Infinite Variance Errors Research Articles

Asymptotics of M-estimators for moderate deviations from a unit root model with possibly infinite variance

In this article, the M-estimator of the moderate deviations from a unit root model with possibly infinite variance errors is proposed. When the autoregressive coefficients are located on both sides of stationary and explosion, the asymptotic normal properties of the corresponding estimators for unknown parameters are proved, respectively. The asymptotic properties of the M-estimators are obtained in both the mildly integrated case and the mildly explosive case.

On partial-sum processes of ARMAX residuals

We establish general and versatile results regarding the limit behavior of the partial-sum process of ARMAX residuals. Illustrations include ARMA with seasonal dummies, misspecified ARMAX models with autocorrelated errors, nonlinear ARMAX models, ARMA with a structural break, a wide range of ARMAX models with infinite-variance errors, weak GARCH models and the consistency of kernel estimation of the density of ARMAX errors. Our results identify the limit distributions, and provide a general algorithm to obtain pivot statistics for CUSUM tests.

Quantile inference for moderate deviations from a unit root model with infinite variance

The limit distribution of the quantile estimate for the moderate deviations from a unit root model with possibly infinite variance errors is derived. Both the mildly integrated case and the mildly explosive case are investigated. Quicker convergence rates than the ordinary least squares estimation are gained. Numerical simulations are conducted to illustrate the performance of our estimation procedure.

On consistency of the least squares estimators in linear errors-in-variables models with infinite variance errors

This paper deals simultaneously with linear structural and functional errors-in-variables models (SEIVM and FEIVM), revisiting in this context the ordinary least squares estimators (LSE) for the slope and intercept of the corresponding simple linear regression. It has been known that, subject to some model conditions, these estimators become weakly and strongly consistent in the linear SEIVM and FEIVM with the measurement errors having finite variances when the explanatory variables have an infinite variance in the SEIVM, and a similar infinite spread in the FEIVM, while otherwise, the LSE’s require an adjustment for consistency with the so-called reliability ratio. In this paper, weak and strong consistency, with and without the possible rates of convergence being determined, is proved for the LSE’s of the slope and intecept, assuming that the measurement errors are in the domain of attraction of the normal law (DAN) and thus are, for the first time, allowed to have infinite variances. Moreover, these results are obtained under the conditions that the explanatory variables are in DAN, have an infinite variance, and dominate the measurement errors in terms of variation in the SEIVM, and under appropriately matching versions of these conditions in the FEIVM. This duality extends a previously known interplay between SEIVM’s and FEIVM’s.

Weighted quantile regression for AR model with infinite variance errors

Autoregressive (AR) models with finite variance errors have been well studied. This paper is concerned with AR models with heavy-tailed errors, which is useful in various scientific research areas. Statistical estimation for AR models with infinite variance errors is very different from those for AR models with finite variance errors. In this paper, we consider a weighted quantile regression for AR models to deal with infinite variance errors. We further propose an induced smoothing method to deal with computational challenges in weighted quantile regression. We show that the difference between weighted quantile regression estimate and its smoothed version is negligible. We further propose a test for linear hypothesis on the regression coefficients. We conduct Monte Carlo simulation study to assess the finite sample performance of the proposed procedures. We illustrate the proposed methodology by an empirical analysis of a real-life data set.

Empirical processes for infinite variance autoregressive models

The paper proposes new procedures for diagnostic checking of fitted models under the assumption of infinite-variance errors which are in the domain of attraction of a stable law. These procedures are functional of residual-based empirical processes. First, the asymptotic distributions of the empirical processes based on residuals are derived. Then two important applications in time series diagnostics are discussed. A goodness-of-fit test is developed using a functional of the empirical process based on residuals. Tests of independence of innovations are also considered. The finite-sample behavior of these tests are studied by simulation and comparison with the classical Portmanteau tests for ARMA models with infinite-variance developed recently by Lin and McLeod (2008) [25] is provided.

Empirical Processes for Infinite Variance Autoregressive Models

The paper proposes new procedures for diagnostic checking of fitted models under the assumption of infinite-variance errors which are in the domain of attraction of a stable law. These procedures are functional of residual-based empirical processes. First, the asymptotic distributions of the empirical processes based on residuals are derived. Then two important applications in time series diagnostics are discussed. A Goodness-of-fit test is developed using a functional of the empirical process based on residuals. Tests of independence of innovations are also considered. The finite-sample behavior of these tests are studied by simulation and comparison with the classical Portmanteau tests for ARMA models with infinite-variance developed recently by Lin and McLeod (2008) is provided.

