Analysis of Multi-Factor Affine Yield Curve Models
35 Pages Posted: 21 Oct 2008
Date Written: September 1, 2008
Abstract
In finance and economics, there is a great deal of work on the theoretical modeling and statistical estimation of the yield curve (defined as the relation between $-\frac{1}{\tau }\log p_{t}(\tau )$ and $\tau$, where $p_{t}(\tau )$ is the time $t$ price of the zero-coupon bond with payoff 1 at maturity date $t \tau$). Of much current interest are models in which the bond prices are derived from a stochastic discount factor (SDF) approach that enforces an important no-arbitrage condition. The log of the SDF is assumed to be an affine function of latent and observed factors, where these factors are assumed to follow a stationary Markov process. In this paper we revisit the question of how such multi-factor affine models of the yield curve should be fit. Our discussion, like that of \cite% {AngDongPiazzesi07}, is from the Bayesian MCMC viewpoint, but our implementation of this viewpoint is different and novel. Key aspects of the inferential framework include (i) a prior on the parameters of the model that is motivated by economic considerations, in particular, those involving the slope of the implied yield curve; (ii) posterior simulation of the parameters in ways to improve the efficiency of the MCMC output, for example, through sampling of the parameters marginalized over the factors, and through tailoring of the proposal densities in the Metropolis-Hastings steps using information about the mode and curvature of the current target based on the output of a simulating annealing algorithm; and (iii) measures to mitigate numerical instabilities in the fitting through reparameterizations and square root filtering recursions. We apply the techniques to explain the monthly yields on nine US Treasuries (with maturities ranging from 1 to 120 months) over the period January 1986 to December 2005. The model contains three factors, one latent and two observed. We also consider the problem of predicting the nine yields for each month of 2006. We show that the (multi-step ahead) prediction regions properly bracket the actual yields in those months, thus highlighting the practical value of the fitted model.
Keywords: Term structure, Yield curve, No-arbitrage condition, Markov chain Monte Carlo, Simulated annealing, Square-root filter, Forecasting
JEL Classification: C11, E43, G12
Suggested Citation: Suggested Citation
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