Bayesian estimation of short-rate models.

Abstract:

Estimating continuous-time short-rate models is challenging since the likelihood function for most popular models is unknown. While approximate likelihood functions are often used, this practice induces bias into the estimation process. This paper explores a Bayesian method of estimating short-rate models. While the approach also employs an approximate likelihood, data augmentation is utilised to mitigate discretisation bias. The results suggest that Bayesian estimates of posterior densities for model parameters closely resemble true posterior densities. While nonessential for point estimation, a small degree of data augmentation is useful in recovering accurate posterior densities and reducing the bias in estimates of bond price. These findings are encouraging for the many cases where exact likelihood-based estimation is impossible and approximations must be relied upon.

Keywords:

SHORT-RATE MODELS: CONTINUOUS-TIME FINANCE; BAYESIAN ESTIMATION; VASICEK.

1. Introduction

In theoretical modelling, finance literature makes regular use of continuous-time stochastic differential equations (SDEs). Nowhere is this more apparent than in the modelling of interest-rate dynamics. Examples of short-rate models include Cox (1975), Vasicek (1977), Dothan (1978), Brennan and Schwartz (1980), Marsh and Rosenfeld (1983), and Cox, Ingersoll, and Ross (1985), to name but a few.

While continuous-time models are popular in theoretical work, empirical estimation of model parameters presents a number of challenges. First, estimation techniques based on the likelihood function of the observed sample are problematic since the transition density of a stochastic process is unknown for all but the most simple models. While empirical research often employs an approximate likelihood function in such cases, this practice induces bias in parameter estimates. Second, even when the exact likelihood function is known, classical methods such as maximum-likelihood (ML) estimation typically resort to large-sample results to derive the joint distribution of model parameters. The finite-sample performance of these asymptotic approximations is largely unknown. Finally, there is a significant discrepancy between the frequency with which we sample data (often daily, weekly or monthly) and the infinitesimal time-interval assumed under a continuous-time process. Estimation using discrete samples can also induce bias.

This paper explores a method of estimating the parameters of continuous-time short-rate models when the exact likelihood function is unknown. An Euler discretisation of the process is employed to yield an approximate likelihood function. Within a Bayesian framework, the joint posterior density of model parameters is estimated using a Markov chain Monte Carlo (MCMC) scheme based on Gibbs-sampling and Metropolis-Hastings steps. The Bayesian framework also facilitates a means of mitigating the problem associated with the discretisation of the continuous-time process. Using the data-augmentation algorithm of Tanner and Wong (1987), the observed sample of interest rates is augmented by simulating values of the process between each pair of observations. By increasing the frequency of data, data-augmentation has the potential to reduce the discretisation bias.

The paper examines the potential usefulness of the Bayesian methodology for estimating short-rate models. In the convenient setting of Vasicek (1977), the properties of point estimates from the true Bayesian posterior densities are compared to ML estimates via a series of simulation experiments. The accuracy of the Euler-approximated and data-augmented posterior densities is also documented using the true densities as the benchmark. The robustness of findings to different sample sizes and degrees of data augmentation is also examined.

The remainder of the paper is structured as follows. Section 2 reviews Bayesian estimation of the parameters of a continuous-time process when the exact likelihood function is known. Vasicek's (1977) popular model of the short-rate is used as an illustration. Section 3 outlines the Euler approximation to the Vasicek likelihood function, as well as the data augmentation procedure. Section 4 examines the accuracy of the Euler approximation. The true posterior densities of parameters from section 2 (based on the exact likelihood function) are used as the benchmark against which the approximation technique is judged. An assessment is also made of the degree of data augmentation necessary to overcome discretisation bias, and the sensitivity of results to various parameter values is evaluated. The economic impact of data augmentation is documented via its reduction in the bias of estimated bond prices. In section 5, the methodology is applied to estimate the parameters of two popular short-rate models using several Australian data series. Concluding remarks are made in section 6.

2. Bayesian Estimation with a Known Likelihood Function

Denote by [r.sub.t] the instantaneous short-rate at time t. The evolution of the short-rate is modelled with a continuous-time SDE of the form:

(1) [dr.sub.t] = [mu]([r.sub.t], [theta])dt + [sigma]([r.sub.t], [theta])d[W.sub.t]

where [W.sub.t] is a standard univariate Brownian motion. The instantaneous drift of the process, ([r.sub.t], [theta]), and the instantaneous volatility, [sigma]([r.sub.t], [theta]), are general functions that can encompass both linear and nonlinear transformations of rt and the parameter vector [theta].

The complexity of the functional form of [mu]([r.sub.t], [theta]) and [sigma]([r.sub.t], [theta]) determines whether the transition density of the stochastic process [[phi].sub.t] ([r.sub.t]|[r.sub.t-1]) can be derived. For parameterisations of [mu]([r.sub.t], [theta]) and [sigma]([r.sub.t], [theta]) common in short-rate modelling, tractable transition densities are the exception rather than the norm. One case where the transition density is known is Vasicek (1977), who models the short-rate [r.sub.t] as an Ornstein-Uhlenbeck (mean-reverting) process:

(2) [dr.sub.t] = [kappa]([gamma] - [r.sub.t])dt + [beta]dW, [kappa], [gamma], [beta] > 0,

where [gamma] is the long-run mean interest rate, [kappa] is the speed of mean reversion, and [beta] is the instantaneous volatility of the short-rate. Since the transition density of the process is known to be Normal, the exact likelihood function can be written. (1) For ease of exposition, assume that the short-rate is sampled at n equally-spaced intervals of length h, r [equivalent to] ([r.sub.h], [r.sub.2h], ..., [r.sub.nh]), with [r.sub.0] assumed fixed. The likelihood function L([theta]|r), with [theta] [equivalent to] ([kappa], [gamma], [[beta].sup.2]), is:

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Let p([theta])denote the prior density of model parameters. Using Bayes' rule, the prior is combined with the data evidence in the sample L([theta]|r) to estimate the joint posterior density of the parameters conditional on the observed data: p([theta]|r) [varies] L([theta]|r)p([theta]). To estimate the posterior density for the Vasicek parameters, MCMC simulations involving Gibbs-sampling and Metropolis-Hastings schemes are employed. The method generates random numbers from carefully chosen candidate densities and, under quite weak conditions, the draws from the candidate densities converge to draws from the desired joint posterior density. In choosing a candidate density for each parameter, a Gibbs-sampling scheme is adopted where the conditional density of the parameter is known, and a Metropolis-Hastings step is employed when the conditional density is nonstandard.

