Parameter uncertainty in Kalman filter estimation of the CIR term structure model

Abstract

The Cox, Ingersoll and Ross (1985) term structure model describes the stochastic evolution of government bond yield curves over time using a square root Orstein-Uhlenbeck diffusion process, whilst imposing cross-sectional no-arbitrage restrictions between yields of different maturities. A Kalman filter approach can be used to estimate the parameters of the CIR model from panel data consisting of a time series of bonds of different maturities. The parameters are estimated by optimising a quasi log-likelihood function that results from the prediction error decomposition of the Kalman filter. The quasi log-likelihood function is usually optimised with a deterministic gradient based optimisation technique such as a quadratic hill climbing optimiser. This paper uses an evolutionary optimiser known as differential evolution (DE) to optimise over the parameter space. The DE optimiser is more likely to find the global maximum than a deterministic optimiser in the presence of a non-convex objective function which may be the case in multifactor term structure models with non-negativity constraints and parameter constraints. The method is applied to estimate parameters from a one and two-factor Cox, Ingersoll and Ross (1985) model. It is shown that in the two factor model the problem of local maxima arises whereby a number of different parameter vectors perform equally well in the estimation procedure. Fixed income derivative prices are particular sensitive to term structure parameters such as the volatility, the rate of mean reversion, and the market price of risk of each factor. The effect of different optimal parameter vectors on fixed income derivatives is examined and is found to be significant.