Asymptotic behaviour of M‐estimators in AR(p) models under nonstandard conditions

The authors derive the limiting distribution of M‐estimators in AR(p) models under nonstandard conditions, allowing for discontinuities in score and density functions. Unlike usual regularity assumptions, these conditions are satisfied in the context of L₁‐estimation and autoregression quantiles. The asymptotic distributions of the resulting estimators, however, are not generally Gaussian. Moreover, their bootstrap approximations are consistent along very specific sequences of bootstrap sample sizes only.     

Bootstrapping Noncausal Autoregressions: With Applications to Explosive Bubble Modeling

In this article, we develop new bootstrap-based inference for noncausal autoregressions with heavy-tailed innovations. This class of models is widely used for modeling bubbles and explosive dynamics in economic and financial time series. In the noncausal, heavy-tail framework, a major drawback of asymptotic inference is that it is not feasible in practice as the relevant limiting distributions depend crucially on the (unknown) decay rate of the tails of the distribution of the innovations. In addition, even in the unrealistic case where the tail behavior is known, asymptotic inference may suffer from small-sample issues. To overcome these difficulties, we propose bootstrap inference procedures using parameter estimates obtained with the null hypothesis imposed (the so-called restricted bootstrap). We discuss three different choices of bootstrap innovations: wild bootstrap, based on Rademacher errors; permutation bootstrap; a combination of the two (“permutation wild bootstrap”). Crucially, implementation of these bootstraps do not require any a priori knowledge about the distribution of the innovations, such as the tail index or the convergence rates of the estimators. We establish sufficient conditions ensuring that, under the null hypothesis, the bootstrap statistics estimate consistently particular conditionaldistributions of the original statistics. In particular, we show that validity of the permutation bootstrap holds without any restrictions on the distribution of the innovations, while the permutation wild and the standard wild bootstraps require further assumptions such as symmetry of the innovation distribution. Extensive Monte Carlo simulations show that the finite sample performance of the proposed bootstrap tests is exceptionally good, both in terms of size and of empirical rejection probabilities under the alternative hypothesis. We conclude by applying the proposed bootstrap inference to Bitcoin/USD exchange rates and to crude oil price data. We find that indeed noncausal models with heavy-tailed innovations are able to fit the data, also in periods of bubble dynamics. Supplementary materials for this article are available online.     

The limiting distribution of the least-squares estimator in nearly integrated seasonal models

We consider the least-squares estimator of the autoregressive parameter in a nearly integrated seasonal model. Building on the study by Chan (1989), who obtained the limiting distribution, we derive a closed-form expression for the appropriate limiting joint moment generating function. We use this function to tabulate percentage points of the asymptotic distribution for various seasonal periods via numerical integration. The results are extended by deriving a stochastic asymptotic expansion to order O_p(T^-l), whose percentage points are also obtained by numerically integrating the appropriate limiting joint moment generating function. The adequacy of the approximation to the finite-sample distribution is discussed.     

Asymptotic expansion for nonparametric M-estimator in a nonlinear regression model with long-memory errors

We consider asymptotic expansion of the nonparametric M-estimator in a fixed-design nonlinear regression model when the errors are generated by long-memory linear processes. Under mild conditions, we show that the nonparametric M-estimator is first-order equivalent to the Nadaraya-Watson (NW) estimator, which implies that the nonparametric M-estimator has the same asymptotic distribution as that of the NW estimator. Furthermore, we study the second-order asymptotic expansion of the nonparametric M-estimator and show that the difference between the nonparametric M-estimator and the NW estimator has a limiting distribution after suitable standardization. The nature of the limiting distribution depends on the range of long-memory parameter α. We also compare the finite sample behavior of the two estimators through a numerical example when the errors are long-memory.     