For the Vasicek model, we seek the posterior density p([theta]|r), where [theta] [equivalent to] ([kappa], [gamma], [[beta].sup.2]). (2) The conditional densities p([gamma]|[kappa], [gamma], r) and p([[beta].sup.2]|[kappa], r) are straightforward and Gibbs-sampling steps suffice. The conditional density p([kappa]|[gamma], [[beta].sup.2], r), however, is non-standard necessitating a Metropolis-Hastings step. The chosen candidate density for [kappa] follows from a second-order Taylor series expansion of the log-likelihood function l = ln L around the value of [kappa] that maximises the log-likelihood function. Using flat priors on ln [[beta].sup.2], [gamma] and [kappa], the candidate densities adopted in the MCMC scheme are: (3)

(4) [[beta].sup.2]|[kappa], [gamma], r ~ IG(n/2, ([kappa]/1 - [e.sup.-2[kappa]h])[n.summation over j=1]([r.sub.jh] - [r.sub.(j-1)h] [e.sup.-[kappa]h] - [gamma][(1 - [e.sup.-[kappa]h])).sup.2]),

(5) [gamma]|[kappa], [[beta].sup.2], r ~ N(1/n[n.summation j=1]([r.sub.jh] - [r.sub.(j-1)h][e.sup.-[kappa]h]/1 - [e.sup.-[kappa]h]), [[beta].sup.2](1 - [e.sup.-2[kappa]h)/2[kappa] n[(1 - [e.sup.-[kappa]h]).sup.2]), [gamma] > 0,

(6) [kappa]|[[beta].sup.2], [gamma], r [congruent to] N([[kappa].sub.max], [DELTA]), [kappa]>0,

where [[kappa].sub.max] is log-likelihood maximising value, and [DELTA] [equivalent to] 1/l"([[kappa].sub.max])). Since the Vasicek SDE (2) requires [kappa], [gamma], [beta] >0, these restrictions are imposed on the prior densities of each parameter and flow through to the posteriors (5) and (6), as shown by the truncation of the respective Normal distributions to positive values. In fact, imposition of a positivity constraint proves vital to obtaining sensible parameter estimates.

3. Bayesian Estimation with an Approximate Likelihood Function and Data Augmentation

Unlike Vasicek, the transition density and exact likelihood function for most short-rate models are unknown. In such cases, the transition density can be approximated by discretising the continuous-time process and making a distributional assumption over the stochastic component. Estimation follows in the usual manner from the approximate likelihood function. The approximation, however, induces error into the likelihood function and the extent to which this error flows-on to estimates of posterior densities of parameters is unclear.

In this section, Bayesian estimation is implemented using a simple Euler approximation of the likelihood function. The resulting (approximate) posterior densities for the Vasicek parameters can be compared to the true posteriors obtained using the exact likelihood function from section 2. Hence, we can assess the degree to which the Euler approximation is a useful tool in estimating short-rate models whose exact likelihood function is unknown. In addition, a procedure known as data augmentation, which involves simulating data between each pair of observed data points, is explored. The rationale for data augmentation is that the approximation error that arises when a continuous-time process is discretised is mitigated if the data employed in estimation are of a sufficiently high frequency.

3.1 Euler Approximation and Data Augmentation

For the general SDE (1) sampled at equally-spaced intervals h, the Euler approximation is:

[r.sub.jh] = [r.sub.(j-1)h] + [mu]([r.sub.(j-1)h], [theta])h + [sigma]([r.sub.(j-1)h], [theta])[square root of h][[epsilon].sub.jh],

where [[epsilon].sub.jh] ~ N(0,1). An approximate likelihood function can be written using the Normal transition density of the process. Empirical literature has used this approximation for a variety of purposes, including quasi-ML and GMM estimation and numerical simulation of continuous-time processes (e.g. Brailsford & Maheswaran 1998; Treepongkaruna & Gray 2003).

The validity of the Euler approximation stems from the fact that, as h approaches zero: (i) the discretised approximation converges weakly to the continuous-time SDE; and (ii) the likelihood function for the Euler approximation converges to that of the SDE (see Kloeden & Platen 1992; Jones 1999; Pedersen 1995). However, Elerian, Chib and Shephard (2001) note that this simple Euler approximation is usually too coarse to approximate the true transition density adequately. While the approximation implicitly assumes the path taken by the stochastic process is linear between two observations, many short-rate models follow a non-linear path (Vasicek included). In addition to this curvature bias, Elerian, Chib and Shephard show that the Euler approximisation of the log-likelihood function may not closely resemble the true log-likelihood. For these reasons, a simple Euler approximation which discretises the process at the same frequency as the observed data is unlikely to be adequate.

To mitigate these problems, this paper follows Elerian, Chib and Shephard (2001), Eraker (2001), and Jones (1999, 2003) by augmenting the observed data with a series of simulated points. In order to minimise the discretisation bias, the path of the stochastic process is simulated between observed data points to 'manufacture' a sample at a sufficiently-high frequency. Assume that we commence with a sample of n observations of the short-rate at equally-spaced intervals h. The approximate transition density is used to simulate m augmented data points between each pair of observations. The augmented data are also assumed to be at equally-spaced intervals [tau] = h/(m+1) and, henceforth, all data are indexed with reference to [tau] rather than h. The total sample (observed plus augmented data) consists of v = nm + n points, plus a starting value ([r.sub.0] is again assumed fixed). Denote by [r.sub.obs] and [r.sub.aug] the observed and augmented data respectively. Observed (augmented) data points correspond to integer (non-integer) values of j[tau]/h,j = 0, ..., v.

3.2 Bayesian Estimation of Parameters

To estimate the posterior density of model parameters ([theta]) conditional on the observed data ([r.sub.obs]), the augmented data ([r.sub.aug]) are essentially treated as unknown parameters. The joint density of the parameters and the augmented data is obtained by, iterating between draws from the conditional densities p([theta]|[r.sub.aug], [r.sub.obs]) and p([r.sub.aug]|[theta], [r.sub.obs]). Under regularity conditions, the iterates of these conditional densities converge to the true joint distribution p([theta], [r.sub.aug]| [r.sub.obs]). The desired posterior density p([theta]|[r.sub.obs]) is then obtained by integrating out the augmented data.

The Euler approximation of the Vasicek model (2) is:

(7) [r.sub.j[tau]] = [r.sub.(j-1)[tau]] + [kappa]([gamma] - [r.sub.(j-1)[tau])[tau] + [beta][square root of [tau]][[epsilon].sub.j[tau]] = [b.sub.1] + [b.sub.2][r.sub.(j-1)[tau]] + [sigma][[epsilon].sub.jt],

where [b.sub.1] [equivalent] [kappa][gamma][tau], [b.sub.2] [equivalent to] 1 - [kappa][tau], [sigma] [equivalent to] [beta][square root of [tau]], and [theta] [equivalent to] ([b.sub.1], [b.sub.2], [[sigma].sup.2]). Equation (7) casts the Vasicek model into the standard linear regression framework, where the posterior densities of the parameters are well-known (Eraker 2001; Jones 1999, 2003 use the same transformation). With flat priors on [b.sub.1], [b.sub.2], and ln [[sigma].sup.2], Zellner (1971, p. 67) shows that the relevant conditional densities are: (4)

(8) [b.sub.1],[b.sub.2]|[[sigma].sup.2], [r.sub.aug], [r.sub.obs] ~ N(b, [[sigma].sup.2][(X'X).sup.-1]), [b.sub.1] > 0, 0 [less than or equal to] [b.sub.2] [less than or equal to] 1

(9) [[sigma].sup.2]|[b.sub.1], [b.sub.2], [r.sub.aug], [r.sub.obs] ~ IG(v/2, 1/2(r - Xb)'(r - Xb)),

where b = [(X'X).sup.-1] X'r and X is an v x 2 data matrix with ones and lagged short-rate data ([r.sub.0], ..., [r.sub.(v-1)[tau]])' in the first and second columns respectively. Given the augmented data, p([b.sub.1], [b.sub.2], [[sigma].sup.2]|[r.sub.aug], [r.sub.obs]) can be constructed using a series of Gibbs-sampling steps.

