M–Estimtors and l–estimators of location: uniform integrability and asymptotic risk–efficient sequential versions

In the context of asymptotically minimum risk (sequential) point estimation of location of a symmetric distribution, M-and L-estimators are considered, and various properties of their sequential versions are studied. Asymptotic distributions of he allied stopping times are also derived. In this study, uniform integrability and moment convergence of (non-sequential) M- and L-estimators are established. These results have interest of therir own and provide the main tools for the proof of the other results presented. For the sequential estimators, their asymptotic risk efficiencies are shown to coincide with the asymptotic efficiencies of the respective non-sequential estimators; this enables one to construct the asymptotically minimax sequential M- and L-estimators in the model of contaminancy. Parallel results also hold for the rank estimators of location.     

Robust estimators under semi‐parametric partly linear autoregression: Asymptotic behaviour and bandwidth selection

Abstract. In this article, under a semi‐parametric partly linear autoregression model, a family of robust estimators for the autoregression parameter and the autoregression function is studied. The proposed estimators are based on a three‐step procedure, in which robust regression estimators and robust smoothing techniques are combined. Asymptotic results on the autoregression estimators are derived. Besides combining robust procedures with M‐smoothers, predicted values for the series and detection residuals, which allow to detect anomalous data, are introduced. Robust cross‐validation methods to select the smoothing parameter are presented as an alternative to the classical ones, which are sensitive to outlying observations. A Monte Carlo study is conducted to compare the performance of the proposed criteria. Finally, the asymptotic distribution of the autoregression parameter estimator is stated uniformly over the smoothing parameter.     

Fixed‐b asymptotic approximation of the sampling behaviour of nonparametric spectral density estimators

Abstract. We propose a new asymptotic approximation for the sampling behaviour of nonparametric estimators of the spectral density of a covariance stationary time series. According to the standard approach, the truncation lag grows more slowly than the sample size. We derive first‐order limiting distributions under the alternative assumption that the truncation lag is a fixed proportion of the sample size. Our results extend the approach of Neave (1970) , who derived a formula for the asymptotic variance of spectral density estimators under the same truncation lag assumption. We show that the limiting distribution of zero‐frequency spectral density estimators depends on how the mean is estimated and removed. The implications of our zero‐frequency results are consistent with exact results for bias and variance computed by Ng and Perron (1996) . Finite sample simulations indicate that the new asymptotics provides a better approximation than the standard one.     

A Note on the Behaviour of Nonparametric Density and Spectral Density Estimators at Zero Points of their Support

The asymptotic behaviour of nonparametric estimators of the stationary density and of the spectral density function of a stationary process have been studied in some detail in the last 50–60years. Nevertheless, less is known about the behaviour of these estimators when the target function happens to vanish at the point of interest. In the article at hand, we fill this gap and show that asymptotic normality still holds true but with super‐efficient and different rates of convergence for the density and for the spectral density estimators that are affected also by the dependence structure of the process.     

Asymptotics of Quantiles and Rank Scores in Nonlinear Time Series

This paper extends the concept of regression and autoregression quantiles and rank scores to a very general nonlinear time series model. The asymptotic linearizations of these nonlinear quantiles are then used to obtain the limiting distributions of a class of L-estimators of the parameters. In particular, the limiting distributions of the least absolute deviation estimator and trimmed estimators are obtained. These estimators turn out to be asymptotically more efficient than the widely used conditional least squares estimator for heavy-tailed error distributions. The results are applicable to linear and nonlinear regression and autoregressive models including self-exciting threshold autoregressive models with known threshold.     

Local power of likelihood‐based tests for cointegrating rank: Comparative analysis of full and partial systems

This note investigates local power properties of likelihood‐based cointegrating rank tests for partial and full vector autoregressive systems. The asymptotic distributions of partial likelihood‐based tests under local alternatives are derived, depending on various specifications of deterministic terms. A simulation study is then performed using both the full and partial systems. It is demonstrated that the rank tests based on the partial system, if a required parametric condition is fulfilled, can be more powerful than those based on the full system. This finding encourages testing cointegrating rank using a partial system as well as a full system, in such circumstances as the parametric condition could be satisfied.     

A geometric time series model with dependent Bernoulli counting series

A new stationary first‐order integer‐valued autoregressive process with geometric marginal distribution based on the generalized binomial thinning is introduced. The model involves dependent count variables. Some properties of the process are determined. A set of estimators are obtained, and their asymptotic distributions are considered. Some numerical results of the estimates are presented. Possible application of the process is discussed through the real data example.     