Testing for Structural Instability in Moment Restriction Models: An Info-Metric Approach

In this paper, we develop an info-metric framework for testing hypotheses about structural instability in nonlinear, dynamic models estimated from the information in population moment conditions. Our methods are designed to distinguish between three states of the world: (i) the model is structurally stable in the sense that the population moment condition holds at the same parameter value throughout the sample; (ii) the model parameters change at some point in the sample but otherwise the model is correctly specified; and (iii) the model exhibits more general forms of instability than a single shift in the parameters. An advantage of the info-metric approach is that the null hypotheses concerned are formulated in terms of distances between various choices of probability measures constrained to satisfy (i) and (ii), and the empirical measure of the sample. Under the alternative hypotheses considered, the model is assumed to exhibit structural instability at a single point in the sample, referred to as the break point; our analysis allows for the break point to be either fixed a priori or treated as occuring at some unknown point within a certain fraction of the sample. We propose various test statistics that can be thought of as sample analogs of the distances described above, and derive their limiting distributions under the appropriate null hypothesis. The limiting distributions of our statistics are nonstandard but coincide with various distributions that arise in the literature on structural instability testing within the Generalized Method of Moments framework. A small simulation study illustrates the finite sample performance of our test statistics.     

Asymptotics for L1-estimators of regression parameters under heteroscedasticityY

We consider the asymptotic behaviour of L₁ -estimators in a linear regression under a very general form of heteroscedasticity. The limiting distributions of the estimators are derived under standard conditions on the design. We also consider the asymptotic behaviour of the bootstrap in the heteroscedastic model and show that it is consistent to first order only if the limiting distribution is normal.     

UNIT ROOT TESTING IN THE PRESENCE OF HEAVY‐TAILED GARCH ERRORS

We derive the asymptotic distributions of the Dickey–Fuller (DF) and augmented DF (ADF) tests for unit root processes with Generalized Autoregressive Conditional Heteroscedastic (GARCH) errors under fairly mild conditions. We show that the asymptotic distributions of the DF tests and ADF t‐type test are the same as those obtained in the independent and identically distributed Gaussian cases, regardless of whether the fourth moment of the underlying GARCH process is finite or not. Our results go beyond earlier ones by showing that the fourth moment condition on the scaled conditional errors is totally unnecessary. Some Monte Carlo simulations are provided to illustrate the finite‐sample‐size properties of the tests.     

A simple condition for the multivariate CLT and the attraction to the Gaussian of Lévy processes at long and short times

We show that a necessary and sufficient condition for the sum of iid random vectors to converge (under appropriate centering and scaling) to a multivariate Gaussian distribution is that the truncated second moment matrix is slowly varying at infinity. This is more natural than the standard conditions, and allows for the possibility that the limiting Gaussian distribution is degenerate (so long as it is not concentrated at a point). We also give necessary and sufficient conditions for a d-dimensional Lévy process to converge (under appropriate centering and scaling) to a multivariate Gaussian distribution as time approaches zero or infinity.     

Consistent Nonparametric Tests for Lorenz Dominance

This article proposes consistent nonparametric methods for testing the null hypothesis of Lorenz dominance. The methods are based on a class of statistical functionals defined over the difference between the Lorenz curves for two samples of welfare-related variables. We present two specific test statistics belonging to the general class and derive their asymptotic properties. As the limiting distributions of the test statistics are nonstandard, we propose and justify bootstrap methods of inference. We provide methods appropriate for case where the two samples are independent as well as the case where the two samples represent different measures of welfare for one set of individuals. The small sample performance of the two tests is examined and compared in the context of a Monte Carlo study and an empirical analysis of income and consumption inequality.     

Inference of Seasonal Long‐memory Time Series with Measurement Error

We consider the Whittle likelihood estimation of seasonal autoregressive fractionally integrated moving‐average models in the presence of an additional measurement error and show that the spectral maximum Whittle likelihood estimator is asymptotically normal. We illustrate by simulation that ignoring measurement errors may result in incorrect inference. Hence, it is pertinent to test for the presence of measurement errors, which we do by developing a likelihood ratio (LR) test within the framework of Whittle likelihood. We derive the non‐standard asymptotic null distribution of this LR test and the limiting distribution of LR test under a sequence of local alternatives. Because in practice, we do not know the order of the seasonal autoregressive fractionally integrated moving‐average model, we consider three modifications of the LR test that takes model uncertainty into account. We study the finite sample properties of the size and the power of the LR test and its modifications. The efficacy of the proposed approach is illustrated by a real‐life example.     