To obtain the conditional density of the augmented data p([r.sub.aug]|[b.sub.1], [b.sub.2], [[sigma].sup.2], [r.sub.obs]), the Metropolis-Hastings algorithm is employed. The approach employed here follows the example of Jacquier, Polson and Rossi (1994) and Jones (1999, 2003). Consider drawing a single augmented data point [r.sub.j[tau]], where j[tau]/h is non-integer. Noting that the Euler approximation of the Vasicek model is a Markov process, we need only condition the draw on adjacent data points. That is, we wish to draw from:

(10) p([r.sub.j[tau])|[r.sub.(j+1)[tau]], [r.sub.(j-1)[tau]], [theta]).

By Bayes' rule, this is proportional to:

p([r.sub.j[tau]), [r.sub.(j+1)[tau]]|[r.sub.(j-1)[tau]], [theta]).

which, again by Bayes' rule, is proportional to:

(11) p([r.sub.(j+1)[tau])|[r.sub.j[tau]], [theta])p([r.sub.j[tau]]|[r.sub.(j-1)[tau], [theta]).

Jones (2003, p, 12) suggests obtaining a draw from (10) as follows: (i) from p([r.sub.j[tau]]|[r.sub.(j-1)[tau], [theta]), generate an iterate of [r.sup.[i].sub.j[tau]] which is likely to follow [r.sub.(j-1)[tau]]; then (ii) only accept the iterate of [r.sup.[i].sub.j[tau]] if it is likely to have preceded the adjacent [r.sub.(j+1)[tau]] .The conditional density p([r.sub.(j+1)[tau]]|[r.sub.j[tau], [theta]) determines the acceptance probability:

(12) [alpha] = min(1, p([r.sub.(j+1)[tau])|[r.sup.[i].sub.j[tau]], [theta])/ [p([r.sub.(j+1)[tau]]|[r.sup.[i-1].sub.j[tau]], [theta])).

Generating from the densities in (11) is trivial given that the approximate transition density for (7) is Normal. Implementation of the Metropolis-Hastings algorithm to generate one iterate of each of the nm augmented data points could simply repeat the above process to cycle through each component of the augmented dataset. However, by relying on the Markov property that only adjacent observations are relevant, the cycling can be reduced to two steps: (i) simultaneously draw [r.sub.j[tau]] for j = 1, 3, 5, ..., v - 1 conditioning on [r.sub.k[tau]] for k = 0, 2, 4,... , v; then (ii) simultaneously draw [r.sub.k[tau]] for k = 0,2,4, ..., v conditioning on the iterates [r.sub.j[tau]] for j = 1,3,5,... , v - 1 from step (i).

To summarise the implementation of the numerical MCMC algorithm described in this section, the desired joint posterior density p([theta], [r.sub.aug]|[r.sub.obs]) is constructed as follows:

1. from equations (8) and (9), draw an iterate of p([[theta].sup.[i]|[q[[r.sub.aug], [r.sub.obs]);

2. conditioning on [[theta].sup.[i]], draw an iterate of p([r.sup.[i].sub.aug]|[[theta].sup.[i], [r.sub.obs]) using the two-step Metropolis-Hastings algorithm outlined in (11);

3. accept the iterate [r.sup.[i].sub.aug] with probability [alpha] given in (12); and

4. repeat this process many times to obtain a series of iterates of ([theta], [r.sub.aug]).

Under regularity conditions, this series converges to draws from the true joint posterior density p([theta], [r.sub.aug]|[r.sub.obs]). The desired posterior density p([theta]|[r.sub.obs])is obtained by integrating out the augmented data ([r.sub.aug]).

4. Simulation Results

This section assesses the extent to which the discrete-time Euler approximation of the likelihood function, both with and without data-augmentation, is useful for estimation continuous-time short-rate models. The reported results follow from the joint posterior density of the Vasicek parameters 0 [equivalent to] ([kappa], [gamma], [[beta].sup.2]), estimated using the MCMC methods described in section 2 and section 3. Several related issues are explored. First, point estimates of model parameters are examined. The mean and median of the posterior density of 0 serve as point estimates and are compared to the true parameters. Second, comparisons are made of the density of Vasicek parameters under ML (asymptotic) and Bayesian (exact) methods, as well the true and Euler-approximated Bayesian posterior densities. Third, a sensitivity analysis is conducted to examine the impact of samples size and degree of data augmentation. Finally, the economic impact of data augmentation is assessed by examining the bias in Bayesian estimates of a range of bond prices.

4.1 Parameter Estimates

To assess the statistical properties of the Bayesian methods described in sections 2 and 3, a time-series of 300 monthly interest rates is simulated under the Vasicek model using its true transition density. This 25-year timeframe is similar to those often encountered (see Chan, Karolyi, Longstaff & Sanders (CKLS) 1992). CKLS's (1992) estimates for the Vasicek model are used as 'population' parameters; specifically, [kappa] = 0.18, [gamma] = 0.08, and [beta] = 0.02. Posterior densities are estimated and the process of simulating a 300-month time-series of interest rates is repeated 1000 times.

Since the exact Vasicek transition density is known, the MCMC scheme described in equations (4)-(6) yields the true posterior density p([theta]|r). Accordingly, this density is used as the benchmark against which the accuracy of the approximation techniques, both with and without data augmentation, is judged. The MCMC scheme is run with 5,500 iterates, the first 500 discarded as warmups leaving 5,000 post-warmup iterates of ([kappa], [gamma], [[beta].sup.2]) from which to estimate the posterior density.

The approximate posterior density p([theta]|[r.sub.obs]) under the Euler approximation of the likelihood function is estimated using equations (8)-(12). As an initial examination of the adequacy of the discretisation, no data augmentation is employed (m = 0). In addition, the observed data are augmented by simulating m points between each observed pair of data points. Data augmentation for m = 1, 2, 5, 10, 20 is investigated. The MCMC scheme for the approximate posterior density also employs 5,000 post-warmup iterates.

The point estimates of [theta] [equivalent to] ([kappa], [gamma], [beta]) reported in table 1 are the average point estimates over the 1000 independent simulations. (5) Panel A suggests that the speed of mean-reversion [kappa] is estimated imprecisely, even with 25 years of monthly data. While the Bayesian estimates (all around 0.29) are less-biased than the ML estimate (0.36), no method yields an estimate near the true value (0.18). Before drawing conclusions regarding the bias of the various estimators, however, note that the numerical standard errors of these estimates are quite high. In all cases, the point estimate is within one standard error of the true value. Regardless, the imprecision of the [kappa] estimate makes it difficult to draw strong inferences about the true value of [kappa].

