Improving the Estimation and Predictions of Small Time Series Models

Gareth Liu-Evans

doi:10.1515/jtse-2021-0051

Abstract

A new approach is developed for improving the point estimation and predictions of parametric time-series models. The method targets performance criteria such as estimation bias, root mean squared error, variance, or prediction error, and produces closed-form estimators focused towards these targets via a computational approximation method. This is done for an autoregression coefficient, for the mean reversion parameter in Vasicek and CIR diffusion models, for the binomial thinning parameter in integer-valued autoregressive (INAR) models, and for predictions from a CIR model. The success of the prediction targeting approach is shown in Monte Carlo simulations and in out-of-sample forecasting of the US Federal Funds rate.

Keywords: Bias correction; Forecasting; likelihood-free estimation; Time series; Diffusions; Count data

JEL Classification: C13; C22; C15; C53

Corresponding author: Gareth Liu-Evans, Management School, University of Liverpool, Chatham Street, Liverpool, L69 7ZH, UK, E-mail: gareth.liu-evans@liverpool.ac.uk

The data and code for the application and methods used in Section 5 are available at the author’s GitHub page.
Conflict of interest statement: There is no conflict of interest relating to the author and this paper.

Appendix A

A.1 Implementation of the Relative Performance Constraint

The implementation in the examples of Sections 2 and 3 and the Supplementary Appendix requires that the bias and RMSE values of the new estimator be no greater than those of the original at each point in the training set T . Let ab _o and RMSE_o denote the vectors of absolute bias and RMSE values for the original estimator corresponding to points in T , while ab and RMSE are similar vectors for the new estimator. Let d _ab and d _RMSE then denote the maximal (signed) elements of the vectors ab − ab _o and RMSE − RMSE_o, respectively. It is required that d _ab ≤ 0 and d _RMSE ≤ 0, and to achieve this the parameter vector w, which defines G in (9) once G is chosen, is selected to minimise the value of the penalised objective function:

L = ‖ E ‖ + λ max ( d a b , 0 ) 2 + max ( d RMSE , 0 ) 2

where λ > 0 is large. This simple penalty function method was sufficient for the applications that were considered, using the subplex global optimisation algorithm.

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Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/jtse-2021-0051).

Received: 2021-11-11

Revised: 2022-02-21

Accepted: 2022-03-03

Published Online: 2022-04-13

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Supplementary Material Details