scholarly journals Limit Theory for Moderate Deviations From a Unit Root Under Weak Dependence

2009 ◽  
pp. 123-162 ◽  
Author(s):  
Peter C.B. Phillips ◽  
Tassos Magdalinos
Keyword(s):  
Unit Root ◽  
Weak Dependence ◽  
Moderate Deviations ◽  
Limit Theory

scholarly journals Limit theory for moderate deviations from a unit root

2007 ◽  
Vol 136 (1) ◽  
pp. 115-130 ◽  
Author(s):  
Peter C.B. Phillips ◽  
Tassos Magdalinos
Keyword(s):  
Unit Root ◽  
Moderate Deviations ◽  
Limit Theory

scholarly journals UNIT ROOT AND COINTEGRATING LIMIT THEORY WHEN INITIALIZATION IS IN THE INFINITE PAST

2009 ◽  
Vol 25 (6) ◽  
pp. 1682-1715 ◽  
Author(s):  
Peter C.B. Phillips ◽  
Tassos Magdalinos
Keyword(s):  
Normal Distribution ◽  
Least Squares ◽  
Rate Of Convergence ◽  
Unit Root ◽  
Initial Conditions ◽  
Cauchy Distribution ◽  
Least Squares Estimator ◽  
Initial Condition ◽  
Limit Theory ◽  
Regression Theory

It is well known that unit root limit distributions are sensitive to initial conditions in the distant past. If the distant past initialization is extended to the infinite past, the initial condition dominates the limit theory, producing a faster rate of convergence, a limiting Cauchy distribution for the least squares coefficient, and a limit normal distribution for the t-ratio. This amounts to the tail of the unit root process wagging the dog of the unit root limit theory. These simple results apply in the case of a univariate autoregression with no intercept. The limit theory for vector unit root regression and cointegrating regression is affected but is no longer dominated by infinite past initializations. The latter contribute to the limiting distribution of the least squares estimator and produce a singularity in the limit theory, but do not change the principal rate of convergence. Usual cointegrating regression theory and inference continue to hold in spite of the degeneracy in the limit theory and are therefore robust to initial conditions that extend to the infinite past.

scholarly journals GAUSSIAN INFERENCE IN AR(1) TIME SERIES WITH OR WITHOUT A UNIT ROOT

2008 ◽  
Vol 24 (3) ◽  
pp. 631-650 ◽  
Author(s):  
Peter C.B. Phillips ◽  
Chirok Han
Keyword(s):  
Time Series ◽  
Central Limit ◽  
Unit Root ◽  
Initial Conditions ◽  
Finite Sample ◽  
Linear Processes ◽  
Limit Theory ◽  
Sample Bias ◽  
Dynamic Panels ◽  
Finite Sample Bias

This paper introduces a simple first-difference-based approach to estimation and inference for the AR(1) model. The estimates have virtually no finite-sample bias and are not sensitive to initial conditions, and the approach has the unusual advantage that a Gaussian central limit theory applies and is continuous as the autoregressive coefficient passes through unity with a uniform $\sqrt{n}$ rate of convergence. En route, a useful central limit theorem (CLT) for sample covariances of linear processes is given, following Phillips and Solo (1992, Annals of Statistics, 20, 971–1001). The approach also has useful extensions to dynamic panels.

HETEROSKEDASTIC TIME SERIES WITH A UNIT ROOT

2009 ◽  
Vol 25 (5) ◽  
pp. 1228-1276 ◽  
Author(s):  
Giuseppe Cavaliere ◽  
A.M. Robert Taylor
Keyword(s):  
Unit Root ◽  
Innovation Process ◽  
Partial Solution ◽  
Weak Dependence ◽  
Martingale Differences ◽  
Inference Problem ◽  
Finite Samples ◽  
Bootstrap Tests ◽  
Spot Volatility ◽  
Innovation Processes

In this paper we provide a unified theory, and associated invariance principle, for the large-sample distributions of the Dickey–Fuller class of statistics when applied to unit root processes driven by innovations displaying nonstationary stochastic volatility of a very general form. These distributions are shown to depend on both the spot volatility and the integrated variation associated with the innovation process. We propose a partial solution (requiring any leverage effects to be asymptotically negligible) to the identified inference problem using a wild bootstrap–based approach. Results are initially presented in the context of martingale differences and are later generalized to allow for weak dependence. Monte Carlo evidence is also provided that suggests that our proposed bootstrap tests perform very well in finite samples in the presence of a range of innovation processes displaying nonstationary volatility and/or weak dependence.

scholarly journals IV AND GMM INFERENCE IN ENDOGENOUS STOCHASTIC UNIT ROOT MODELS

2017 ◽  
Vol 34 (5) ◽  
pp. 1065-1100 ◽  
Author(s):  
Offer Lieberman ◽  
Peter C.B. Phillips
Keyword(s):  
Stock Returns ◽  
Unit Root ◽  
Instrumental Variable ◽  
Nonlinear Least Squares ◽  
Asymptotic Distributions ◽  
Small Samples ◽  
Time Varying ◽  
Limit Theory ◽  
The One ◽  
Unit Root Models

Lieberman and Phillips (2017; LP) introduced a multivariate stochastic unit root (STUR) model, which allows for random, time varying local departures from a unit root (UR) model, where nonlinear least squares (NLLS) may be used for estimation and inference on the STUR coefficient. In a structural version of this model where the driver variables of the STUR coefficient are endogenous, the NLLS estimate of the STUR parameter is inconsistent, as are the corresponding estimates of the associated covariance parameters. This paper develops a nonlinear instrumental variable (NLIV) as well as GMM estimators of the STUR parameter which conveniently addresses endogeneity. We derive the asymptotic distributions of the NLIV and GMM estimators and establish consistency under similar orthogonality and relevance conditions to those used in the linear model. An overidentification test and its asymptotic distribution are also developed. The results enable inference about structural STUR models and a mechanism for testing the local STUR model against a simple UR null, which complements usual UR tests. Simulations reveal that the asymptotic distributions of the NLIV and GMM estimators of the STUR parameter as well as the test for overidentifying restrictions perform well in small samples and that the distribution of the NLIV estimator is heavily leptokurtic with a limit theory which has Cauchy-like tails. Comparisons of STUR coefficient and standard UR coefficient tests show that the one-sided UR test performs poorly against the one-sided STUR coefficient test both as the sample size and departures from the null rise. The results are applied to study the relationships between stock returns and bond spread changes.

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