Difficulties in estimating the drift component of SDEs, particularly when a discrete approximation is involved, are well known (see Merton 1980; Phillips 1973; Lo 1988). In contrast to estimators of the diffusion component which become more precise as the number of observations increases, drift estimators gain precision only as the time-span involved in the estimation lengthens. Accordingly, estimation of the diffusion benefits from higher-frequency data, while drift estimation does not. The improvement in precision in estimating [kappa] over longer periods is an issue explored in the sensitivity analysis of section 4.3.

In table 1 panel B, the long-run equilibrium interest rate [gamma] is estimated more precisely. Both Bayesian and ML approaches estimate the true value (0.08) with little bias. Similarly, in panel C, the instantaneous volatility [beta] is estimated with high precision under all methods.

Adopting the Euler approximation of the Vasicek process does not appear to affect the accuracy of point estimates, which are close to estimates from the true posterior densities. This finding is encouraging for models whose exact likelihood function is unknown, leaving no alternative but to rely on an approximation. Similarly, the degree of data augmentation seems unimportant. Point estimates do not appear to become less-biased as the level of data-augmentation m increases. Finally, there is limited evidence that the median of the posterior density provides a better point estimate than the mean. The median values of [kappa] and [gamma] are marginally less-biased than the mean.

Table 1 also reports simulation inefficiency factors (SIF) of Kim, Shephard, and Chib (1998). The SIF is a diagnostic for the MCMC sampling scheme which assesses how well the chain mixes. The SIFs reported in table 1 are typical of those reported in MCMC literature. Importantly, the SIFs are not sensitive to the degree of data augmentation.

4.2 Posterior Density Estimates

The results reported in section 4.1 are the average estimates across 1000 independent simulations. This section examines a single simulated time-series of interest rates. (6) Figure 1 (left-side) compares the asymptotic Normal densities for [theta] [equivalent to] ([kappa],[gamma],[beta]) obtained under ML estimation (solid line) to the true Bayesian posterior densities (dotted line). The latter follow from 20,000 post-warmup MCMC iterates and a kernel density estimator employing a Normal kernel with Scott's optimal bandwidth.

The plots suggest there are some minor differences between the asymptotic Normal densities and the exact Bayesian posteriors. In particular, the truncation of [kappa] to positive values under the Bayesian method results in a less disperse and right-skewed density. The Bayesian posterior density for [gamma] is also slightly less disperse. The diffusion parameter [beta] is estimated so precisely under both methods that the densities are near identical.

The right-side plots in figure 1 assess the extent to which the Euler approximation with data augmentation recovers the true Bayesian posterior. The solid line represents the true posterior density based on the exact likelihood function, while the dotted line is the approximate density with m = 5. In all cases, the approximate density closely resembles the true posterior. Again, this gives confidence that the Euler method can be used for models where the true transition density and likelihood function are unknown. Although not shown to preserve clarity, the approximate densities with no data augmentation (m = 0) are marginally less accurate, suggesting there is some value in augmenting the observed data with simulated points.

To summarise, the results suggest that the Euler approximation is a useful tool for estimation and inference relating to a continuous-time model when the exact likelihood function is unknown. Point estimates of model parameters obtained from approximate posterior densities closely match those from the true posterior density. While seemingly unnecessary to obtain accurate point estimates, some degree of data augmentation is useful to obtain approximate posterior densities which closely resemble the true posterior density.

4.3 Sensitivity Analysis

This section explores the sensitivity of the previous findings to variations in the assumed population parameters and timeframes used for estimation. Apart from the stated variations, the methodology is identical to that described in section 4.1. (7)

The most striking feature of table 1 is the imprecision with which the speed of mean-reversion [kappa] is estimated under all methods. The assumed population value [kappa] = 0.18 implies that reversion to the equilibrium interest rate takes approximately five years (if there are no random components). A priori, we might expect the diffusion component of the process to swamp the deterministic drift, resulting in an estimate of [kappa] near zero. Table 1, however, reports that [kappa] is overestimated (all estimates around 0.29).

To determine whether the estimation methods are more precise when the mean-reversion of the interest-rate process is stronger, the 1000 independent simulations are re-run with [kappa] = 0.50. Perhaps unrealistically, this value implies the process returns to equilibrium levels in approximately two years (if there are no random components). Table 2 reports that both the true and approximate posterior densities of [kappa] are now centred around the population value. While this indicates a reduced bias in these estimates, the standard errors of the estimates have increased (relative to table 1). Note also that the ML estimate of [kappa] remains overestimated.

As an alternative to increasing [kappa], table 3 reports the results of decreasing the random component of the interest-rate process. The 1000 independent simulations are re-run reducing [beta] to 0.01, while leaving [kappa]=0.18 and [gamma]=0.08. Several interesting points emerge. First, [kappa] is still overestimated despite the reduction in 'noise' in the short-rate process. While the point estimates (around 0.28) are slightly less biased than those in table 1, the standard errors are of similar magnitude to table 1 suggesting [kappa] estimates remain imprecise. Second, even halving the diffusion term [beta] to 0.01, the Bayesian methods estimate [beta] with high precision.

Finally, rather than altering the population values for [theta] [equivalent to] ([kappa],[gamma],[beta]), the length of the time-series used to estimate the posteriors is varied. Sample sizes of 200, 400, and 600 months are reported in table 4 panels A, B and C respectively. As noted earlier, Merton (1980) has shown that the drift of a geometric Brownian motion process can be estimated with greater precision as the time-interval for estimation increases. Table 4 suggests this is also true for the Vasicek model of the short-rate. The estimates of [kappa] decrease notably as sample size increase. The standard errors of these estimates are also notably lower, indicating greater precision. The efficiency of the MCMC sampling for [kappa] also improves rapidly with sample size.

4.4 Bayesian Estimation of Bond Prices

To measure the economic impact of Bayesian estimation of short-rate parameters with data augmentation, consider the price of a pure discount bond under Vasicek short-rate dynamics. The time t price of a bond with $1 face value and a maturity of t + [tau], B(t,t + [tau])is:

B(t,t + [tau]) = exp(-m([r.sub.t],[tau]) + 1/2 v([tau])),

Where

M([r.sub.t],[tau]) = [gamma][tau] + (1 - [e.sup.[kappa][tau]])([r.sub.t]-[gamma]/[kappa])

V([tau] = [[beta].sup.2]/2[[kappa].sup.3] (4[e.sup.-[kappa][tau]] - e[sup.-2[kappa][tau] + 2[kappa][tau] - 3).

A time-series of 300 monthly interest rates is simulated under the Vasicek model with [kappa] = 0.18, [gamma] = 0.08 and [beta] = 0.02. Parameter estimates are obtained under ML and Bayesian approaches (both with and without data augmentation). The Vasicek bond price is calculated under each estimation method and compared to the true (known) price. The procedure is repeated 1000 times to establish the average price and bias.

