scholarly journals Quantile inference for nonstationary processes with infinite variance innovations

2021 ◽  
Vol 36 (3) ◽  
pp. 443-461
Author(s):  
Qi-meng Liu ◽  
Gui-li Liao ◽  
Rong-mao Zhang
Keyword(s):  
Unit Root ◽  
Brownian Bridge ◽  
Consumer Price Index ◽  
Infinite Variance ◽  
Finite Variance ◽  
Ratio Test ◽  
Finite Sample ◽  
Kolmogorov Smirnov ◽  
Von Mises ◽  
Quantile Inference

AbstractBased on the quantile regression, we extend Koenker and Xiao (2004) and Ling and McAleer (2004)’s works from finite-variance innovations to infinite-variance innovations. A robust t-ratio statistic to test for unit-root and a re-sampling method to approximate the critical values of the t-ratio statistic are proposed in this paper. It is shown that the limit distribution of the statistic is a functional of stable processes and a Brownian bridge. The finite sample studies show that the proposed t-ratio test always performs significantly better than the conventional unit-root tests based on least squares procedure, such as the Augmented Dick Fuller (ADF) and Philliphs-Perron (PP) test, in the sense of power and size when infinite-variance disturbances exist. Also, quantile Kolmogorov-Smirnov (QKS) statistic and quantile Cramer-von Mises (QCM) statistic are considered, but the finite sample studies show that they perform poor in power and size, respectively. An application to the Consumer Price Index for nine countries is also presented.

Asymptotic Theory of LAD Estimation in a Unit Root Process with Finite Variance Errors

1996 ◽  
Vol 12 (1) ◽  
pp. 129-153 ◽  
Author(s):  
Miguel A. Herce
Keyword(s):  
Unit Root ◽  
Infinite Variance ◽  
Finite Variance ◽  
Test Statistics ◽  
Finite Sample ◽  
Least Absolute Deviations ◽  
Autoregressive Parameter ◽  
Finite Sample Properties ◽  
Heavy Tailed ◽  
Lad Estimation

In this paper we derive the asymptotic distribution of the least absolute deviations (LAD) estimator of the autoregressive parameter under the unit root hypothesis, when the errors are assumed to have finite variances, and present LAD-based unit root tests, which, under heavy-tailed errors, are expected to be more powerful than tests based on least squares. The limiting distribution of the LAD estimator is that of a functional of a bivariate Brownian motion, similar to those encountered in cointegrating regressions. By appropriately correcting for serial correlation and other distributional parameters, the test statistics introduced here are found to have either conditional or unconditional normal limiting distributions. The results of the paper complement similar ones obtained by Knight (1991, Canadian Journal of Statistics 17, 261-278) for infinite variance errors. A simulation study is conducted to investigate the finite sample properties of our tests.

scholarly journals A POWERFUL TEST OF THE AUTOREGRESSIVE UNIT ROOT HYPOTHESIS BASED ON A TUNING PARAMETER FREE STATISTIC

2009 ◽  
Vol 25 (6) ◽  
pp. 1515-1544 ◽  
Author(s):  
Morten Ørregaard Nielsen
Keyword(s):  
Asymptotic Distribution ◽  
Unit Root ◽  
Linear Time ◽  
Nonparametric Test ◽  
Tuning Parameter ◽  
Ratio Test ◽  
Local Power ◽  
Finite Sample ◽  
Power Of The Test ◽  
The Family

This paper presents a family of simple nonparametric unit root tests indexed by one parameter,d, and containing the Breitung (2002,Journal of Econometrics108, 342–363) test as the special cased= 1. It is shown that (a) each member of the family withd> 0 is consistent, (b) the asymptotic distribution depends ondand thus reflects the parameter chosen to implement the test, and (c) because the asymptotic distribution depends ondand the test remains consistent for alld> 0, it is possible to analyze the power of the test for different values ofd. The usual Phillips–Perron and Dickey–Fuller type tests are indexed by bandwidth, lag length, etc., but have none of these three properties.It is shown that members of the family withd< 1 have higher asymptotic local power than the Breitung (2002) test, and whendis small the asymptotic local power of the proposed nonparametric test is relatively close to the parametric power envelope, particularly in the case with a linear time trend. Furthermore, generalized least squares (GLS) detrending is shown to improve power whendis small, which is not the case for the Breitung (2002) test. Simulations demonstrate that when applying a sieve bootstrap procedure, the proposed variance ratio test has very good size properties, with finite-sample power that is higher than that of the Breitung (2002) test and even rivals the (nearly) optimal parametric GLS detrended augmented Dickey–Fuller test with lag length chosen by an information criterion.