Robust Estimation For Periodic Autoregressive Time Series

Abstract. A robust estimation procedure for periodic autoregressive (PAR) time series is introduced. The asymptotic properties and the asymptotic relative efficiency are discussed by the estimating equation approach. The performance of the robust estimators for PAR time‐series models with order one is illustrated by a simulation study. The technique is applied to a real data analysis.     

SEMI‐PARAMETRIC ESTIMATION OF LINEAR COINTEGRATING MODELS WITH NONLINEAR CONTEMPORANEOUS ENDOGENEITY

This article considers linear cointegrating models with unknown nonlinear short‐run contemporaneous endogeneity. Two estimators are proposed to estimate the linear cointegrating parameter after the nonlinear endogenous component is estimated by local linear regression approach. Both the proposed estimators are shown to have the same mixed normal limiting distribution with zero mean and smaller asymptotic variance than the fully modified ordinary least squares and instrumental variables estimators. Monte Carlo simulations are used to evaluate the finite sample performance of our proposed estimators, and an empirical application is also included.     

Combining Cumulative Sum Change‐Point Detection Tests for Assessing the Stationarity of Univariate Time Series

We derive tests of stationarity for univariate time series by combining change‐point tests sensitive to changes in the contemporary distribution with tests sensitive to changes in the serial dependence. The proposed approach relies on a general procedure for combining dependent tests based on resampling. After proving the asymptotic validity of the combining procedure under the conjunction of null hypotheses and investigating its consistency, we study rank‐based tests of stationarity by combining cumulative sum change‐point tests based on the contemporary empirical distribution function and on the empirical autocopula at a given lag. Extensions based on tests solely focusing on second‐order characteristics are proposed next. The finite‐sample behaviors of all the derived statistical procedures for assessing stationarity are investigated in large‐scale Monte Carlo experiments, and illustrations on two real datasets are provided. Extensions to multi‐variate time series are briefly discussed as well.     

YULE-WALKER ESTIMATES FOR CONTINUOUS-TIME AUTOREGRESSIVE MODELS

Abstract. I consider continuous-time autoregressive processes of order p and develop estimators of the model parameters based on Yule-Walker type equations. For continuously recorded data, it is shown that these estimators are least squares estimators and have the same asymptotic distribution as maximum likelihood estimators.
In practice, though, data can only be observed discretely. For discrete data, I consider approximations to the continuous-time estimators. It is shown that some of these discrete-time estimators are asymptotically biased. Alternative estimators based on the autocovariance function are suggested. These are asymptotically unbiased and are a fast alternative to the maximum likelihood estimators described by Jones. They may also be used as starting values for maximum likelihood estimation.     

Adaptive procedures for the two-sample location model

Adaptive estimates and tests based on ranks are proposed in the two-sample location shiftmodel. Beran (1974) describes the construction of uniformly asymptotically efficient rank estimates; he developed a Fourier series estimator of the score-generating unction based on the whole sample and considered the linearized rank estimators corresponding to the estimator of the optimal choice of the score-generating function. Here other estimators of the score-generating function are developed. They differ in two respects: General type of Fourier series and an estimator of the Fourier coefficient based on the linearity of rank statistics are used.     

VECTOR AUTOREGRESSIVE MODELS WITH UNIT ROOTS AND REDUCED RANK STRUCTURE:ESTIMATION. LIKELIHOOD RATIO TEST, AND FORECASTING

Abstract. The nonstationary multivariate autoregressive (AR) model Φ ( L ) Y _t =ε _t is considered for an m -dimensional process { Y _t }, where it is assumed that det {Φ( L )}= 0 has d < m unit roots and all other roots are outside the unit circle, and also that rank {Φ(1)}= r ( r = m – d ). Limiting distribution results obtained by Ahn and Reinsel for the least-squares and the Gaussian reduced rank (unit roots imposed) estimators for this AR model are extended to a model where the AR parameters possess additional structure such as nested reduced rank, and based on these results the asymptotic distribution of the likelihood ratio test statistic for testing the number d of unit roots is obtained. An analysis of three US monthly interest rate series is presented to illustrate the testing and estimation procedures. A small simulation study is also performed to examine the finite-sample properties of the likelihood ratio test and the prediction performance of models which impose different numbers of unit roots.     