Table 5 shows that, by severely over-estimating [kappa], ML produces large bias in estimates of bond price. The magnitude of bias increases with bond maturity ([tau]). The bias under all Bayesian approaches is less than that under ML. Curiously, bond price estimates under the approximate Bayesian densities are less biased than under the true Bayesian density. Table 5 also shows that a small degree of data augmentation is beneficial. The bias in estimated bond price with one augmented data point (m = 1) is roughly half that with no data augmentation (m = 0). Further data augmentation (m = 5, 10) adds no further bias reduction

5. Estimation with Real Data

This section estimates the models of Vasicek (1977) and CKLS (1992) using common Australian data series. CKLS (1992) propose a general short-rate model that incorporates mean reversion and conditional heteroscedasticity:

(13) [d.sub.t] = [kappa]([gamma] - [r.sub.t])dt + [beta][r.sup.[xi].sub.t] d[W.sub.t].

The continuous-time model (13) is discretised using the Euler approximation. Bayesian estimation of [theta] = ([kappa], [gamma], [[beta].sup.2], [xi]) follows a similar procedure to that outlined for the Vasicek model, with one complication. The conditional density for [xi] is nonstandard prohibiting the use of a Gibbs-sampling scheme. As in section 2, the Metropolis-Hastings procedure is adopted with a Normal candidate density obtained from a second-order Taylor-series expansion of the log-likelihood function around the value of [xi] that maximises the log-likelihood function ([[xi].sub.max]). (8)

With flat priors and transformations similar to those used earlier, the conditional densities are:

(14) [b.sub.1],[b.sub.2]|[[beta].sup.2],[xi],r ~ N(b, [[beta].sup.2][(X'X).sup.-1]) [b.sub.1] > 0,0 [less than or equal] [b.sub.2] [less than or equal] 1

(15) [[beta].sup.2]|[b.sub.1],[b.sub.2],[xi],r ~ IG(v/2, 1/2 (y - Xb)' (y - Xb))

(16) [xi]|[b.sub.1],[b.sub.2],[[beta].sup.2], r [congruent to] N([[xi].sub.max], 1/l"([[xi].sub.max)), [xi] [greater than or equal] 0.

Data series for 30-day, 90-day, and 180-day bank accepted bills (BABs) and 13-week and 26-week Treasury notes (T-notes) were obtained from the Reserve Bank of Australia. The BAB data is from July 1976 through February 2002, while the T-notes are from June 1986 through February 2002. The daily, weekly, and monthly series contain 6512, 1526, and 307 observations for the BABs and 3992, 938, and 190 and for the T-notes. Posterior densities for the Vasicek and CKLS parameters are estimated using 5000 post-warmup MCMC iterates and m=2 data augmentation.

Table 6 reports the mean and standard error of the posterior densities of model parameters. Under the Vasicek model, estimates of the speed of mean reversion ([kappa]) range from 0.09 to 0.25. The long-run mean rate is around 8% for the BABs and slightly lower (around 5%) for government-issued T-notes. Estimates of volatility ([beta]) are reasonably consistent within each class, with the T-notes being slightly less volatile.

Similar conclusions can be drawn for [kappa] and [gamma] under the CKLS model. Estimates of [beta] are consistent for the two T-note series, but vary considerably for the different BAB series. Using similar data series, Brailsford and Maheswaran (1998) present a similar finding. Perhaps most interesting is the magnitude of [xi]. Chan et al. (1992) estimate [xi] to be 1.50, and in model comparisons, conclude that models with [xi] > 1 capture short-rate dynamics best. Estimates of [xi] in table 6 range from 1.25 to 1.66 and the posterior density oft in figure 2 clearly indicates that this parameter is well in excess of unity. These results are broadly consistent with prior Australian estimates of 1.55 (Gray 1996), 0.93 to 1.55 (Treepongkaruna & Gray 2003), and 1.14 to 1.70 (Brailsford & Maheswaran 1998). Finally, note that the conditional volatility of the T-note series is clearly lower than that of the BAB series.

Figure 2 plots the posterior densities model parameters estimated using the 30-day BAB series. The truncation on the posterior density of [kappa] is evident under both models. Note also the lower dispersion of [kappa] iterates under the CKLS model. The bottom-right plot is also striking. In light of the focus in the literature on the magnitude of [xi], Figure 2 clearly shows that most of the posterior mass lies between 1.6 [less than or equal] [xi] [less than or equal] 1.7. For this data series, therefore, models with constant volatility (e.g. Vasicek 1977) or low levels of heteroscedasticity (e.g. Cox, Ingersoll & Ross 1985) are unlikely to adequately capture short-rate dynamics.

Table 7 compares estimates for the 30-day BAB series using different sampling frequencies. Brailsford and Maheswaran (1998) and Treepongkaruna and Gray (2003) report widely varying parameter estimates under different sampling frequencies. In stark contrast, the Bayesian approach produces Vasicek model estimates that are remarkably stable across sampling frequencies. CKLS estimates are also reasonably consistent, with the exception that the volatility for the daily series is smaller than series sampled less often.

6. Conclusions

Continuous-time SDEs are used extensively in theoretical finance to model variables of interest. Empirical estimation of these models, however, is plagued by a variety of problems. Often, the complexity of the drift and diffusion components of the process makes the transition density intractable, precluding exact likelihood-based estimation. In such cases, approximation methods based on a discretisation of the continuous-time process are often employed. In addition to the likelihood approximation error involved, this practice also induces a discretisation bias.

This paper evaluates a method of conducting estimation and inference for continuous-time processes when the transition density, and hence the exact likelihood function, is unknown. The procedure adopts the Euler discretisation of the process to write an approximate transition density and likelihood function. Within a Bayesian framework, the joint posterior density of model parameters is estimated. The framework also facilitates the practice of augmenting observed data points with simulated data. This procedure reduces the bias caused by discretisation by manufacturing a higher-frequency dataset than that actually observed.

In general, the findings are encouraging. The results suggest that the joint posterior density of model parameters flowing from an Euler discretisation is a close approximation to the true posterior density. Bayesian point estimates compare favourably to ML estimates, especially for the speed of mean reversion ([kappa]) which is notoriously difficult to estimate. Sensitivity analysis shows that precise estimation of [kappa] requires a very long time-series. While data-augmentation does not appear to increase the accuracy of point estimates, it is shown to be useful in recovering accurate approximations of the true posterior densities of parameters. Data augmentation also produces less-biased estimates of bond price. One caveat is that, while data augmentation is straightforward to implement, it significantly increases the computation time and, as such, may not be justifiable.

There are several interesting areas for further research. First, the Vasicek model examined in this paper was an obvious starting point since it admits a closed-form transition density and likelihood function. However, since the Vasicek transition density is Normal, the employment of the Euler approximation (which also assumes Normal innovations) may paint an unduly favourable picture of the Bayesian approach. Future research could examine a short-rate model such as Cox, Ingersoll, and Ross (1985) where the transition density is known, but non-Normal. Implementing Bayesian estimation in such a case will present a number of technical challenges, since the transition density is non-central chi-squared and the likelihood function will require the product of Bessel functions. Second, the Bayesian and ML approaches in this paper could be compared to alternate estimation procedures for short-rate models. For example, Singleton (2001) suggests estimating a diffusion process by Fourier inversion of the conditional characteristic function of the SDE. Third, alternate sampling procedures might be considered in order to improve the SIFs documented in this study.

