# Quantile double autoregression

## TL;DR

This paper introduces the quantile double autoregression model, a novel approach for financial time series that captures varying structures at different quantiles and conditional heteroscedasticity simultaneously.

## Contribution

It proposes a new model that combines features of quantile regression and autoregression, with methods for estimation, model checking, and demonstration on real financial data.

## Key findings

- Model is strictly stationary under certain conditions.
- Self-weighted quantile estimation performs well in finite samples.
- Application to S&P 500 data demonstrates model's necessity.

## Abstract

Many financial time series have varying structures at different quantile levels, and also exhibit the phenomenon of conditional heteroscedasticity at the same time. In the meanwhile, it is still lack of a time series model to accommodate both of the above features simultaneously. This paper fills the gap by proposing a novel conditional heteroscedastic model, which is called the quantile double autoregression. The strict stationarity of the new model is derived, and a self-weighted conditional quantile estimation is suggested. Two promising properties of the original double autoregressive model are shown to be preserved. Based on the quantile autocorrelation function and self-weighting concept, two portmanteau tests are constructed, and they can be used in conjunction to check the adequacy of fitted conditional quantiles. The finite-sample performance of the proposed inference tools is examined by simulation studies, and the necessity of the new model is further demonstrated by analyzing the S&P500 Index.