RDP 9702: The Implementation of Monetary Policy in Australia Appendix B: Estimating the Bias from Ordinary Least Squares
April 1997
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It is convenient to rewrite the model for non-farm GDP growth, Equation (1) in the text, as
where Z_{t}_{−1} = y_{t}_{−1} − χ^{*}w_{t}_{−1} (χ^{*} is the cointegrating vector between y and w) and W_{t} is the matrix of exogenous variables, W_{t} = [1 Δw_{t} Δw_{t}_{−1} Δf_{t}_{−2} Δf_{t}_{−4}]. Equation (B1) may be further simplified to
where X_{t} = [r_{t} r_{t}_{−1} r_{t}_{−2} r_{t}_{−3} r_{t}_{−4} r_{t}_{−5} r_{t}_{−6} Z_{t}_{−1}] is the matrix of regressors presumed to be correlated with the disturbance term, ε_{t}.
OLS on Equation (B2) yields the following estimate for α,
where M_{W} = I − W(W′W)^{−1}W′.
Now,
as plim 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, r_{t}, can be expressed as
where ‘exogenous’ implies uncorrelated with the error term in Equation (B2) and u_{t} 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 Z_{t−1} and ε_{t} (which is identical to the covariance between y_{t−1} and ε_{t}) is denoted σ_{ε} σ_{u} θ and is derived below. In symbols we have
where σ_{ε} and σ_{u} are estimates of the standard deviations of the errors in Equations (B1) and (B5). For the underlying model, σ_{ε} = 0.56, while for σ_{u}, we use the value derived from estimating Equation (3) in the text (which is a simple version of Equation (B5)). This gives the estimate σ_{u}=1.32.
Now define the variables C_{i}, i = 0,...,6 by
Denote the covariance between ε_{t} and y_{t−i} as PL_{i}, and between ε_{t} and Δy_{t−i} as PC_{i}. The model, Equation (B2), implies the recursive structure,
We require θ = PL_{1}/σ_{ε}σ_{u}, 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 as our estimate of plim. (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, to generate an estimate via Equation (B8). We now have an estimate for plim via Equation (B6). This enables us to generate an estimate, , of the ‘true’ vector α via Equation (B4). We now iterate: implies a new estimate for θ, , which, in turn, implies a new estimate for α, . This process is continued until it converges, yielding . 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 .