# Estimation and Inference in the Presence of Fractional d=1/2 and Weakly Nonstationary Processes

## TL;DR

This paper develops new limit theory for weakly nonstationary processes, including fractional processes with d=1/2, and introduces a robust specification test for regression models that remains valid regardless of the regressor's persistence.

## Contribution

It provides the first comprehensive limit theory for WNPs and proposes a new chi-squared based specification test that is robust to the persistence level of regressors.

## Key findings

- The new test maintains size control across various data generating processes.
- Simulation results show the test outperforms existing methods like Wang and Phillips (2012).
- The theory applies to fractional and autoregressive processes near nonstationarity.

## Abstract

We provide new limit theory for functionals of a general class of processes lying at the boundary between stationarity and nonstationarity -- what we term weakly nonstationary processes (WNPs). This includes, as leading examples, fractional processes with d=1/2, and arrays of autoregressive processes with roots drifting slowly towards unity. We first apply the theory to study inference in parametric and nonparametric regression models involving WNPs as covariates. We then use these results to develop a new specification test for parametric regression models. By construction, our specification test statistic has a chi-squared limiting distribution regardless of the form and extent of persistence of the regressor, implying that a practitioner can validly perform the test using a fixed critical value, while remaining agnostic about the mechanism generating the regressor. Simulation exercises confirm that the test controls size across a wide range of data generating processes, and outperforms a comparable test due to Wang and Phillips (2012, Ann. Stat.) against many alternatives.