# Non-standard inference for augmented double autoregressive models with null volatility coefficients

## TL;DR

This paper develops a robust statistical inference method for augmented double autoregressive models with null volatility coefficients, addressing challenges posed by heavy-tailed data and boundary parameters.

## Contribution

It introduces a self-weighted GQMLE approach that works under weaker moment conditions and provides non-standard asymptotic results for various test statistics.

## Key findings

- The proposed method is valid for stationary data with heavy tails.
- Simulation studies demonstrate the effectiveness of the inference procedure.
- Application to real data illustrates practical utility.

## Abstract

This paper considers an augmented double autoregressive (DAR) model, which allows null volatility coefficients to circumvent the over-parameterization problem in the DAR model. Since the volatility coefficients might be on the boundary, the statistical inference methods based on the Gaussian quasi-maximum likelihood estimation (GQMLE) become non-standard, and their asymptotics require the data to have a finite sixth moment, which narrows applicable scope in studying heavy-tailed data. To overcome this deficiency, this paper develops a systematic statistical inference procedure based on the self-weighted GQMLE for the augmented DAR model. Except for the Lagrange multiplier test statistic, the Wald, quasi-likelihood ratio and portmanteau test statistics are all shown to have non-standard asymptotics. The entire procedure is valid as long as the data is stationary, and its usefulness is illustrated by simulation studies and one real example.