# The Hyv\"arinen scoring rule in Gaussian linear time series models

## TL;DR

This paper explores score-matching estimators based on the Hyv"arinen scoring rule for Gaussian linear time series models, offering an alternative to likelihood-based methods that avoids the normalising constant.

## Contribution

It introduces and analyzes score-matching estimators for stationary time-series models, comparing their performance with traditional likelihood-based estimators across different models.

## Key findings

- Score-matching estimators perform well for MA and ARFIMA models.
- Performance varies depending on the model, with less effectiveness for AR models.
- The approach is useful when the normalising constant is difficult to compute.

## Abstract

Likelihood-based estimation methods involve the normalising constant of the model distributions, expressed as a function of the parameter. However in many problems this function is not easily available, and then less efficient but more easily computed estimators may be attractive. In this work we study stationary time-series models, and construct and analyse "score-matching'' estimators, that do not involve the normalising constant. We consider two scenarios: a single series of increasing length, and an increasing number of independent series of fixed length. In the latter case there are two variants, one based on the full data, and another based on a sufficient statistic. We study the empirical performance of these estimators in three special cases, autoregressive (\AR), moving average (MA) and fractionally differenced white noise (\ARFIMA) models, and make comparisons with full and pairwise likelihood estimators. The results are somewhat model-dependent, with the new estimators doing well for $\MA$ and \ARFIMA\ models, but less so for $\AR$ models.