The Moran coefficient for non-normal data

This paper summarizes findings that extend statistical distribution properties of the Moran coefficient index measuring spatial autocorrelation to non-normal random variables. Pitman–Koopmans theorem results are extended for the mean and the variance of this index. This summary includes a corollary to this theorem, as well as a new theorem (with its proof) and two conjectures implied by it. The first of these statements is supported by asymptotic heuristics; the second is supported by simulation experiment results. Mixture random variables that include heteroscedasticity or overdispersion also are explored. In addition, a simple asymptotic variance for the Moran coefficient is presented, assessed, and found to be very precise for sample sizes as small as 25–100. The principal conclusion is that independence and sample size are the most relevant properties for Pitman–Koopmans theorem results to be extended to non-normal random variables. The independent and identically distributed property reduces the necessary sample size for this extension, as do the properties of symmetry and normal approximation.     

A central limit theorem for the sample autocorrelations of a Lévy driven continuous time moving average process

In this article we consider Lévy driven continuous time moving average processes observed on a lattice, which are stationary time series. We show asymptotic normality of the sample mean, the sample autocovariances and the sample autocorrelations. A comparison with the classical setting of discrete moving average time series shows that in the last case a correction term should be added to the classical Bartlett formula that yields the asymptotic variance. An application to the asymptotic normality of the estimator of the Hurst exponent of fractional Lévy processes is also deduced from these results.     

Expansion and estimation of Lévy process functionals in nonlinear and nonstationary time series regression

In this article, we develop a series estimation method for unknown time-inhomogeneous functionals of Lévy processes involved in econometric time series models. To obtain an asymptotic distribution for the proposed estimators, we establish a general asymptotic theory for partial sums of bivariate functionals of time and nonstationary variables. These results show that the proposed estimators in different situations converge to quite different random variables. In addition, the rates of convergence depend on various factors rather than just the sample size. Finite sample simulations are provided to evaluate the finite sample performance of the proposed model and estimation method.     

Asymptotic Inefficiency of Mean-Correction on Parameter Estimation for a Periodic First-Order Autoregressive Model

A common practice in time series analysis is to fit a centered model to the mean-corrected data set. For stationary autoregressive moving-average (ARMA) processes, as far as the parameter estimation is concerned, fitting an ARMA model without intercepts to the mean-corrected series is asymptotically equivalent to fitting an ARMA model with intercepts to the observed series. We show that, related to the parameter least squares estimation of periodic ARMA models, the second approach can be arbitrarily more efficient than the mean-corrected counterpart. This property is illustrated by means of a periodic first-order autoregressive model. The asymptotic variance of the estimators for both approaches is derived. Moreover, empirical experiments based on simulations investigate the finite sample properties of the estimators.     

Componentwise B-spline estimation for varying coefficient models with longitudinal data

A componentwise B-spline method is proposed for estimating the unknown functions in the varying-coefficient models with longitudinal data. Different amounts of smoothing are used for different individual coefficient functions and the estimators of different coefficient functions are obtained by different minimization operations. The local asymptotic bias and variance of the estimators are derived. It is shown that our estimators achieve the local and global optimal convergence rates even if the coefficient functions belong to different smoothness families. The asymptotic distributions of the estimators are also established and are used to construct approximate pointwise confidence intervals for coefficient functions. Finite sample properties of our procedures are studied through Monte Carlo simulations.     

Conservative confidence intervals for the intraclass correlation coefficient for clustered binary data

Asymptotic approaches are traditionally used to calculate confidence intervals for intraclass correlation coefficient in a clustered binary study. When sample size is small to medium, or correlation or response rate is near the boundary, asymptotic intervals often do not have satisfactory performance with regard to coverage. We propose using the importance sampling method to construct the profile confidence limits for the intraclass correlation coefficient. Importance sampling is a simulation based approach to reduce the variance of the estimated parameter. Four existing asymptotic limits are used as statistical quantities for sample space ordering in the importance sampling method. Simulation studies are performed to evaluate the performance of the proposed accurate intervals with regard to coverage and interval width. Simulation results indicate that the accurate intervals based on the asymptotic limits by Fleiss and Cuzick generally have shorter width than others in many cases, while the accurate intervals based on Zou and Donner asymptotic limits outperform others when correlation and response rate are close to their boundaries.     

