A Numerical Algorithm For The Efficient Estimation Of Band-Tar Models

This paper proposes an efficient estimation method for Band Threshold Autoregressive (Band-TAR) models. Standard maximum-likelihood algorithms cannot be used here because the log-likelihood function is not differentiable with respect to the threshold parameter, and one commonly uses a grid search over the threshold space. We propose a novel numerical algorithm to estimate Band-TARs that improves the ratio of accuracy to computation time. This procedure first divides the continuous threshold space into non-overlapping intervals and solves the ML/OLS problems of the piecewise linear Band-TAR for the first interval using the QR decomposition. It then moves from interval to interval solving the associated ML/OLS problems by updating the QR decomposition using Givens rotations. This updating process is stable and requires significantly less calculation than doing the QR anew. We then show that the RSS is a rational function of the threshold. This function is of degree (4,2), and its coefficients are readily identified using a rational interpolation method with just seven arbitrary points in each interval. Using the LS principle, we determine the optimal threshold over each interval by minimizing a continuous RSS function. Finally we estimate the threshold for the Band-TAR model as the locally optimal threshold that minimizes an Akaike criterion. Using typical sample sizes in economics and finance with varying degrees of volatility, simulations in GAUSS 3.2 show that our method displays considerable gains in accuracy to computation over conventional grid-search methods.