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

C.1 Formulas for aτ and bτ

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


with Inline Equation 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 xt, compute the one-step-ahead forecast of xt, given by Inline Equation, and its variance matrix Inline Equation.
  2. Compute the one-step-ahead forecast of yt, given by Inline Equation, and its variance matrix Inline Equation, where R is the zero-coupon bond measurement error variance matrix.
  3. Compute the forecast errors of Inline Equation, given by Inline Equation.
  4. Update the prediction of xt+1 with Inline Equation and the variance with Inline Equation.

For times t when we have analysts' forecasts, replace yt, a, B and R by Inline Equation and Inline Equation, 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 Inline Equation as s → ∞, which ensures the stability of the model (see Equation (9) and the formulas for aτ and bτ). We also require that the σi, 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).