RDP 2008-09: A Term Structure Decomposition of the Australian Yield Curve Appendix C: Model Implementation

C.1 Formulas for a_τ and b_τ

From Kim and Orphanides (2005) we take the following formulas (with corrections):

where

with vec taking a matrix to a vector column-wise, and vec⁻¹ doing the opposite.

C.2 The Kalman Filter

Our implementation of the Kalman filter is based on that used by Duffee and Stanton (2004). The recursion goes from t = 1 forward, and is as follows:

1. Using the current value of x_t, compute the one-step-ahead forecast of x_t, given by , and its variance matrix .
2. Compute the one-step-ahead forecast of y_t, given by , and its variance matrix , where R is the zero-coupon bond measurement error variance matrix.
3. Compute the forecast errors of , given by .
4. Update the prediction of x_t+1 with and the variance with .

For times t when we have analysts' forecasts, replace y_t, a, B and R by and , respectively.

We then choose our parameter vector Θ to solve

where the sample is n periods long, and the period-t approximate log-likelihood is given by

C.3 Implementation

To implement the model we restrict the parameters to those which result in K and K^* having positive eigenvalues. This results in e^−Ks → 0 and as s → ∞, which ensures the stability of the model (see Equation (9) and the formulas for a_τ and b_τ). We also require that the σ_i, being variances, are positive.

By way of parameter choices that must be made, we set x₁ = [0.005, 0.03, 0.01]′ (with an initial standard deviation of 10 per cent) and then discard the first six months of the estimation. The standard deviation of zero-coupon yield measurement errors is set to 10 basis points, while those of the survey forecasts are set to 50 basis points per square root year.

Finally, the standard errors in Tables 1 and 2 are calculated using a random walk chain Metropolis-Hastings algorithm – for details see Geweke (1992).