Inference in the presence of redundant moment conditions and the impact of government health expenditure on health outcomes in England

In his 1999 article with Breusch, Qian, and Wyhowski in the Journal of Econometrics, Peter Schmidt introduced the concept of “redundant” moment conditions. Such conditions arise when estimation is based on moment conditions that are valid and can be divided into two subsets: one that identifies the parameters and another that provides no further information. Their framework highlights an important concept in the moment-based estimation literature, namely, that not all valid moment conditions need be informative about the parameters of interest. In this article, we demonstrate the empirical relevance of the concept in the context of the impact of government health expenditure on health outcomes in England. Using a simulation study calibrated to this data, we perform a comparative study of the finite performance of inference procedures based on the Generalized Method of Moment (GMM) and info-metric (IM) estimators. The results indicate that the properties of GMM procedures deteriorate as the number of redundant moment conditions increases; in contrast, the IM methods provide reliable point estimators, but the performance of associated inference techniques based on first order asymptotic theory, such as confidence intervals and overidentifying restriction tests, deteriorates as the number of redundant moment conditions increases. However, for IM methods, it is shown that bootstrap procedures can provide reliable inferences; we illustrate such methods when analysing the impact of government health expenditure on health outcomes in England.     

Weak Instrumental Variables Models for Longitudinal Data

This article considers the estimation and testing of a within-group two-stage least squares (TSLS) estimator for instruments with varying degrees of weakness in a longitudinal (panel) data model. We show that adding the repeated cross-sectional information into a regression model can improve the estimation in weak instruments. Moreover, the consistency and limiting distribution of the TSLS estimator are established when both N and T tend to infinity. Some asymptotically pivotal tests are extended to a longitudinal data model and their asymptotic properties are examined. A Monte Carlo experiment is conducted to evaluate the finite sample performance of the proposed estimators.     

Robust Inference for Near-Unit Root Processes with Time-Varying Error Variances

The autoregressive Cauchy estimator uses the sign of the first lag as instrumental variable (IV); under independent and identically distributed (i.i.d.) errors, the resulting IV t-type statistic is known to have a standard normal limiting distribution in the unit root case. With unconditional heteroskedasticity, the ordinary least squares (OLS) t statistic is affected in the unit root case; but the paper shows that, by using some nonlinear transformation behaving asymptotically like the sign as instrument, limiting normality of the IV t-type statistic is maintained when the series to be tested has no deterministic trends. Neither estimation of the so-called variance profile nor bootstrap procedures are required to this end. The Cauchy unit root test has power in the same 1/T neighborhoods as the usual unit root tests, also for a wide range of magnitudes for the initial value. It is furthermore shown to be competitive with other, bootstrap-based, robust tests. When the series exhibit a linear trend, however, the null distribution of the Cauchy test for a unit root becomes nonstandard, reminiscent of the Dickey-Fuller distribution. In this case, inference robust to nonstationary volatility is obtained via the wild bootstrap.     

Behaviour of skewness, kurtosis and normality tests in long memory data

We establish the limiting distributions for empirical estimators of the coefficient of skewness, kurtosis, and the Jarque–Bera normality test statistic for long memory linear processes. We show that these estimators, contrary to the case of short memory, are neither ${\sqrt{n}}We establish the limiting distributions for empirical estimators of the coefficient of skewness, kurtosis, and the Jarque–Bera normality test statistic for long memory linear processes. We show that these estimators, contrary to the case of short memory, are neither ?n{\sqrt{n}}-consistent nor asymptotically normal. The normalizations needed to obtain the limiting distributions depend on the long memory parameter d. A direct consequence is that if data are long memory then testing normality with the Jarque–Bera test by using the chi-squared critical values is not valid. Therefore, statistical inference based on skewness, kurtosis, and the Jarque–Bera normality test, needs a rescaling of the corresponding statistics and computing new critical values of their nonstandard limiting distributions.     

