ROBUST INFERENCE IN AUTOREGRESSIONS WITH MULTIPLE OUTLIERS

2009 ◽  
Vol 25 (6) ◽  
pp. 1625-1661 ◽  
Author(s):  
Giuseppe Cavaliere ◽  
Iliyan Georgiev
Keyword(s):  
Unit Root ◽  
Ordinary Least Squares ◽  
Test Statistics ◽  
Robust Methods ◽  
Finite Sample ◽  
Dummy Variables ◽  
Asymptotic Size ◽  
Finite Sample Size ◽  
Quasi Maximum Likelihood ◽  
Consistent Parameter

We consider robust methods for estimation and unit root (UR) testing in autoregressions with infrequent outliers whose number, size, and location can be random and unknown. We show that in this setting standard inference based on ordinary least squares estimation of an augumented Dickey–Fuller (ADF) regression may not be reliable, because (a) clusters of outliers may lead to inconsistent estimation of the autoregressive parameters and (b) large outliers induce a jump component in the asymptotic distribution of UR test statistics. In the benchmark case of known outlier location, we discuss why the augmentation of the ADF regression with appropriate dummy variables not only ensures consistent parameter estimation but also gives rise to UR tests with significant power gains, growing with the number and the size of the outliers. In the case of unknown outlier location, the dummy-based approach is compared with a robust, mixed Gaussian, quasi maximum likelihood (QML) approach, novel in this context. It is proved that, when the ordinary innovations are Gaussian, the QML and the dummy-based approach are asymptotically equivalent, yielding UR tests with the same asymptotic size and power. Moreover, as a by-product of QML the outlier dates can be consistently estimated. When the innovations display tails fatter than Gaussian, the QML approach ensures further power gains over the dummy-based method. Simulations show that the QML ADF-typet-test, in conjunction with standard Dickey–Fuller critical values, yields the best combination of finite-sample size and power.

Near Observational Equivalence and Theoretical size Problems with Unit Root Tests

1996 ◽  
Vol 12 (4) ◽  
pp. 724-731 ◽  
Author(s):  
Jon Faust
Keyword(s):  
Unit Root ◽  
Sufficient Conditions ◽  
Unit Root Test ◽  
Asymptotic Distributions ◽  
Unit Root Tests ◽  
Finite Sample ◽  
Test Procedures ◽  
Asymptotic Size ◽  
Finite Sample Size ◽  
Topological Sense

Said and Dickey (1984,Biometrika71, 599–608) and Phillips and Perron (1988,Biometrika75, 335–346) have derived unit root tests that have asymptotic distributions free of nuisance parameters under very general maintained models. Under models as general as those assumed by these authors, the size of the unit root test procedures will converge to one, not the size under the asymptotic distribution. Solving this problem requires restricting attention to a model that is small, in a topological sense, relative to the original. Sufficient conditions for solving the asymptotic size problem yield some suggestions for improving finite-sample size performance of standard tests.

THE NONSTATIONARY FRACTIONAL UNIT ROOT

1999 ◽  
Vol 15 (4) ◽  
pp. 549-582 ◽  
Author(s):  
Katsuto Tanaka
Keyword(s):  
Unit Root ◽  
Asymptotic Properties ◽  
Test Statistics ◽  
Finite Sample ◽  
Asymptotic Results ◽  
Simulation Experiments ◽  
Normality Assumption ◽  
Asymptotically Efficient ◽  
Uniformly Most Powerful ◽  
Fractional Unit

This paper deals with a scalar I(d) process {yj}, where the integration order d is any real number. Under this setting, we first explore asymptotic properties of various statistics associated with {yj}, assuming that d is known and is greater than or equal to ½. Note that {yj} becomes stationary when d < ½, whose case is not our concern here. It turns out that the case of d = ½ needs a separate treatment from d > ½. We then consider, under the normality assumption, testing and estimation for d, allowing for any value of d. The tests suggested here are asymptotically uniformly most powerful invariant, whereas the maximum likelihood estimator is asymptotically efficient. The asymptotic theory for these results will not assume normality. Unlike in the usual unit root problem based on autoregressive models, standard asymptotic results hold for test statistics and estimators, where d need not be restricted to d ≥ ½. Simulation experiments are conducted to examine the finite sample performance of both the tests and estimators.

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.

ASYMPTOTIC DISTRIBUTIONS FOR UNIT ROOT TEST STATISTICS IN NEARLY INTEGRATED SEASONAL AUTOREGRESSIVE MODELS

2000 ◽  
Vol 16 (2) ◽  
pp. 200-230 ◽  
Author(s):  
Seiji Nabeya
Keyword(s):  
Least Squares ◽  
Unit Root ◽  
Linear Trend ◽  
Unit Root Test ◽  
Ordinary Least Squares ◽  
Asymptotic Distributions ◽  
Autoregressive Models ◽  
Test Statistics ◽  
Autoregressive Parameter ◽  
Ordinary Least Squares Estimator

Seasonal autoregressive models with an intercept or linear trend are discussed. The main focus of this paper is on the models in which the intercept or trend parameters do not depend on the season. One of the most important results from this study is the asymptotic distribution for the ordinary least squares estimator of the autoregressive parameter obtained under nearly integrated condition, and another is the approximation to the limiting distribution of the t-statistic under the null for testing the unit root hypothesis.