(Date of receipt of final transcript: November 9, 2004. Accepted by Garry Twite and Doug Foster, Area Editors.)

Table 1

Parameter Estimates

Point estimates of parameters using true posterior densities,
approximate posterior densities, and ML estimation. A time-series of
300 monthly observations is simulated under the Vasicek model with
[kappa] = 0.18, [gamma] = 0.08, and [beta] = 0.02. Posterior densities
are estimated using 5000 post-warmup MCMC iterates. Approximate
densities are obtained by simulating m augmented data points between
each pair of observed data points. The simulation inefficiency factors
(SIF) for the sampling scheme are calculated according to Kim,
Shephard, and Chib (1998). The reported estimates are the average over
1000 independent simulations.

                                      Approximate Densities
                     True
                    Density      m = 0       m = 1       m = 2

                            Panel A: [kappa] = 0.18

Mean                0.2946      0.2892      0.2867      0.2877
Median              0.2850      0.2803      0.2769      0.2779
(standard error)   (0.1521)    (0.1485)    (0.1515)    (0.1522)
SIF                84.29       81.37       80.84       81.33

                            Panel B. [gamma] = 0.08

Mean                0.0792      0.0792      0.0825      0.0825
Median              0.0804      0.0804      0.0816      0.0816
(standard error)   (0.0228)    (0.0228)    (0.0268)    (0.0268)
SIF                 5.96        5.95        6.39        6.40

                            Panel C. [beta] = 202

Mean                0.0201      0.0199      0.0200      0.0200
Median              0.0201      0.0198      0.0200      0.0200
(standard error)   (0.0008)    (0.0008)    (0.0008)    (0.0008)
SIF                 1.30        1.30        1.30        1.30

                              Approximate Densities

                     m = 5      m = 10      m = 20        MLE

                            Panel A: [kappa] = 0.18

Mean                0.2887      0.2893      0.2891      0.3638
Median              0.2788      0.2794      0.2792      0.3206
(standard error)   (0.1527)    (0.1530)    (0.1530)    (0.2055)
SIF                81.87       82.23       82.10

                            Panel B. [gamma] = 0.08

Mean                0.0825      0.0825      0.0825      0.0806
Median              0.0816      0.0816      0.0816      0.0793
(standard error)   (0.0268)    (0.0268)    (0.0268)    (0.0231)
SIF                 6.39        6.40        6.39

                            Panel C. [beta] = 202

Mean                0.0201      0.0201      0.0201      0.0200
Median              0.0200      0.0201      0.0201      0.0200
(standard error)   (0.0008)    (0.0008)    (0.0008)    (0.0008)
SIF                 1.30        1.30        1.30

Table 2

Parameter Estimates: Sensitivity Analysis ([kappa] = 0.50)

Point estimates of parameters using true posterior densities,
approximate posterior densities, and ML estimation. A time-series of
300 monthly observations is simulated under the Vasicek model with
[kappa]=0.50, [gamma]=0.08, and [beta]=0.02. Posterior densities are
estimated using 5000 post-warmup MCMC iterates. Approximate densities
are obtained by simulating m augmented data points between each pair of
observed data points. The simulation inefficiency factors (SIF) for the
sampling scheme are calculated according to Kim, Shephard, and Chib
(1998). The reported estimates are the average over 1000 independent
simulations.

                                       Approximate Densities
                      True
                     Density      m = 0       m = 1       m = 2

                                Panel A: [kappa] = 0.5

Mean                 0.5077      0.5004      0.5011      0.5031
Median               0.5168      0.5093      0.5108      0.5128
(standard error)    (0.2060)    (0.2017)    (0.2057)    (0.2062)
SIF                213.12      207.90      208.63      209.98

                                Panel B: [gamma] = 0.08

Mean                 0.0793      0.0793      0.0802      0.0802
Median               0.0799      0.0799      0.0801      0.0801
(standard error)    (0.0142)    (0.0141)    (0.0161)    (0.0160)
SIF                  5.77        5.77        5.87        5.87

                                Panel C: [beta] = 0.02

Mean                 0.0200      0.0196      0.0198      0.0199
Median               0.0200      0.0196      0.0198      0.0198
(standard error)    (0.0008)    (0.0008)    (0.0008)    (0.0008)
SIF                  1.30        1.29        1.30        1.30

                           Approximate Densities

                      m = 5      m = 10        MLE

                          Panel A: [kappa] = 0.5

Mean                 0.5042      0.5048      0.6499
Median               0.5141      0.5147      0.6279
(standard error)    (0.2070)    (0.2073)    (0.2160)
SIF                 210.83     211.31

                          Panel B: [gamma] = 0.08

Mean                 0.0801      0.0802      0.0803
Median               0.0801      0.0801      0.0802
(standard error)    (0.0160)    (0.0160)    (0.0082)
SIF                  5.87        5.87

                          Panel C: [beta] = 0.02

Mean                 0.0199      0.0200      0.0200
Median               0.0199      0.0199      0.0200
(standard error)    (0.0008)    (0.0008)    (0.0008)
SIF                  1.30        1.30

Table 3

Parameter Estimates: Sensitivity Analysis ([beta] = 0.01)

Point estimates of parameters using true posterior densities,
approximate posterior densities, and ML estimation. A time-series of
300 monthly observations is simulated under the Vasicek model with
[kappa] = 0.18, [gamma] = 0.08, and [beta] = 0.01. Posterior densities
are estimated using 5000 post-warmup MCMC iterates. Approximate
densities are obtained by simulating m augmented data points between
each pair of observed data points. The simulation inefficiency factors
(SIF) for the sampling scheme are calculated according to Kim,
Shephard, and Chib (1998). The reported estimates are the average
over 1000 independent simulations.

                                      Approximate Densities
                      True
                     Density      m = 0       m = 1       m = 2

                           Panel A: [kappa] = 0.18

Mean                 0.2827      0.2772      0.2775      0.2788
Median               0.2729      0.2678      0.2677      0.2688
(standard error)    (0.1524)    (0.1486)    (0.1513)    (0.1519)
SIF                 79.27       76.41       76.99       77.61

                           Panel B. [gamma] = 0.08

Mean                 0.0781      0.0781      0.0794      0.0794
Median               0.0799      0.0798      0.0801      0.0801
(standard error)    (0.0175)    (0.0176)    (0.0200)    (0.0200)
SIF                  5.66        5.66        5.82        5.82

                           Panel C. [beta] = 0.01

Mean                 0.0101      0.0099      0.0100      0.0100
Median               0.0100      0.0099      0.0100      0.0100
(standard error)    (0.0004)    (0.0004)    (0.0004)    (0.0004)
SIF                  1.08        1.07        1.08        1.08

                         Approximate Densities

                      m = 5      m = 10        MLE

                        Panel A: [kappa] = 0.18

Mean                 0.2799      0.2805      0.3635
Median               0.2699      0.2705      0.3216
(standard error)    (0.1525)    (0.1530)    (0.1987)
SIF                 78.19       78.50

                        Panel B. [gamma] = 0.08

Mean                 0.0794      0.0794      0.0801
Median               0.0801      0.0801      0.0806
(standard error)    (0.0199)    (0.0199)    (0.0123)
SIF                  5.82        5.82

                        Panel C. [beta] = 0.01

Mean                 0.0100      0.0101      0.0100
Median               0.0100      0.0100      0.0100
(standard error)    (0.0004)    (0.0004)    (0.0004)
SIF                  1.08        1.08

Table 4

Parameter Estimates: Sensitivity Analysis (n = 200, 400, 600)

Point estimates of parameters using true posterior densities,
approximate posterior densities, and ML estimation. A time-series of
n monthly observations is simulated under the Vasicek model with
[kappa] = 0.18, [gamma] = 0.08, and [beta] = 0.02. Posterior densities
are estimated using 5000 post-warmup MCMC iterates. Approximate
densities are obtained by simulating m augmented data points between
each pair of observed data points. The simulation inefficiency factors
(SIF) for the sampling scheme are calculated according to Kim,
Shephard, and Chib (1998). The reported estimates are the average over
1000 independent simulations.

