Time Series Analysis of Rankings: A GARCH-Type Approach

TL;DR

This paper introduces a novel GARCH-inspired model for analyzing time series of ranking data, capturing temporal dynamics and dependencies with theoretical guarantees and practical estimation methods.

Contribution

It develops a new class of time-varying ranking models based on GARCH principles, incorporating autoregressive components and handling missing data effectively.

Findings

Model demonstrates stationarity and ergodicity.

Estimation via maximum likelihood and Monte Carlo EM algorithms.

Successful application to tennis player rankings from 2015 to 2019.

Abstract

Ranking data are frequently obtained nowadays but there are still scarce methods for treating these data when temporally observed. The present paper contributes to this topic by proposing and developing novel models for handling time series of ranking data. We introduce a class of time-varying ranking models inspired by the Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) models. More specifically, the temporal dynamics are defined by the conditional distribution of the current ranking given the past rankings, which are assumed to follow a Mallows distribution, which implicitly depends on a distance. Then, autoregressive and feedback components are incorporated into the model through the conditional expectation of the associated distances. Theoretical properties of our ranking GARCH models such as stationarity and ergodicity are established. The estimation of parameters is performed via maximum likelihood estimation when data is fully observed. We develop a Monte Carlo Expectation-Maximisation algorithm to deal with cases involving missing data. Monte Carlo simulation studies are presented to study the performance of the proposed estimators under both non-missing and missing data scenarios. A real data application about the weekly ranking of professional tennis players from 2015 to 2019 is presented under our proposed ranking GARCH models.