Median‐unbiased Estimation and Exact Inference Methods for First‐order Autoregressive Models with Conditional Heteroscedasticity of Unknown Form

Abstract

Abstract. Consider the first‐order autoregressive model yt = φyt−1 + ɛt, t = 1,…, T, with arbitrary initial non‐zero value y0. Assuming that the error terms ɛt are independently distributed according to median‐zero distributions [Zieliński (1999)Journal of Time Series Analysis, Vol. 20, p. 477] shows that the estimator conjectured by Hurwicz (1950)Statistical Inference in Dynamic Economic Models. New York, NY: Wiley – the median of the consecutive ratios yt/yt−1– is an exactly median‐unbiased estimator of the autoregressive parameter φ. This paper shows that the Hurwicz estimator remains median‐unbiased under more general distributional assumptions, without assuming statistical independence. In particular, no restrictions are placed on the degree of heterogeneity and dependence of the conditional variance process. A computationally efficient method is also proposed to build exact confidence intervals for the autoregressive parameter which are valid in finite samples for any value of φ on the real line.