Asymptotic Normality of M-Estimators for Varying Coefficient Models with Longitudinal Data

This article considers a nonparametric varying coefficient regression model with longitudinal observations. The relationship between the dependent variable and the covariates is assumed to be linear at a specific time point, but the coefficients are allowed to change over time. A general formulation is used to treat mean regression, median regression, quantile regression, and robust mean regression in one setting. The local M-estimators of the unknown coefficient functions are obtained by local linear method. The asymptotic distributions of M-estimators of unknown coefficient functions at both interior and boundary points are established. Various applications of the main results, including estimating conditional quantile coefficient functions and robustifying the mean regression coefficient functions are derived. Finite sample properties of our procedures are studied through Monte Carlo simulations.     

Structural Change Monitoring for Random Coefficient Autoregressive Time Series

A monitoring scheme is proposed to sequentially detect a structural change in random coefficient autoregressive time series of order p (RCA(p)) after a training period of size T. It extends structural change monitoring to RCA(p) time series. The asymptotic properties of our monitoring statistic are established under both the null of no change in parameters and the alternative of a change in coefficient. The finite sample properties are investigated by a simulation study.     

Variance estimation for sparse ultra-high dimensional varying coefficient models

This paper considers the problem of variance estimation for sparse ultra-high dimensional varying coefficient models. We first use B-spline to approximate the coefficient functions, and discuss the asymptotic behavior of a naive two-stage estimator of error variance. We also reveal that this naive estimator may significantly underestimate the error variance due to the spurious correlations, which are even higher for nonparametric models than linear models. This prompts us to propose an accurate estimator of the error variance by effectively integrating the sure independence screening and the refitted cross-validation techniques. The consistency and the asymptotic normality of the resulting estimator are established under some regularity conditions. The simulation studies are carried out to assess the finite sample performance of the proposed methods.     

Variance Shifts,Structural Breaks,and Stationarity Tests

This article considers the problem of testing the null hypothesis of stochastic stationarity in time series characterized by variance shifts at some (known or unknown) point in the sample. It is shown that existing stationarity tests can be severely biased in the presence of such shifts, either oversized or undersized, with associated spurious power gains or losses, depending on the values of the breakpoint parameter and on the ratio of the prebreak to postbreak variance. Under the assumption of a serially independent Gaussian error term with known break date and known variance ratio, a locally best invariant (LBI) test of the null hypothesis of stationarity in the presence of variance shifts is then derived. Both the test statistic and its asymptotic null distribution depend on the breakpoint parameter and also, in general, on the variance ratio. Modifications of the LBI test statistic are proposed for which the limiting distribution is independent of such nuisance parameters and belongs to the family of Cramér–von Mises distributions. One such modification is particularly appealing in that it is simultaneously exact invariant to variance shifts and to structural breaks in the slope and/or level of the series. Monte Carlo simulations demonstrate that the power loss from using our modified statistics in place of the LBI statistic is not large, even in the neighborhood of the null hypothesis, and particularly for series with shifts in the slope and/or level. The tests are extended to cover the cases of weakly dependent error processes and unknown breakpoints. The implementation of the tests are illustrated using output, inflation, and exchange rate data series.     

Estimated Generalized Least Squares for Random Coefficient Regression Models

ABSTRACT. This paper considers a general class of random coefficient regression (RCR) models to represent pooled cross-sectional and time series data. A new method is given to estimate the covariance matrix of the error component in these RCR models. Also, the asymptotic and small sample properties of the estimated generalized least squares estimator of the regression coefficient vector are established. Procedures for testing a linear restriction on the mean vector of the random coefficients are derived. Finally, a test for non-randomness in the RCR model is devised, and the asymptotic distribution of the test statistic is obtained.     

Forecasting Vector ARMA Processes With Systematically Missing Observations

The following two predictors are compared for time series with systematically missing observations: (a) A time series model is fitted to the full series X_t , and forecasts are based on this model, (b) A time series model is fitted to the series with systematically missing observations Y _τ, and forecasts are based on the resulting model. If the data generation processes are known vector autoregressive moving average (ARMA) processes, the first predictor is at least as efficient as the second one in a mean squared error sense. Conditions are given for the two predictors to be identical. If only the ARMA orders of the generation processes are known and the coefficients are estimated, or if the process orders and coefficients are estimated, the first predictor is again, in general, superior. There are, however, exceptions in which the second predictor, using seemingly less information, may be better. These results are discussed, using both asymptotic theory and small sample simulations. Some economic time series are used as illustrative examples.     

