Almacén

The models to be estimated in this paper are based on the following parameterization of (1a)

dr =(α +βr)dt+σrγdZ. (1b) The specification of each model can easily be implemented by setting a number of restrictions on the parameters (α, β, γ and σ). The first model under consideration has the form:

when Merton (1973) is considered. Thus, the riskless rate is a Gaussian process. Also, The movements of the short-term risk free rates at different maturities are perfectly correlated as the difference between any two of them is deterministic. The second model is introduced by Vasicek (1977) and the stochastic differential equation (SDE) takes the form:

dr=(α +βr)dt+σdZ (3) where α, β and σ are constants. The SDE is an Ornstien-Uhlenbeck process composed of Brownian motion and a restoring drift that pushes it downwards when the process is

above α/β and upwards when it is below. Thus, The distribution of the process is mean

reverting and converges in equilibrium to a normally distributed mean α/β and variance

σ2/2β22. As in Merton (1973), the only source of randomness is the Brownian motion,

which is a process over time not over maturity. Thus, the Merton model can be

considered as a special case of the Vasicek model when the parameter β is restricted to

zero.

The third model is the mean reverting process introduced by Cox, Ingresoll and Ross (CIR SR) (1985). The instantaneous rate’s stochastic differential equation is

dr _{=(α +β}r₎dt_+σr1/2dZ (4)

where α, β, and σ are deterministic functions of time. The process is composed of

Brownian motion and a restoring force, the drift term, that moves toward the expected value of α/β. The volatility term is decreasing with r, so allowing α to prevent r from

going below zero. This condition holds as long as α ≥ 1/2σ2. The CIR model has been

22 _{The convergence of the Ornstien-Uhlenbeck distribution to its mean does not imply that the short-term}

widely used in pricing interest contingent claims, as in Ramaswamy and Sundaresan’s (1986) futures pricing model and Longstaff’s (1990) yield option valuation model.

The fourth model appears in Brennan and Schwartz (1977) in modeling savings and callable bonds and has also been used for discount bond valuation in Dothan (1978). The SDE is a driftless Brownian motion that allows the volatility term to be proportional to riskless rate. The SDE is

dr=σrdZ (5) The fifth model is used by March and Rosenfeld (1983) in deriving an equilibrium model for bond prices. The stochastic process of the riskless rate is simply the geometric Brownian motion (GBM) process introduced by Black and Scholes (1973). The SDE of the riskless rate takes the form

dr=βrdt+σr dZ (6)

where the riskless rate follows an arithmetic random walk with i.i.d. increments. In

contrast, we expect that the GBM model will perform better when it is estimated using the level of interest rates as compared to the stationary component. The idea behind this is related Bierens (2000) who argues that a nonlinear stochastic process is likely to act as a unit root process.

The sixth model is considered by Brennan and Schwartz (1980) in deriving a model for pricing discount bond options. The stochastic differential equation for the model is

As in both the Black and Scholes (1973) and Dothan (1978) models, the volatility term is

deterministic and proportional to r. Thus, the Brennan and Schwartz, GBM, and Dothan

models, can be nested within the CIR model by the following parameter restrictions γ=1,

α = 0 and γ=1, and α = 0, β = 0 and γ=1, respectively.

The seventh model is used by CIR (1980) in analyzing variable rate loan contracts. The SDE of the model is a driftless Brownian motion that takes the form

dr _=σr3/2dZ (8)

The volatility term is set to smaller at a decreasing rate as r approaches zero, thus

preventing r from going below zero. The model also appears in Costantinides and

Ingersoll (1984) in pricing taxable bonds. The eighth model is introduced by Cox (1975) and Cox and Ross (1976). The SDE of the model is constant elasticity of variance diffusion process that takes the form

dr =βrdt+σrγdZ (9) This model can be nested within CIR (1980) by parameters restrictions β = 0 and γ=3/2.

The ninth model is introduced by Black and Karasinski (1991) in pricing discount bond options when the riskless rate is log normally distributed. The stochastic process of the natural logarithm of the riskless rate, denoted by X, is

dX =(α +βX)dt+σdZ (10a) Unlike the Vasicek model, the parameters α, β and σ are deterministic functions of time and the instantaneous riskless rate is

The SDE of the model is an Ornstien-Uhlenbeck process and the logarithm of the riskless rate is normally distributed and drifts towards the current mean of α/β. Additionally, the riskless rate itself is mean reverting and always positive.

III. Methodologies

This section describes the structural break tests of Andrews (1993) and Andrews and Ploberger (1994) and the generalized method of moments (GMM) of Hansen (1982) adopted in our paper for testing and comparing continuous-time models of the short term interest rate. First, I briefly review the structural break tests introduced by Andrews (1993) and Andrews and Ploberger (1994). Next, the GMM is developed.