                                       Approximate Densities
                      True
                     Density      m = 0         m = 1         m = 2

                           Panel A: Sample Size n = 200

Mean                 0.3559       0.3506        0.3459        0.3472
Median               0.3464       0.3416        0.3359        0.3372
(standard error)    (0.1911)     (0.1871)      (0.1908)      (0.1914)
SIF                116.88       114.20        111.83        112.62
Mean                 0.0797       0.0798        0.0833        0.0833
Median               0.0812       0.0811        0.0825        0.0825
(standard error)    (0.0231)     (0.0231)      (0.0274)      (0.0273)
SIF                  6.06         6.07          6.55          6.55
Mean                 0.0201       0.0198        0.0200        0.0200
Median               0.0201       0.0198        0.0199        0.0200
(standard error)    (0.0010)     (0.0010)      (0.0010)      (0.0010)
SIF                  1.30         1.30          1.30          1.30

                           Panel B: Sample Size n = 4110

Mean                 0.2617       0.2568        0.2556        0.2564
Median               0.2540       0.2498        0.2481        0.2488
(standard error)    (0.1299)     (0.1268)      (0.1295)      (0.1301)
SIF                 65.92        63.32         63.47         63.83
Mean                 0.0782       0.0783        0.0809        0.0809
Median               0.0791       0.0791        0.0800        0.0800
(standard error)    (0.0218)     (0.0218)      (0.0255)      (0.0255)
SIF                  5.79         5.80          6.14          6.14
Mean                 0.0201       0.0199        0.0200        0.0200
Median               0.0200       0.0198        0.0199        0.0200
(standard error)    (0.0007)     (0.0007)      (0.0007)      (0.0007)
SIF                  1.30         1.30          1.30          1.30

                           Panel C: Sample Size n = 600

Mean                 0.2292       0.2255        0.2252        0.2259
Median               0.2250       0.2218        0.2214        0.2219
(standard error)    (0.1031)     (0.1008)      (0.1029)      (0.1032)
SIF                 49.29        47.51         47.84         48.15
Mean                 0.0788       0.0788        0.0807        0.0807
Median               0.0795       0.0795        0.0801        0.0801
(standard error)    (0.0199)     (0.0199)      (0.0228)      (0.0227)
SIF                  5.81         5.81          6.05          6.05
Mean                 0.0201       0.0199        0.0200        0.0200
Median               0.0200       0.0198        0.0199        0.0200
(standard error)    (0.0006)     (0.0006)      (0.0006)      (0.0006)
SIF                  1.30         1.30          1.30          1.30

                    Approximate Densities                 Population
                                                           Parameter
                      m = 5       m = 10        MLE

                           Panel A: Sample Size n = 200

Mean                 0.3479       0.3484      0.4744
Median               0.3378       0.3384      0.4162    [kappa] = 0.18
(standard error)    (0.1921)     (0.1924)    (0.2716)
SIF                112.80       113.17
Mean                 0.0833       0.0833      0.0808
Median               0.0825       0.0825      0.0817    [gamma] = 0.08
(standard error)    (0.0274)     (0.0273)    (0.0264)
SIF                  6.55         6.55
Mean                 0.0201       0.0201      0.0201
Median               0.0200       0.0201      0.0201     [beta] = 0.02
(standard error)    (0.0010)     (0.0010)    (0.0010)
SIF                  1.30         1.30

                           Panel B: Sample Size n = 4110

Mean                 0.2572       0.2575      0.3156
Median               0.2494       0.2497      0.2831    [kappa] = 0.18
(standard error)    (0.1306)     (0.1308)    (0.1602)
SIF                 64.24        64.34
Mean                 0.0809       0.0810      0.0792
Median               0.0800       0.0800      0.0802    [gamma] = 0.08
(standard error)    (0.0255)     (0.0255)    (0.0196)
SIF                  6.14         6.14
Mean                 0.0200       0.0200      0.0200
Median               0.0200       0.0200      0.0200     [beta] = 0.02
(standard error)    (0.0007)     (0.0007)    (0.0007)
SIF                  1.30         1.30

                           Panel C: Sample Size n = 600

Mean                 0.2265       0.2269      0.2694
Median               0.2225       0.2228      0.2483    [kappa] = 0.18
(standard error)    (0.1037)     (0.1038)    (1.1185)
SIF                 48.49        48.66
Mean                 0.0807       0.0807      0.0798
Median               0.0801       0.0801      0.0801    [gamma] = 0.08
(standard error)    (0.0227)     (0.0227)    (0.0157)
SIF                  6.05         6.05
Mean                 0.0200       0.0200      0.0200
Median               0.0200       0.0200      0.0200    [beta] = 0.02
(standard error)    (0.0006)     (0.0006)    (0.0006)
SIF                  1.30         1.30

Table 5

Estimates of Bond Price

Vasicek bond prices are calculated using point estimates of parameters
using true posterior densities, approximate posterior densities (with
and without data augmentation), and ML estimation. The maturity of the
bond (in years) is indicated by [tau]. All point estimates follow from
a time-series of 300 monthly observations is simulated under the
Vasicek model with [kappa] = 0.18, [gamma] = 0.08, and [beta] = 0.02.
Posterior densities are estimated using 5000 post-warmup MCMC iterates.
Approximate densities are obtained by simulating m augmented data
points between each pair of observed data points. The reported estimates
are the average over 1000 independent simulations.