Effect of Violation of the Normal Assumption on MI and ML Estimators in the Analysis of Incomplete Data

Asymptotic distributions of normal-theory-based ML/MI estimators are studied in a simple regression model under general distributions with MAR missing data. The asymptotic variance of the ML/MI estimator of residuals’ variance is explicitly derived, from which it follows that the kurtosis of the error distribution primarily affects the asymptotic variance. Results of numerical simulations conducted to study finite sample properties of the estimators, conformed largely to the asymptotic results, and they also indicated interesting findings particularly for small samples, which do not follow from the asymptotic property. It is concluded that the ML estimators perform best in the situation studied here.     

Statistical inference in partially time-varying coefficient models

A partially time-varying coefficient time series model is introduced to characterize the nonlinearity and trending phenomenon. To estimate the regression parameter and the nonlinear coefficient function, the profile least squares approach is applied with the help of local linear approximation. The asymptotic distributions of the proposed estimators are established under mild conditions. Meanwhile, the generalized likelihood ratio test is studied and the test statistics are demonstrated to follow asymptotic χ²-distribution under the null hypothesis. Furthermore, some extensions of the proposed model are discussed and several numerical examples are provided to illustrate the finite sample behavior of the proposed methods.     

Asymptotic variance–covariance matrix of sample autocorrelations for threshold-asymmetric GARCH processes

In the field of financial time series, threshold-asymmetric conditional variance models can be used to explain asymmetric volatilities [C.W. Li and W.K. Li, On a double-threshold autoregressive heteroscedastic time series model, J. Appl. Econometrics 11 (1996), pp. 253–274]. In this paper, we consider a broad class of threshold-asymmetric GARCH processes (TAGARCH, hereafter) including standard ARCH and GARCH models as special cases. Since sample autocorrelation function provides a useful information to identify an appropriate time-series model for the data, we derive asymptotic distributions of sample autocorrelations both for original process and for squared process. It is verified that standard errors of sample autocorrelations for TAGARCH models are significantly different from unity for lower lags and they are exponentially converging to unity for higher lags. Furthermore they are shown to be asymptotically dependent while being independent of standard GARCH models. These results will be interesting in the light of the fact that TAGARCH processes are serially uncorrelated. A simulation study is reported to illustrate our results.     

Robust estimation for functional coefficient regression models with spatial data

A global smoothing procedure is developed using B-spline function approximation for estimating the unknown functions of a functional coefficient regression model with spatial data. A general formulation is used to treat mean regression, median regression, quantile regression and robust mean regression in one setting. The global convergence rates of the estimators of unknown coefficient functions are established. Various applications of the main results, including estimating conditional quantile coefficient functions and robustifying the mean regression coefficient functions are given. Finite sample properties of our procedures are studied through Monte Carlo simulations. A housing data example is used to illustrate the proposed methodology.     

The Sample Coefficient of Determination in Simple Linear Regression

The behavior of the sample coefficient of determination is examined for some arrangements of independent variable values in a simple linear regression with normally distributed error terms. Numerical values of means and standard deviations are presented that provide some insight into the influence of range and arrangement of independent variable values and sample size on the size of the sample coefficient of determination. Some asymptotic results are given.     

Testing for changes in autocovariances of nonparametric time series models

In the literature on change-point analysis, much attention has been paid to detecting changes in certain marginal characteristics, such as mean, variance, and marginal distribution. For time series data with nonparametric time trend, we study the change-point problem for the autocovariance structure of the unobservable error process. To derive the asymptotic distribution of the cumulative sum test statistic, we develop substantial theory for uniform convergence of weighted partial sums and weighted quadratic forms. Our asymptotic results improve upon existing works in several important aspects. The performance of the test statistic is examined through simulations and an application to interest rates data.     

Nonparametric quasi-likelihood for right censored data

Quasi-likelihood was extended to right censored data to handle heteroscedasticity in the frame of the accelerated failure time (AFT) model. However, the assumption of known variance function in the quasi-likelihood for right censored data is usually unrealistic. In this paper, we propose a nonparametric quasi-likelihood by replacing the specified variance function with a nonparametric variance function estimator. This nonparametric variance function estimator is obtained by smoothing a function of squared residuals via local polynomial regression. The rate of convergence of the nonparametric variance function estimator and the asymptotic limiting distributions of the regression coefficient estimators are derived. It is demonstrated in simulations that for finite samples the proposed nonparametric quasi-likelihood method performs well. The new method is illustrated with one real dataset.     

On the construction of confidence limits for the regression coefficient when the residuals are dependent

A time series model with a linear trend and with residuals following either a first—order auto—regressive or a first—order moving—average model is considered. An asymptotic expression for the variance of the ordinary least square estimator of the regression coefficient is derived and used in a modified method for the construction of confidence intervals. Simulated results and guidelines for the use of the modified method are given.