Robust M-Estimation for Heavy Tailed Nonlinear AR-GARCH

We develop new tail-trimmed M-estimation methods for heavy tailed Nonlinear AR-GARCH models. Tail-trimming allows both identification of the true parameter and asymptotic normality for nonlinear models with asymmetric errors. In heavy tailed cases the rate of convergence is infinitesimally close to the highest possible amongst M-estimators for a particular loss function, hence super- root(n)-convergence can be achieved in nonlinear AR models with infinite variance errors, and arbitrarily near root(n)-convergence for GARCH with errors that have an infinite fourth moment. We present a consistent estimator of the covariance matrix that permits classic inference without knowledge of the rate of convergence, and explore asymptotic covariance and bootstrap mean-squared-error methods for selecting trimming parameters. A simulation study shows the estimator trumps existing ones for AR and GARCH models based on sharpness, approximate normality, rate of convergence, and test accuracy. We then use the estimator to study asset returns data.

COMMENT ON “WEAK CONVERGENCE TO A MATRIX STOCHASTIC INTEGRAL WITH STABLE PROCESSES”

In this comment we identify a lacuna in a proof in the paper by M. Caner published in 1997 in Econometric Theory concerning the weak limit behavior of various expressions involving heavy-tailed multivariate vectors and the convergence of stochastic integrals. In a later paper (Caner, 1998) the results for these limit relations are used to formulate tests for cointegration with infinite variance errors.

Asymptotic inference in time series regressions with a unit root and infinite variance errors

In this paper, we study the limiting distribution of the OLS estimators and t-statistics of the null hypothesis of a unit root in various regression models when the true generating mechanism is either a driftless random walk or a random walk with drift and the distribution of the error term belongs to the normal domain of attraction of a stable law with characteristic exponent less than two. We show that in the former data generating processes (DGP) the same functional form as in the finite variance case applies with the Lévy process replacing the standard Wiener process. On the other hand, under the random walk with drift DGP, we show that different limiting distributions are obtained according to the magnitude of the maximal moment exponent α. Finally, we investigate the consequences of a “local” departure from the finite variance setup and provide simulation evidence on the robustness of the limiting distributions (derived under finite variance) to heavy tails in finite samples.

Rank tests of unit root hypothesis with infinite variance errors

We consider a family of rank tests based on the regression rank score process introduced by Gutenbrunner and Jurečková (Ann. Statist. 20 (1992) 305) to test the unit root hypothesis under infinite variance innovations. Unlike the finite variance case as studied by Hasan and Koenker (Econometrica 65 (1997) 133) the original rankscore test statistics ( T n ) exhibit a simple Gaussian limiting behavior. However, finite sample investigations suggest a correction similar to what HK proposed. This corrected version ( S ̂ T ) has reliable size, and exhibits remarkable power even in near unit root cases under a variety of α-stable distributions. Also, the test statistics do not depend on the α parameter.

New tests for unit roots in autoregressive processes with possibly infinite variance errors

For autoregressive processes with possibly infinite variance innovations, tests for unit roots are constructed. The limiting null distributions of the test statistics are standard normal both for finite variance innovations and for infinite variance innovations. The test statistics are the pivotal statistics of modified M-estimators in which the signs of regressors rather than the regressors themselves are used as instrumental variables in estimating unit roots. A Monte-Carlo experiment compares the proposed tests favorably with tests based on the OLSE and tests based on the M-estimators for several innovations.

Tests for cointegration with infinite variance errors

This paper develops the asymptotic theory for residual-based tests and quasi-likelihood ratio tests for cointegration under the assumption of infinite variance errors. This article extends the results of Phillips and Ouliaris (1990)and Johansen (1988, 1991)which are derived under the assumption of square-integrable errors. Here the limit laws are expressed in terms of functionals of symmetric stable laws rather than Brownian motion. Critical values of the residual-based tests of Phillips and Ouliaris (1990)and likelihood-ratio-based tests of Johansen (1991)are calculated and tabulated. We also investigate whether these tests are robust to infinite variance errors. We found that regardless of the index of stability α, the residual-based tests are more robust to infinite variance errors than the likelihood-ratio-based tests.

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

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.

A Shortcut to LAD Estimator Asymptotics

Using generalized functions of random variables and generalized Taylor series expansions, we provide quick demonstrations of the asymptotic theory for the LAD estimator in a regression model setting. The approach is justified by the smoothing that is delivered in the limit by the asymptotics, whereby the generalized functions are forced to appear as linear functionals wherein they become real valued. Models with fixed and random regressors, and autoregressions with infinite variance errors are studied. Some new analytic results are obtained including an asymptotic expansion of the distribution of the LAD estimator.

The Durbin-Watson ratio under infinite-variance errors

Abstract This paper studies the properties of the von Neumann ratio for time series with infinite variance. Our asymptotics cover the null of iid variates and general moving-average (MA) alternatives. Regression residuals are also considered. In the static regression model the Durbin-Watson statistic has the same limit distribution as the von Neumann ratio. In dynamic models the results are more complex. In finite-variance models our results specialize to those of the Durbin h-statistic and equivalent LM test asymptotics. Some Monte Carlo results are reported, illustrating the effects of infinite-variance errors and regressors in finite samples.

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

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.