                                                  Approximate Densities
  Bond                    True         True
Maturity                  Price       Density      m = 0        m = 1

[tau] = 0.5    Price    $0.95165    $0.95188     $0.95187     $0.95181
               % Bias                0.0244       0.0231       0.0157
[tau] = 1      Price    $0.90643    $0.90722     $0.90718     $0.90692
               % Bias                0.0877       0.0834       0.0546
[tau] = 3      Price    $0.75191    $0.75594     $0.75574     $0.75404
               % Bias                0.5362       0.5086       0.2836
[tau] = 5      Price    $0.63036    $0.63705     $0.63670     $0.63325
               % Bias                1.0606       1.0049       0.4576

                            Approximate Densities
  Bond
Maturity                  m = 5        m = 10       MLE

[tau] = 0.5    Price    $0.95180     $0.95181     $0.95196
               % Bias    0.0161       0.0161       0.0321
[tau] = 1      Price    $0.90614     $0.90694     $0.90745
               % Bias    0.0561       0.0561       0.1131
[tau] = 3      Price    $0.75411     $0.75411     $0.75672
               % Bias    0.2926       0.2926       0.6391
[tau] = 5      Price    $0.63336     $0.63335     $0.63776
               % Bias    0.4754       0.4746       1.1743

Table 6

Parameter Estimates Across Different Data Series

The bank accepted bills (BAB) data is from July 1976 through February
2002 (1526 weekly observations). The Treasury note (T-note) data is
from June 1986 through February 2002 (938 weekly observations). Point
estimates of Vasicek and CKLS parameters using approximate posterior
densities with 5000 post-warmup MCMC iterates. Approximate densities
are obtained by simulating m = 2 augmented data points between
each pair of observed data points.

   Data Series                [kappa]    [gamma]     [beta]      [xi]

Vasicek
  30-day BAB        Mean       0.2519     0.0892     0.0355
                    Std Err   (0.1372)   (0.0291)   (0.0006)
  90-day BAB        Mean       0.1136     0.0814     0.0238
                    Std Err   (0.0723)   (0.0348)   (0.0004)
  180-day BAB       Mean       0.0961     0.0810     0.0211
                    Std Err   (0.0618)   (0.0354)   (0.0004)
  13-week T-notes   Mean       0.0912     0.0496     0.0140
                    Std Err   (0.0515)   (0.0299)   (0.0003)
  26-week T-notes   Mean       0.0942     0.0512     0.0146
                    Std Err   (0.0545)   (0.0310)   (0.0003)
CKLS
  30-day BAB        Mean       0.1250     0.0742     1.2443     1.6606
                    Std Err   (0.0767)   (0.0282)   (0.1211)   (0.0397)
  90-day BAB        Mean       0.0882     0.0573     0.7019     1.5751
                    Std Err   (0.0537)   (0.0269)   (0.0806)   (0.0478)
  180-day BAB       Mean       0.0892     0.0617     0.3774     1.3331
                    Std Err   (0.0511)   (0.0278)   (0.0380)   (0.0409)
  13-week T-notes   Mean       0.1615     0.0408     0.3479     1.3821
                    Std Err   (0.0769)   (0.0150)   (0.0428)   (0.0473)
  26-week T-notes   Mean       0.1654     0.0440     0.2780     1.2561
                    Std Err   (0.0802)   (0.0453)   (0.0394)   (0.0533)

Table 7

Parameter Estimates Across Different Sampling Frequencies

Point estimates of parameters using approximate posterior densities
with 5000 post-warmup MCMC iterates. Approximate densities are obtained
by simulating m = 2 augmented data points between each pair of observed
data points. Data are 1526 weekly observations of 30-day BAB yields
from 1976 to February 2002.

Sampling Frequency     [kappa]     [gamma]     [beta]       [xi]

Vasicek
  Daily     Mean       0.2376      0.0870      0.0348
            Std Err   (0.1338)    (0.0307)    (0.0003)
  Weekly    Mean       0.2519      0.0892      0.0355
            Std Err   (0.1372)    (0.0291)    (0.0006)
  Monthly   Mean       0.2441      0.0893      0.0349
            Std Err   (0.1387)    (0.0301)    (0.0014)
CKLS
  Daily     Mean       0.1716      0.0778      0.4196      1.1922
            Std Err   (0.0926)    (0.0267)    (0.0237)    (0.0233)
  Weekly    Mean       0.1250      0.0742      1.2443      1.6606
            Std Err   (0.0767)    (0.0282)    (0.1211)    (0.0397)
  Monthly   Mean       0.1086      0.0652      1.8717      1.8546
            Std Err   (0.0708)    (0.0287)    (0.3620)    (0.0865)

Figure 1

Densities of Parameters [theta] [equivalent to] ([kappa], [gamma],
[beta])

A single time-series of 300 monthly interest rates are simulated under
the Vasicek model with [kappa] = 0.18, [gamma] = 0.08 and [beta] = 0.02.
The left-side plots compare the asymptotic ML (solid line) and true
Bayesian posterior densities (dotted line) of the parameters [theta]
[equivalent to] ([kappa], [kappa], [beta]). The right-side plots compare
the true Bayesian densities (solid line) to densities under the Euler
approximation (dotted line) with data augmentation (m = 5). A Normal
kernel density estimator is used in all cases. The MCMC algorithm was
run with 20,000 post-warmup MCMC iterates.

[GRAPHIC OMITTED]

Figure 2

Posterior Densities of Vasicek/CKLS Parameters

Posterior densities of Vasicek (left side) and CKLS (right side)
parameters estimated for 30-day BABs. The time-series consists of 1526
weekly observations from July 1976 through February 2002. Bayesian
density estimates follow from an Euler approximation with data
augmentation (m = 2) and 20,000 post-warmup MCMC iterates.

[GRAPHIC OMITTED]

(1.) Bayesian estimation for the Vasicek model with a known likelihood function is detailed in Gray (2002). The main ideas are reproduced here to facilitate comparison with section 3 where the likelihood function is assumed to be unknown.

(2.) Zellner (1971, p. 44) shows that it makes no difference whether [beta] or [[beta].sup.2] is modeled, and the latter is chosen in this paper.

(3.) Derivations of results presented in this paper are contained in a technical appendix available from the author on request.

(4.) By integrating (8) with respect to [[sigma].sup.2], Zellner (1971, p. 67) shows that the marginal posterior p([b.sub.1], [b.sub.2]|[r.sub.aug], [r.sub.obs]) is multivariate Student t. Similarly, integrating (9) over ([b.sub.1], [b.sub.2]) gives the marginal p([[sigma].sup.2]|[r.sub.aug], [r.sub.obs]). However, for consistency with section 2, we simply use the full conditional density in the MCMC scheme.

(5.) Note that, while sections 2 and 3 model [[beta].sup.2], table 1 reports estimates of [beta] for convenience.

(6.) The time-series chosen has point estimates of [theta] [equivalent to] ([kappa], [kappa], [beta])close to the assumed population parameters, for no other reason than to have the densities centered around the population values.

(7.) Tables 2-4 do not report results for data augmentation with m = 20. The computation time for m = 20 in table 1, where the sample size was n 300, was 38 hours. Given that the results are insensitive to the degree of data augmentation, there is little point computing m = 20 for each sensitivity analysis.

(8.) Again, all derivations are contained in the appendix.

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The author gratefully acknowledges the comments of Jamie Alcock, Glen Barnett, Tim Brailsford, Carl Chiarella, Greg Clinch, Doug Foster, the late Egon Kalotay, Robert Kohn, Tom Mollee, Tom Smith, Garry Twite, an anonymous referee, and seminar participants at the Department of Mathematics at the University of Queensland, the 2000 AAANZ conference, and the 2000 APFA conference.

Philip Gray, UQ Business School, The University of Queensland, St Lucia QLD 4072. Email: [e-mail]