y and w) and Wt is the matrix of exogenous variables, Wt = [1 Δwt Δwt−1 Δft−2 Δft−4]. Equation (B1) may be further simplified to

where Xt = [rt rt−1 rt−2 rt−3 rt−4 rt−5 rt−6 Zt−1] is the matrix of regressors presumed to be correlated with the disturbance term, εt.

OLS on Equation (B2) yields the following estimate for α,

where MW = IW(WW)−1W′.


as plimInline Equation since W is exogenous. In the limit, as the sample size increases, the true value of the vector α is then

We presume that the short-term real interest rate, rt, can be expressed as

where ‘exogenous’ implies uncorrelated with the error term in Equation (B2) and ut is determined by the policy-maker on the basis of information about current and future output not available to the econometrician estimating the output equation (B2).

As explained in the text, the correlation coefficient between real interest rates and the error term in Equation (B2) is assumed to be a geometrically declining function of the lag of the real interest rate, with no correlation after the sixth lag. The covariance between Zt−1 and εt (which is identical to the covariance between yt−1 and εt) is denoted σε σu θ and is derived below. In symbols we have

where σε and σu are estimates of the standard deviations of the errors in Equations (B1) and (B5). For the underlying model, σε = 0.56, while for σu, we use the value derived from estimating Equation (3) in the text (which is a simple version of Equation (B5)). This gives the estimate σu=1.32.

Now define the variables Ci, i = 0,...,6 by

Denote the covariance between εt and yt−i as PLi, and between εt and Δyt−i as PCi. The model, Equation (B2), implies the recursive structure,

We require θ = PL1εσu, which is a function of the true vector α. For given γ in the range zero to 0.75, we proceed as follows. First, we use the sample value of Inline Equation as our estimate of plimInline Equation. (This requires an estimate of the cointegrating vector, χ*, between y and w; we use the OLS estimate for this.) Next, we use the OLS estimate, Inline Equation to generate an estimate Inline Equation via Equation (B8). We now have an estimate for plimInline Equation via Equation (B6). This enables us to generate an estimate, Inline Equation, of the ‘true’ vector α via Equation (B4). We now iterate: Inline Equation implies a new estimate for θ, Inline Equation, which, in turn, implies a new estimate for α, Inline Equation. This process is continued until it converges, yielding Inline Equation. The estimated response to a permanent 1 per cent increase in the real interest rate on year-ended growth shown in Figure 4 and on the average lag of monetary policy shown in Figure 5 are generated using Inline Equation.