scholarly journals UNIT ROOT TESTING IN PRACTICE: DEALING WITH UNCERTAINTY OVER THE TREND AND INITIAL CONDITION

2009 ◽  
Vol 25 (3) ◽  
pp. 587-636 ◽  
Author(s):  
David I. Harvey ◽  
Stephen J. Leybourne ◽  
A.M. Robert Taylor
Keyword(s):  
Decision Rule ◽  
Unit Root ◽  
Ordinary Least Squares ◽  
Unit Root Tests ◽  
Trend Detection ◽  
Initial Condition ◽  
Finite Sample ◽  
Testing Procedures ◽  
Unit Root Testing ◽  
Deterministic Trend

In this paper we focus on two major issues that surround testing for a unit root in practice, namely, (i) uncertainty as to whether or not a linear deterministic trend is present in the data and (ii) uncertainty as to whether the initial condition of the process is (asymptotically) negligible or not. In each case simple testing procedures are proposed with the aim of maintaining good power properties across such uncertainties. For the first issue, if the initial condition is negligible, quasi-differenced (QD) detrended (demeaned) Dickey–Fuller-type unit root tests are near asymptotically efficient when a deterministic trend is (is not) present in the data generating process. Consequently, we compare a variety of strategies that aim to select the detrended variant when a trend is present, and the demeaned variant otherwise. Based on asymptotic and finite-sample evidence, we recommend a simple union of rejections-based decision rule whereby the unit root null hypothesis is rejected whenever either of the detrended or demeaned unit root tests yields a rejection. Our results show that this approach generally outperforms more sophisticated strategies based on auxiliary methods of trend detection. For the second issue, we again recommend a union of rejections decision rule, rejecting the unit root null if either of the QD or ordinary least squares (OLS) detrended/demeaned Dickey–Fuller-type tests rejects. This procedure is also shown to perform well in practice, simultaneously exploiting the superior power of the QD (OLS) detrended/demeaned test for small (large) initial conditions.

scholarly journals SEMI-PARAMETRIC SEASONAL UNIT ROOT TESTS

2017 ◽  
Vol 34 (2) ◽  
pp. 447-476 ◽  
Author(s):  
Tomás del Barrio Castro ◽  
Paulo M.M. Rodrigues ◽  
A.M. Robert Taylor
Keyword(s):  
Unit Root ◽  
Moving Average ◽  
Size Control ◽  
Unit Root Tests ◽  
Finite Sample ◽  
Oxford University ◽  
Finite Sample Size ◽  
Carry Over ◽  
Image Position ◽  
Economic Studies

We extend the ${\cal M}$ class of unit root tests introduced by Stock (1999, Cointegration, Causality and Forecasting. A Festschrift in Honour of Clive W.J. Granger. Oxford University Press), Perron and Ng (1996, Review of Economic Studies 63, 435–463) and Ng and Perron (2001, Econometrica 69, 1519–1554) to the seasonal case, thereby developing semi-parametric alternatives to the regression-based augmented seasonal unit root tests of Hylleberg, Engle, Granger, and Yoo (1990, Journal of Econometrics 44, 215–238). The success of this class of unit root tests to deliver good finite sample size control even in the most problematic (near-cancellation) case where the shocks contain a strong negative moving average component is shown to carry over to the seasonal case as is the superior size/power trade-off offered by these tests relative to other available tests.

Testing for Cointegration in a System of Equations

1995 ◽  
Vol 11 (5) ◽  
pp. 952-983 ◽  
Author(s):  
In Choi ◽  
Byung Chul Ahn
Keyword(s):  
Single Equation ◽  
Ordinary Least Squares ◽  
Asymptotic Distributions ◽  
Small Scale ◽  
Multiple Time ◽  
System Of Equations ◽  
Finite Sample ◽  
Multivariate Tests ◽  
Working Paper ◽  
Finite Sample Size

This paper introduces various consistent tests for the null of cointegration against the alternative of noncointegration that can be applied to a system of equations as well as to a single equation. The tests are analogs of Choi and Ahn's (1993, Testing the Null of Stationarity for Multiple Time Series, working paper, The Ohio State University) multivariate tests for the null of stationarity and use Park's (1992, Econometrica 60, 119–143) canonical cointegrating regression (CCR) residuals to make the tests free of nuisance parameters in the limit. The asymptotic distributions of the tests are complex but expressed in unified manner by using standard vector Brownian motion. These distributions are tabulated by simulation for some practical cases. Furthermore, the rates of divergence of the tests are reported. Because there are methods for estimating cointegrating matrices other than CCR, it is illustrated for a model without time trends that the tests we introduce work exactly the same way in the limit when Phillips and Hansen's (1990, Review of Economic Studies 57, 99–125) fully modified ordinary least-squares (OLS) procedure is used. Also, is shown that difficulties arise when OLS residuals are used to formulate the tests. Small-scale simulation results are reported to examine the finite sample performance of the tests. The tests are shown to work reasonably wellin finite samples. In particular, it is illustrated that using the multivariate tests introduced in this paper can be a better testing strategy in terms of the finite sample size and power than applying univariate tests several times to each equation in a system of equations.

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