Boundary Limit Theory for Functional Local to Unity Regression

Bykhovskaya, Anna; Phillips, Peter C.B.

doi:10.1111/jtsa.12285

Cited by 11 publications

(1 citation statement)

References 29 publications

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“…Remark Bykhoskaya and Phillips (2018a,b) considered a time series generated by the following model:

X_{t} = θ_{t N} X_{t - 1} + u_{t}, t = 1, \dots, N,

where

u_{t} \sim N (0, σ^{2})

and

θ_{t N} = \exp {\frac{c (t / N)}{N}} \approx 1 + \frac{c (t / N)}{N}

. They called this model as functional local unit root (FLUR).…”

Section: Tempered Functional Local To Unity Root Modelmentioning

confidence: 99%

Tempered functional time series

Sabzikar

Kokoszka

2022

Journal Time Series Analysis

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We propose a broad class of models for time series of curves (functions) that can be used to quantify near long‐range dependence or near unit root behavior. We establish fundamental properties of these models and rates of consistency for the sample mean function and the sample covariance operator. The latter plays a role analogous to sample cross‐covariances for multivariate time series, but is far more important in the functional setting because its eigenfunctions are used in principal component analysis, which is a major tool in functional data analysis. It is used for dimension reduction of feature extraction. We also establish a central limit theorem for functions following our model. Both the consistency rates and the normalizations in the Central Limit Theorem (CLT) are nonstandard. They reflect the local unit root behavior and the long memory structure at moderate lags.

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“…Remark Bykhoskaya and Phillips (2018a,b) considered a time series generated by the following model:

X_{t} = θ_{t N} X_{t - 1} + u_{t}, t = 1, \dots, N,

where

u_{t} \sim N (0, σ^{2})

and

θ_{t N} = \exp {\frac{c (t / N)}{N}} \approx 1 + \frac{c (t / N)}{N}

. They called this model as functional local unit root (FLUR).…”

Section: Tempered Functional Local To Unity Root Modelmentioning

confidence: 99%

Tempered functional time series

Sabzikar

Kokoszka

2022

Journal Time Series Analysis

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Threshold regression asymptotics: From the compound Poisson process to two-sided Brownian motion

Phillips

2018

Economics Letters

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The asymptotic distribution of the least squares estimator in threshold regression is expressed in terms of a compound Poisson process when the threshold e¤ect is …xed and as a functional of two-sided Brownian motion when the threshold e¤ect shrinks to zero. This paper explains the relationship between this dual limit theory by showing how the asymptotic forms are linked in terms of joint and sequential limits. In one case, joint asymptotics apply when both the sample size diverges and the threshold e¤ect shrinks to zero, whereas sequential asymptotics operate in the other case in which the sample size diverges …rst and the threshold e¤ect shrinks subsequently. The two operations lead to the same limit distribution, thereby linking the two di¤erent cases. The proofs make use of ideas involving limit theory for sums of a random number of summands.

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Random coefficient continuous systems: Testing for extreme sample path behavior

Tao

Phillips

2019

Journal of Econometrics

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This paper studies a continuous time dynamic system with a random persistence parameter. The exact discrete time representation is obtained and related to several discrete time random coefficient models currently in the literature. The model distinguishes various forms of unstable and explosive behavior according to specific regions of the parameter space that open up the potential for testing these forms of extreme behavior. A two-stage approach that employs realized volatility is proposed for the continuous system estimation, asymptotic theory is developed, and test statistics to identify the different forms of extreme sample path behavior are proposed. Simulations show that the proposed estimators work well in empirically realistic settings and that the tests have good size and power properties in discriminating characteristics in the data that differ from typical unit root behavior. The theory is extended to cover models where the random persistence parameter is endogenously determined. An empirical application based on daily real S&P 500 index data over 1928-2018 reveals strong evidence against parameter constancy over the whole sample period leading to a long duration of what the model characterizes as extreme behavior in real stock prices.

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Boundary Limit Theory for Functional Local to Unity Regression

Cited by 11 publications

References 29 publications

Tempered functional time series

Tempered functional time series

Threshold regression asymptotics: From the compound Poisson process to two-sided Brownian motion

Random coefficient continuous systems: Testing for extreme sample path behavior

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