A family of nonparametric unit root tests for processes driven by infinite variance innovations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Kemal Caglar Gogebakan
Keyword(s):  
Unit Root ◽  
Unit Root Tests ◽  
Tuning Parameter ◽  
Infinite Variance ◽  
Finite Sample ◽  
Power Performance ◽  
Finite Sample Size ◽  
Heavy Tailed ◽  
The Impact ◽  
Adf Test

Abstract This paper presents extensions to the family of nonparametric fractional variance ratio (FVR) unit root tests of Nielsen (2009. “A Powerful Test of the Autoregressive Unit Root Hypothesis Based on a Tuning Parameter Free Statistic.” Econometric Theory 25: 1515–44) under heavy tailed (infinite variance) innovations. In this regard, we first develop the asymptotic theory for these FVR tests under this setup. We show that the limiting distributions of the tests are free of serial correlation nuisance parameters, but depend on the tail index of the infinite variance process. Then, we compare the finite sample size and power performance of our FVR unit root tests with the well-known parametric ADF test under the impact of the heavy tailed shocks. Simulations demonstrate that under heavy tailed innovations, the nonparametric FVR tests have desirable size and power properties.

scholarly journals On a random-coefficient AR(1) process with heavy-tailed renewal switching coefficient and heavy-tailed noise

2006 ◽  
Vol 43 (02) ◽  
pp. 421-440
Author(s):  
Remigijus Leipus ◽  
Vygantas Paulauskas ◽  
Donatas Surgailis
Keyword(s):  
Unit Root ◽  
Domain Of Attraction ◽  
Stationary Solutions ◽  
Partial Sums ◽  
Infinite Variance ◽  
Finite Variance ◽  
Stable Law ◽  
Stable Lévy Process ◽  
Heavy Tailed ◽  
Independent Identically Distributed

We discuss the limit behavior of the partial sums process of stationary solutions to the (autoregressive) AR(1) equation X t = a t X t−1 + ε t with random (renewal-reward) coefficient, a t , taking independent, identically distributed values A j ∈ [0,1] on consecutive intervals of a stationary renewal process with heavy-tailed interrenewal distribution, and independent, identically distributed innovations, ε t , belonging to the domain of attraction of an α-stable law (0 &lt; α ≤ 2, α ≠ 1). Under suitable conditions on the tail parameter of the interrenewal distribution and the singularity parameter of the distribution of A j near the unit root a = 1, we show that the partial sums process of X t converges to a λ-stable Lévy process with index λ &lt; α. The paper extends the result of Leipus and Surgailis (2003) from the case of finite-variance X t to that of infinite-variance X t .

scholarly journals Time Series Regression With a Unit Root and Infinite-Variance Errors

1990 ◽  
Vol 6 (1) ◽  
pp. 44-62 ◽  
Author(s):  
P.C.B. Phillips
Keyword(s):  
Unit Root ◽  
Stable Process ◽  
Nuisance Parameter ◽  
Domain Of Attraction ◽  
Infinite Variance ◽  
Finite Variance ◽  
Quadratic Functional ◽  
Time Series Regression ◽  
Limit Laws ◽  
Stable Law

In [4] Chan and Tran give the limit theory for the least-squares coefficient in a random walk with i.i.d. (identically and independently distributed) errors that are in the domain of attraction of a stable law. This paper discusses their results and provides generalizations to the case of I(1) processes with weakly dependent errors whose distributions are in the domain of attraction of a stable law. General unit root tests are also studied. It is shown that the semiparametric corrections suggested by the author in other work [22] for the finite-variance case continue to work when the errors have infinite variance. Surprisingly, no modifications to the formulas given in [22] are required. The limit laws are expressed in terms of ratios of quadratic functional of a stable process rather than Brownian motion. The correction terms that eliminate nuisance parameter dependencies are random in the limit and involve multiple stochastic integrals that may be written in terms of the quadratic variation of the limiting stable process. Some extensions of these results to models with drifts and time trends are also indicated.

scholarly journals Asymptotic Theory in Model Diagnostic for General Multivariate Spatial Regression

2016 ◽  
Vol 2016 ◽  
pp. 1-16 ◽  
Author(s):  
Wayan Somayasa ◽  
Gusti N. Adhi Wibawa ◽  
La Hamimu ◽  
La Ode Ngkoimani
Keyword(s):  
Asymptotic Theory ◽  
Spatial Regression ◽  
Test Procedure ◽  
Real Data ◽  
Partial Sums ◽  
Finite Sample ◽  
Partial Sums Processes ◽  
Finite Sample Size ◽  
Kolmogorov Smirnov ◽  
Von Mises

