Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Convergence Of Probability Measures

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

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

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

Lieberman and Phillips (J Time Ser Anal 35(6):592–623 2014, J Econ 196(1):99–110 2017) proposed a stochastic unit root model, where the source of the variation of the autoregressive coefficient is driven by a stationary process. In this paper, we study a new stochastic unit root model, in which the source of the variation of the autoregressive coefficient is driven by a (nearly) non-stationary process. The asymptotic theory for this model is established not only in the presence of an intercept but also under a mis-specification of intercept. Our study reveals some new findings which are different from those established in Lieberman and Phillips (J Time Ser Anal 35(6):592–623 2014, J Econ 196(1):99–110 2017). Our theoretical results demonstrate that the statistical properties of the estimators of the model parameters vary from case to case, and depend not only on whether the parameters are zero or not, and the validity of the model specification, but also on the degree of the persistence of the source of the variation of the autoregressive coefficient. The application to a hypothesis testing with null hypothesis of a deterministic unit root model is briefly discussed. Monte Carlo simulations are conducted to examine the finite-sample performance for the estimators of model parameters. Our theoretical findings are supported by the simulation results.

Tian-Xiao Pang

Papers

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

Asymptotic Theory for a Stochastic Unit Root Model with Intercept and Under Mis-Specification of Intercept