Moment Estimation of the Parameters in Rayleigh Distribution with Two Parameters

In this article, the moment estimation of the parameters in two-parameter Rayleigh distribution is studied. For given sample values, the necessary and sufficient conditions on the estimating equations which have unique solution are obtained, and it is also proved that these conditions are satisfied on asymptotic probability 1. Furthermore, the strong consistency and asymptotic normality are obtained. Finally, some simulation experiments are made to show the conclusions.     

Oracle GMM estimation for misspecified models via thresholding

We propose a thresholding generalized method of moments (GMM) estimator for misspecified time series moment condition models. This estimator has the following oracle property: its asymptotic behavior is the same as of any efficient GMM estimator obtained under the a priori information that the true model were known. We propose data adaptive selection methods for thresholding parameter using multiple testing procedures. We determine the limiting null distributions of classical parameter tests and show the consistency of the corresponding block-bootstrap tests used in conjunction with thresholding GMM inference. We present the results of a simulation study for a misspecified instrumental variable regression model and for a vector autoregressive model with measurement error. We illustrate an application of the proposed methodology to data analysis of a real-world dataset.     

A modified likelihood ratio test for homogeneity in finite mixture models

Testing for homogeneity in finite mixture models has been investigated by many researchers. The asymptotic null distribution of the likelihood ratio test (LRT) is very complex and difficult to use in practice. We propose a modified LRT for homogeneity in finite mixture models with a general parametric kernel distribution family. The modified LRT has a χ-type of null limiting distribution and is asymptotically most powerful under local alternatives. Simulations show that it performs better than competing tests. They also reveal that the limiting distribution with some adjustment can satisfactorily approximate the quantiles of the test statistic, even for moderate sample sizes.     

New modified moment estimators for the two-parameter weighted Lindley distribution

We propose a modification of the moment estimators for the two-parameter weighted Lindley distribution. The modification replaces the second sample moment (or equivalently the sample variance) by a certain sample average which is bounded on the unit interval for all values in the sample space. In this method, the estimates always exist uniquely over the entire parameter space and have consistency and asymptotic normality over the entire parameter space. The bias and mean squared error of the estimators are also examined by means of a Monte Carlo simulation study, and the empirical results show the small-sample superiority in addition to the desirable large sample properties. Monte Carlo simulation study showed that the proposed modified moment estimators have smaller biases and smaller mean-square errors than the existing moment estimators and are compared favourably with the maximum likelihood estimators in terms of bias and mean-square error. Three illustrative examples are finally presented.     

A limit theorem on moment convergence of centered spectral statistics of random matrices

ABSTRACT

The eigenvalues of a random matrix are a sequence of specific dependent random variables, the limiting properties of which are one of interesting topics in probability theory. The aim of the article is to extend some probability limiting properties of i.i.d. random variables in the context of the complete moment convergence to the centered spectral statistics of random matrices. Some precise asymptotic results related to the complete convergence of p-order conditional moment of Wigner matrices and sample covariance matrices are obtained. The proofs mainly depend on the central limit theorem and large deviation inequalities of spectral statistics.     

A new class of models for bivariate joint tails

Summary.  A fundamental issue in applied multivariate extreme value analysis is modelling dependence within joint tail regions. The primary focus of this work is to extend the classical pseudopolar treatment of multivariate extremes to develop an asymptotically motivated representation of extremal dependence that also encompasses asymptotic independence. Starting with the usual mild bivariate regular variation assumptions that underpin the coefficient of tail dependence as a measure of extremal dependence, our main result is a characterization of the limiting structure of the joint survivor function in terms of an essentially arbitrary non-negative measure that must satisfy some mild constraints. We then construct parametric models from this new class and study in detail one example that accommodates asymptotic dependence, asymptotic independence and asymmetry within a straightforward parsimonious parameterization. We provide a fast simulation algorithm for this example and detail likelihood-based inference including tests for asymptotic dependence and symmetry which are useful for submodel selection. We illustrate this model by application to both simulated and real data. In contrast with the classical multivariate extreme value approach, which concentrates on the limiting distribution of normalized componentwise maxima, our framework focuses directly on the structure of the limiting joint survivor function and provides significant extensions of both the theoretical and the practical tools that are available for joint tail modelling.