Cumulative Distribution Function and Probability Density Function of the Continuous Transitional Limiting Distribution of Normalized OLS Estimator

Abstract

This paper derives the cumulative distribution function and probability density function of the continuous transitional distribution of normalized Ordinary Least Squares (OLS) estimator for autoregressive process with local to moderate deviations from unity root. The derivation of these distributions is based on the asymptotic results in Phillips, Magdalinos and Giraitis (2010). The approach is one of first deriving the asymptotic joint characteristic function of the numerator and denominator of the normalized OLS estimator and then inverting it by numerical integration to find these distributions. We present some graphs to illustrate the behavior of smooth transition. We also provide tables of percentile of the limiting distribution of normalized OLS estimator and assess the approximate accuracy of the distribution in detail.