Transformed-Linear Innovations Algorithm for Modeling and Forecasting of Time Series Extremes

TL;DR

This paper introduces a transformed-linear innovations algorithm for modeling and forecasting time series extremes, enabling parameter estimation and capturing tail dependence in nonnegative regularly-varying models.

Contribution

It develops a novel innovations algorithm for transformed-linear time series models and demonstrates its effectiveness in modeling tail dependence and parameter estimation.

Findings

The algorithm provides the best linear predictor for the class of models.

It enables parameter estimation for transformed-linear MA(∞) models.

The models effectively capture tail dependence in GARCH(1,1) and Markov series.

Abstract

The innovations algorithm is a classical recursive forecasting algorithm used in time series analysis. We develop the innovations algorithm for a class of nonnegative regularly varying time series models constructed via transformed-linear arithmetic. In addition to providing the best linear predictor, the algorithm also enables us to estimate parameters of transformed-linear regularly-varying moving average (MA) models, thus providing a tool for modeling. We first construct an inner product space of transformed-linear combinations of nonnegative regularly-varying random variables and prove its link to a Hilbert space which allows us to employ the projection theorem, from which we develop the transformed-linear innovations algorithm. Turning our attention to the class of transformed linear MA($\infty$) models, we give results on parameter estimation and also show that this class of models is dense in the class of possible tail pairwise dependence functions (TPDFs). We also develop an extremes analogue of the classical Wold decomposition. Simulation study shows that our class of models captures tail dependence for the GARCH(1,1) model and a Markov time series model, both of which are outside our class of models.