This article revisits the asymptotic inference for nonstationary AR(1) models of Phillips and Magdalinos (2007a) by incorporating a structural change in the AR parameter at an unknown time k0. Consider the model and the least squares estimator of the change point for the cases above under some mild conditions. Monte Carlo simulations are conducted to examine the finite-sample properties of the estimators. Our theoretical findings are supported by the Monte Carlo simulations.

/pdf/structural-change-in-nonstationary-ar-1-models-ubav6tyyws.pdf

Structural change in nonstationary ar(1) models

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 k₀. Consider the model y_{t}=β₁y_{t-1}I{t≤k₀}+β₂y_{t-1}I{t>k₀}+e_{t}, t=1,2,⋯,T, where I{⋅} denotes the indicator function. We examine two cases: Case (I) |β₁|<1,β₂=β_{2T}=1-c/T; and case (II) β₁=β_{1T}=1-c/T,|β₂|<1, where c is a fixed constant, and {e_{t},t≥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 β₁ and β₂, 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

Asymptotic inferences for an ar(1) model with a change point: stationary and nearly non‐stationary cases

This paper examines the asymptotic inference for AR(1) models with a possible structural break in the AR parameter $\beta $ near the unity at an unknown time $k_{0}$. Consider the model $y_{t}=\beta_{1}y_{t-1}I\{t\leq k_{0}\}+\beta _{2}y_{t-1}I\{t>k_{0}\}+\varepsilon_{t},~t=1,2,\cdots ,T,$ where $I\{\cdot \}$ denotes the indicator function. We examine two cases: Case (I) $|\beta_{1}|<1,\beta_{2}=\beta_{2T}=1-c/T$; and case (II)~$\beta_{1}=\beta_{1T}=1-c/T,|\beta _{2}|<1$, 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 $\beta_{1}$ and $\beta_{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

Asymptotic inferences for an AR(1) model with a change point: stationary and nearly non-stationary cases

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 k₀. Consider the model y_{t}=β₁y_{t-1}I{t≤k₀}+β₂y_{t-1}I{t>k₀}+e_{t}, t=1,2,⋯,T, where I{⋅} denotes the indicator function. We examine two cases: Case (I) |β₁|

Asymptotic Inferences for an AR(1) Model with a Change Point: Stationary and Nearly Non-stationary Cases

An asymptotic distribution is derived for the least squares (LS) estimate of a first‐order autoregression with a mildly explosive root and anti‐persistent errors. While the sample moments depend on the Hurst parameter asymptotically, the Cauchy limiting distribution theory remains valid for the LS estimates in the model without intercept and a model with an asymptotically negligible intercept. Monte Carlo studies are designed to check the precision of the Cauchy distribution in finite samples. An empirical study based on the monthly NASDAQ index highlights the usefulness of the model and the new limiting distribution.

https://ink.library.smu.edu.sg/cgi/viewcontent.cgi?article=3433&context=soe_research

Mildly Explosive Autoregression with Anti‐persistent Errors*

An asymptotic theory was given by Phillips and Magdalinos (J Econom 136(1):115–130, 2007) for autoregressive time series Yt = ρYt−1 + ut, t = 1,...,n, with ρ = ρn = 1 + c/kn, under (2 + δ)-order moment condition for the innovations ut, where δ > 0 when c 0, {ut} is a sequence of independent and identically distributed random variables, and (kn)n ∈ ℕ is a deterministic sequence increasing to infinity at a rate slower than n. In the present paper, we established similar results when the truncated second moment of the innovations $l(x)=\textsf{E} [u_1^2I\{|u_1|\le x\}]$ is a slowly varying function at ∞, which may tend to infinity as x → ∞. More interestingly, we proposed a new pivotal for the coefficient ρ in case c < 0, and formally proved that it has an asymptotically standard normal distribution and is nuisance parameter free. Our numerical simulation results show that the distribution of this pivotal approximates the standard normal distribution well under normal innovations.

/pdf/limit-theory-for-moderate-deviations-from-a-unit-root-under-4ubieeny2u.pdf

Sai-Hua Huang

Papers

Limit Theory for Moderate Deviations from a Unit Root Under Innovations with a Possibly Infinite Variance