We establish an asymptotic approach for checking the appropriateness of an assumed multivariate spatial regression model by considering the set-indexed partial sums process of the least squares residuals of the vector of observations. In this work, we assume that the components of the observation, whose mean is generated by a certain basis, are correlated. By this reason we need more effort in deriving the results. To get the limit process we apply the multivariate analog of the well-known Prohorov’s theorem. To test the hypothesis we define tests which are given by Kolmogorov-Smirnov (KS) and Cramér-von Mises (CvM) functionals of the partial sums processes. The calibration of the probability distribution of the tests is conducted by proposing bootstrap resampling technique based on the residuals. We studied the finite sample size performance of the KS and CvM tests by simulation. The application of the proposed test procedure to real data is also discussed.

scholarly journals NONPARAMETRIC SIGNIFICANCE TESTING IN MEASUREMENT ERROR MODELS

2021 ◽  
pp. 1-43
Author(s):  
Hao Dong ◽  
Luke Taylor
Keyword(s):  
Measurement Error ◽  
Bootstrap Method ◽  
Monte Carlo Study ◽  
Significance Test ◽  
Test Statistics ◽  
Finite Sample ◽  
Null Distributions ◽  
Kolmogorov Smirnov ◽  
Von Mises ◽  
Noisy Measurements

We develop the first nonparametric significance test for regression models with classical measurement error in the regressors. In particular, a Cramér-von Mises test and a Kolmogorov–Smirnov test for the null hypothesis $E\left [Y|X^{*},Z^{*}\right ]=E\left [Y|X^{*}\right ]$ are proposed when only noisy measurements of $X^{*}$ and $Z^{*}$ are available. The asymptotic null distributions of the test statistics are derived, and a bootstrap method is implemented to obtain the critical values. Despite the test statistics being constructed using deconvolution estimators, we show that the test can detect a sequence of local alternatives converging to the null at the $\sqrt {n}$ -rate. We also highlight the finite sample performance of the test through a Monte Carlo study.

Calculation of noncrossing probabilities for Poisson processes and its corollaries

2001 ◽  
Vol 33 (3) ◽  
pp. 702-716 ◽  
Author(s):  
Estate Khmaladze ◽  
Eka Shinjikashvili
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
High Intensity ◽  
Empirical Process ◽  
Brownian Bridge ◽  
Poisson Processes ◽  
Square Root ◽  
Finite Sample ◽  
Kolmogorov Smirnov ◽  
Very High

The paper describes a new numerical method for the calculation of noncrossing probabilities for arbitrary boundaries by a Poisson process. We find the method to be simple in implementation, quick and efficient - it works reliably for Poisson processes of very high intensity n, up to several thousand. Hence, it can be used to detect unusual features in the finite-sample behaviour of empirical process and trace it down to very high sample sizes. It also can be used as a good approximation for noncrossing probabilities for Brownian motion and Brownian bridge, in particular when the boundaries are not regular. As a numerical example we demonstrate the divergence of normalized Kolmogorov-Smirnov statistics from their prescribed limiting distributions (Eicker (1979), Jaeshke (1979)) for quite large n in contrast to very regular behaviour of statistics of Mason (1983). For the Brownian motion case we considered square-root, Daniels' (1969) and Grooneboom's (1989) boundaries.

scholarly journals On a random-coefficient AR(1) process with heavy-tailed renewal switching coefficient and heavy-tailed noise

2006 ◽  
Vol 43 (2) ◽  
pp. 421-440
Author(s):  
Remigijus Leipus ◽  
Vygantas Paulauskas ◽  
Donatas Surgailis
Keyword(s):  
Unit Root ◽  
Domain Of Attraction ◽  
Stationary Solutions ◽  
Partial Sums ◽  
Infinite Variance ◽  
Finite Variance ◽  
Stable Law ◽  
Stable Lévy Process ◽  
Heavy Tailed ◽  
Independent Identically Distributed

We discuss the limit behavior of the partial sums process of stationary solutions to the (autoregressive) AR(1) equation Xt = atXt−1 + εt with random (renewal-reward) coefficient, at, taking independent, identically distributed values Aj ∈ [0,1] on consecutive intervals of a stationary renewal process with heavy-tailed interrenewal distribution, and independent, identically distributed innovations, εt, belonging to the domain of attraction of an α-stable law (0 < α ≤ 2, α ≠ 1). Under suitable conditions on the tail parameter of the interrenewal distribution and the singularity parameter of the distribution of Aj near the unit root a = 1, we show that the partial sums process of Xt converges to a λ-stable Lévy process with index λ < α. The paper extends the result of Leipus and Surgailis (2003) from the case of finite-variance Xt to that of infinite-variance Xt.

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