# Gaussian autoregressive process with dependent innovations. Some asymptotic results

## TL;DR

This paper introduces the DIG-AR(1) model, a Gaussian autoregressive process with dependent innovations, analyzing its properties and asymptotic behavior of estimators, highlighting the inconsistency of OLS and proposing an alternative estimator.

## Contribution

It presents a novel dependent innovations Gaussian AR(1) model and derives its asymptotic properties, including a new estimator with normality results.

## Key findings

- OLS estimator is inconsistent for the new model
- Proposed estimator achieves $\
- The model's dependence structure affects estimator properties

## Abstract

In this paper we introduce a modified version of a gaussian standard first-order autoregressive process where we allow for a dependence structure between the state variable $Y_{t-1}$ and the next innovation $\xi_t$. We call this model dependent innovations gaussian AR(1) process (DIG-AR(1)). We analyze the moment and temporal dependence properties of the new model. After proving that the OLS estimator does not consistently estimate the autoregressive parameter, we introduce an infeasible estimator and we provide its $\sqrt{T}$-asymptotic normality.