### Regime-Switching Univariate Diffusion Models of the Short-Term Interest Rate

Seungmoon Choi
2009 Studies in Nonlinear Dynamics & Econometrics
This article proposes very general regime-switching univariate diffusion models and estimates these using short-term interest rates. There is a large literature supporting the existence of regime changes in the evolution of interest rates. In particular, these papers report strong evidence of high volatility in U.S. short-term interest rates during some periods of historic events. My study extends the existing models in four directions. First, I use a very general parametrization which nests
more » ... tion which nests most of the earlier specifications. Keeping the same constant elasticity of variance diffusion function of Conley et al. (1997) , I add a third order term to the drift function. Second, all parameters of the drift and the volatility components are subject to regime shifts. Third, the transition probabilities governing the dynamics of the regime variable vary with the interest rates. Fourth and finally, when constructing the likelihood function by using the algorithm established by Hamilton (1989) , I use a very accurate approximate transition density function of the diffusion process obtained by the method of Aït-Sahalia (2002b). The maximum likelihood estimates are calculated using the weekly series of U.S. 3-month treasury bill rates. For comparison, the regime-switching Vasicek and CIR models are also estimated. Finally, I derive a system of partial differential equations for the bond price of our model, which can be solved numerically since the analytic solution is not available. The estimation results reveal that there is strong evidence for the existence of two regimes, for the time varying transition matrix, and for the high persistence of both regimes. The volatility, but no the drift, is estimated accurately and plays a key role in explaining the dynamics of the interest rates. The volatility of one regime is about three times that of the other regime over the support of interest rates. Based on the inferred probability of the process being in a regime, I can classify the sample period into high and low volatility states quite distinctively. The likelihood-based test apparently rejects the other two models in favor of our model. This implies that misspecification can result in misleading outcomes particularly regarding the volatility and transition probabilities of the regime index. * I am grateful to Bruce Hansen and Dennis Kristensen for helpful comments and suggestions. There is a large literature supporting the existence of regime changes in the evolution of interest rates (e.g. ). In particular, these papers report strong evidence of high volatility in U.S. short-term interest rates during the episodes of the 1973 and 1979 OPEC oil crises, the 1979-82 Federal Reserve Monetary Experiment, and the 1987 stock market crash. Most of the literature focus on discrete time models, such as autoregressive (AR) and autoregressive conditional heteroskedasticity (ARCH) specifications in order to discover the regimeswitching features of interest rates. On the other hand, the short-term interest rate is often modeled as a continuous time diffusion process. Many earlier models of term structure of interest rates are based on the simplifying assumption that changes in all yields are driven by a single underlying, random factor. Studies of one-factor diffusion model of the instantaneous rate include document that three factors (level, slope and curvature) can fully capture most of the variability of interest rates. In fact, roughly 90 percent of the variation in U.S. yields changes can be explained by the first factor, which is typically identified with the short rate of interest. Recent extensive articles on the univariate diffusion models of the short-term interest rate can be interpreted as focusing on the dominant first factor and attempting to investigate the relationship between the level of interest rates and their expected changes and volatilities in detail. However, empirical evidence proves that additional factors are needed to fit the data better (e.g. Dai and Singleton (2000) , Ahn, Dittmar, and Gallant (2001), Andersen, Benzoni, and und (2003)). My regimeswitching univariate diffusion model is a two-factor model. One is the continuously evolving short rates and the other is the regime variable changing discretely, taking finite number of values. While there have been considerable works in which the instantaneous rate follows a univariate stochastic differential equation (SDE), to my knowledge, there are only a few papers that introduce the regime shifts into the continuous time model to study the dynamics of short-term interest rates. Naik and Lee (1998) analyze the regime-switching Vasicek model in which only the volatility is subject to regime shifts. They use the known transition density function of the Vasicek process to carry out the maximum likelihood (ML) estimation. Three long-term bond yields data sets are inverted with the bond price formula they derived to get the short rate and estimate the model. Using the Markov Chain Monte Carlo (MCMC) method Liechty and Roberts (2001) estimate a regime-switching Gaussian diffusion model where only the drift component is regime-dependent. The Regime-switching CIR model is studied by Driffill, Kenc, Sola, and Spagnolo (2004) (hereafter DKSS). They