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What is Learnable in Valiant's Theory of the Learnable? 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Asymptotic inferences for an AR(1) model with a change point: stationary and nearly non-stationary cases
Pang Tianxiao, Zhang Danna, Chong Terence Tai-Leung · 2013-06-06 · via math.ST updates on arXiv.org

This paper examines the asymptotic inference for AR(1) models with a possible structural break in the AR parameter $β$ near the unity at an unknown time $k_{0}$. Consider the model $y_{t}=β_{1}y_{t-1}I\{t\leq k_{0}\}+β_{2}y_{t-1}I\{t>k_{0}\}+\varepsilon_{t},~t=1,2,\cdots ,T,$ where $I\{\cdot \}$ denotes the indicator function. We examine two cases: Case (I) $|β_{1}|<1,β_{2}=β_{2T}=1-c/T$; and case (II)~$β_{1}=β_{1T}=1-c/T,|β_{2}|<1$, where $c$\ is a fixed constant, and $\{\varepsilon_{t},t\geq 1\}$\ is a sequence of i.i.d. random variables which are in the domain of attraction of the normal law with zero means and possibly infinite variances. We derive the limiting distributions of the least squares estimators of $β_{1}$ and $β_{2} $, and that of the break-point estimator for shrinking break for the aforementioned cases. Monte Carlo simulations are conducted to demonstrate the finite sample properties of the estimators. Our theoretical results are supported by Monte Carlo simulations.\newline

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