Weighted scatter estimation method of the GO‐GARCH models

We propose a new estimation method for the factor loading matrix in modelling multivariate volatility processes. The key step of the method is based on the weighted scatter estimators, which does not involve optimizing any objective function. The method can therefore be easily applied to high‐dimensional systems without running into computational problems. The estimation is proved to be consistent and the asymptotic distribution is derived. The method inherits robust properties in dealing with ‘outlier’ clusters generated by GARCH processes. Through both simulation and real‐world case studies, we show that the method works well.     

A Self‐Normalized Semi‐Parametric Test to Detect Changes in the Long Memory Parameter

We consider the problem of testing for change points in the long memory parameter. The test relies on semi‐parametric estimation of the long memory parameter, which does not require the complete parametric specification of the whole spectrum. A self‐normalizer utilizing a sequence of recursive semi‐parametric estimators is used to make the asymptotic distribution of the test statistic free of the nuisance scale parameter. We study the asymptotic behavior of the proposed test for situations when there is at most one change point and also when there are an unknown number of change points. Monte Carlo simulations are carried out to examine the finite‐sample performance of the proposed test.     

Zero‐Modified Geometric INAR(1) Process for Modelling Count Time Series with Deflation or Inflation of Zeros

In this article, we propose a first‐order integer‐valued autoregressive [INAR(1)] process for dealing with count time series with deflation or inflation of zeros. The proposed process has zero‐modified geometric marginals and contains the geometric INAR(1) process as a particular case. The proposed model is also capable of capturing underdispersion and overdispersion, which sometimes are caused by deflation or inflation of zeros. We explore several statistical and mathematical properties of the process, discuss point estimation of the parameters and find the asymptotic distribution of the proposed estimators. We also propose a test based on our model for checking if the count time series considered is deflated or inflated of zeros. Two empirical illustrations are presented in order to show the potential for practice of our zero‐modified geometric INAR(1) process. This article contains a Supporting Information.     

REDUCTION OF THE ASYMPTOTIC BIAS OF AUTOREGRESSIVE AND SPECTRAL ESTIMATORS BY TAPERING

Abstract. The asymptotic bias to terms of order T ^-1, where T is the observed series length, is studied for estimators of the coefficients and disturbance variance in an AR( p ) model. Reduction of the asymptotic bias by tapering is established and, if the tapering function is defined appropriately to depend on T , not only is the asymptotic bias reduced, but the asymptotic distribution of the estimators is not altered. In addition, the asymptotic biases of other time series parameter estimators constructed from the sample covariance function, such as several types of spectral estimators, can also be reduced by tapering.     

Statistical Inference for Unified Garch–Itô Models with High‐Frequency Financial Data

The existing estimation methods for the model parameters of the unified GARCH–Itô model (Kim and Wang, 2014 ) require long period observations to obtain the consistency. However, in practice, it is hard to believe that the structure of a stock price is stable during such a long period. In this article, we introduce an estimation method for the model parameters based on the high‐frequency financial data with a finite observation period. In particular, we establish a quasi‐likelihood function for daily integrated volatilities, and realized volatility estimators are adopted to estimate the integrated volatilities. The model parameters are estimated by maximizing the quasi‐likelihood function. We establish asymptotic theories for the proposed estimator. A simulation study is conducted to check the finite sample performance of the proposed estimator. We apply the proposed estimation approach to the Bank of America stock price data.     

Asymptotic Behavior of Optimal Weighting in Generalized Self‐Normalization for Time Series

Self‐normalization has been celebrated for its ability to avoid direct estimation of the nuisance long‐run variance and its versatility in handling the mean and other quantities. The self‐normalizer in its original form uses only recursive estimators of one direction, and generalizations involving both forward and backward estimators were recently given. Unlike existing results that weigh the forward and backward estimators in a deterministic manner, the current article focuses on a data‐driven weight that corresponds to confidence intervals with minimal lengths. We study the asymptotic behavior of such a data‐driven weight choice, and find an interesting dichotomy between linear and nonlinear quantities. Another interesting phenomenon is that, for nonlinear quantities, the data‐driven weight typically distributes over an uncountable set in finite‐sample problems but in the limit it converges to a discrete distribution with a finite support.     

An Estimate of the Fractal Index Using Multiscale Aggregates

We study a family of estimators of the fractal index of a Gaussian process based on the quadratic deviations at different aggregation scales. The estimators are convergent and asymptotically Gaussian when suitably normalized. Confidence intervals are provided. These asymptotic results hold for a large family of stationary-increment models including fractional Brownian motions with square-integrable spectral density. The estimates are applied to the analysis of an electrical signal