A.S. Hurn

On the efficacy of simulated maximum likelihood for estimating the parameters of stochastic differential Equations

A method for estimating the parameters of stochastic differential equations (SDEs) by simulated maximum likelihood is presented. This method is feasible whenever the underlying SDE is a Markov process. Estimates are compared to those generated by indirect inference, discrete and exact maximum likelihood. The technique is illustrated with reference to a one-factor model of the term structure of interest rates using 3-month US Treasury Bill data.

Forecasting day-ahead electricity load using a multiple equation time series approach

The quality of short-term electricity load forecasting is crucial to the operation and trading activities of market participants in an electricity market. In this paper, it is shown that a multiple equation time-series model, which is estimated by repeated application of ordinary least squares, has the potential to match or even outperform more complex nonlinear and nonparametric forecasting models. The key ingredient of the success of this simple model is the effective use of lagged information by allowing for interaction between seasonal patterns and intra-day dependencies. Although the model is built using data for the Queensland region of Australia, the method is completely generic and applicable to any load forecasting problem. The model’s forecasting ability is assessed by means of the mean absolute percentage error (MAPE). For day-ahead forecast, the MAPE returned by the model over a period of 11 years is an impressive 1.36%. The forecast accuracy of the model is compared with a number of benchmarks including three popular alternatives and one industrial standard reported by the Australia Energy Market Operator (AEMO). The performance of the model developed in this paper is superior to all benchmarks and outperforms the AEMO forecasts by about a third in terms of the MAPE criterion.

Estimating the parameters of stochastic differential equations

Two maximum likelihood methods for estimating the parameters of stochastic differential equations (SDEs) from time-series data are proposed. The first is that of simulated maximum likelihood in which a nonparametric kernel is used to construct the transitional density of an SDE from a series of simulated trials. The second approach uses a spectral technique to solve the Kolmogorov equation satisfied by the transitional probability density. The exact likelihood function for a geometric random walk is used as a benchmark against which the performance of each method is measured. Both methods perform well with the spectral method returning results which are practically identical to those derived from the exact likelihood. The technique is illustrated by modelling interest rates in the UK gilts market using a fundamental one-factor term-structure equation for the instantaneous rate of interest.

The Devil is in the Detail: Hints for Practical Optimisation

Finding the minimum of an objective function, such as a least squares or negative log-likelihood function, with respect to the unknown model parameters is a problem often encountered in econometrics. Consequently, students of econometrics and applied econometricians are usually well-grounded in the broad differences between the numerical procedures employed to solve these problems. Often, however, relatively little time is given to understanding the practical subtleties of implementing these schemes when faced with illbehaved problems. This paper addresses some of the details involved in practical optimisation, such as dealing with constraints on the parameters, specifying starting values, termination criteria and analytical gradients, and illustrates some of the general ideas with several instructive examples.

Transitional Densities of Diffusion Processes: A New Approach to Solving the Fokker-Planck Equation

Many common option pricing problems require numerical solution techniques. One standard tool is to solve a finite difference approximation to the instruments fundamental partial differential equation (PDE), with appropriate boundary conditions. The approximation will converge to the true solution of the PDE asymptotically as the time and price steps go to zero. In practice, of course, finite step sizes must be selected as a compromise between accuracy and computation time. But this can lead to numerical difficulties in computation. Often these problems are difficult to spot a priori because they only occur for certain parameter values that seem improbable economically. Unfortunately, in searching for correct estimates, a calibration routine can still stumble onto such problem parameter values, and then break down or deliver highly inaccurate results. One such numerical problem occurs at time 0, when the probability density at the initial asset price is a Dirac delta function. In this short article, Hurn, Jeisman and Lindsay offer a simple solution to the problem by reformulating the calculation into one based on the transitional cumulative distribution function (CDF) instead of the transitional density. Simulation analysis suggests that the new approach may not greatly reduce calculation times, although performance can be enhanced with a minor modification. But a plain vanilla real world example of fitting a simple interest rate model on 1-month LIBOR demonstrates the fragility of the standard methodology and the improvement in robustness provided by the new technique.

On the Specification of the Drift and Diffusion Functions for Continuous-Time Models of the Spot Interest Rate

This paper explores the specification of drift and diffusion functions for continuous-time short-term interest rate models. Various forms for the drift and diffusion of 7-day Eurodollar rates are proposed and then estimated by discrete maximum-likelihood. The results suggest that a nonparametric specification of drift and volatility in terms of orthogonal polynomial expansions is effective in eliminating problems of parameter identification encountered previously. Some evidence is found to support the claim that the drift of the short term interest rate is nonlinear.

Using discrete-time techniques to test continuous-time models for nonlinearity in drift

This paper examines whether or not a discrete-time econometric test for nonlinearity in mean may be used in cases where the data are believed to be generated in continuous time. It is demonstrated that appropriate bootstrapping techniques are required to yield a test statistic with sensible statistical properties. The technique is demonstrated by using it to examine 7-day Eurodollar rates for nonlinearity in mean.