Bootstrapping frequency domain tests in multivariate time series with an application to comparing spectral densities

Summary.  We propose a general bootstrap procedure to approximate the null distribution of non-parametric frequency domain tests about the spectral density matrix of a multivariate time series. Under a set of easy-to-verify conditions, we establish asymptotic validity of the bootstrap procedure proposed. We apply a version of this procedure together with a new statistic to test the hypothesis that the spectral densities of not necessarily independent time series are equal. The test statistic proposed is based on an L ₂-distance between the non-parametrically estimated individual spectral densities and an overall, 'pooled' spectral density, the latter being obtained by using the whole set of m time series considered. The effects of the dependence between the time series on the power behaviour of the test are investigated. Some simulations are presented and a real life data example is discussed.     

Empirical likelihood tests for two‐sample problems via nonparametric density estimation

The authors study the problem of testing whether two populations have the same law by comparing kernel estimators of the two density functions. The proposed test statistic is based on a local empirical likelihood approach. They obtain the asymptotic distribution of the test statistic and propose a bootstrap approximation to calibrate the test. A simulation study is carried out in which the proposed method is compared with two competitors, and a procedure to select the bandwidth parameter is studied. The proposed test can be extended to more than two samples and to multivariate distributions.     

Spectral Density Based Goodness-of-Fit Tests for Time Series Models

A new goodness-of-fit test for time series models is proposed. The test statistic is based on the distance between a kernel estimator of the ratio between the true and the hypothesized spectral density and the expected value of the estimator under the null. It provides a quantification of how well a parametric spectral density model fits the sample spectral density (periodogram). The asymptotic distribution of the statistic proposed is derived and its power properties are discussed. To improve upon the large sample (Gaussian) approximation of the distribution of the test statistic under the null, a bootstrap procedure is presented and justified theoretically. The finite sample performance of the test is investigated through a simulation experiment and applications to real data sets are given.     

Testing a linear relationship in varying coefficient spatial autoregressive models

In this article, we propose a test to check a linear relationship in varying coefficient spatial autoregressive models, in which a residual-based bootstrap procedure is suggested to approximate the null distribution of the resulting test statistic. We conduct simulation studies to assess the performance of the test, including the validity of the bootstrap approximation to the null distribution of the test statistic and the power of the test. The simulation results demonstrate that the residual-based bootstrap procedure gives very accurate estimate of the null distribution of the test statistic and the test is of satisfactory power. Furthermore, a real example is given to demonstrate the application of the proposed test.     

Detection and estimation of structural change in heavy-tailed sequence

In this paper, bootstrap detection and ratio estimation are proposed to analysis mean change in heavy-tailed distribution. First, the test statistic is constructed into a ratio form on the CUSUM process. Then, the asymptotic distribution of test statistic is obtained and the consistency of the test is proved. To solve the problem that the null distribution of the test statistic contains unknown tail index, we present a bootstrap approximation method to determine the critical values of the null distribution. We also discuss how to estimate change point based on ratio method. The consistency and rate of convergence for the change-point estimator are established. Finally, the excellent performance of our method is demonstrated through simulations using artificial and real data sets. Especially the simulation results of bootstrap test are better than those of another existing method.     

Testing for Linear Regression Relationships in Randomly Right-Censored Varying-Coefficient Models

A test statistic is constructed to test linear relationships in randomly right-censored varying-coefficient models. A residual-based bootstrap procedure is employed to derive the p-value of the test. The performance of the test is examined by extensive simulations. The simulation results show that the bootstrap estimate of the null distribution of the test statistic is approximately valid and the test method with the residual-based bootstrap works satisfactorily for at least moderate censoring rates of the response. Furthermore, the proposed test is applied to the Stanford heart transplant data for exploring a linear regression relationship between the logrithm of the survival time and the age of the patients.     

Testing and dating structural changes in copula-based dependence measures

This paper is concerned with testing and dating structural breaks in the dependence structure of multivariate time series. We consider a cumulative sum (CUSUM) type test for constant copula-based dependence measures, such as Spearman''s rank correlation and quantile dependencies. The asymptotic null distribution is not known in closed form and critical values are estimated by an i.i.d. bootstrap procedure. We analyze size and power properties in a simulation study under different dependence measure settings, such as skewed and fat-tailed distributions. To date breakpoints and to decide whether two estimated break locations belong to the same break event, we propose a pivot confidence interval procedure. Finally, we apply the test to the historical data of 10 large financial firms during the last financial crisis from 2002 to mid-2013.     

Nonlinear regression models with general distortion measurement errors

This paper considers nonlinear regression models when neither the response variable nor the covariates can be directly observed, but are measured with both multiplicative and additive distortion measurement errors. We propose conditional variance and conditional mean calibration estimation methods for the unobserved variables, then a nonlinear least squares estimator is proposed. For the hypothesis testing of parameter, a restricted estimator under the null hypothesis and a test statistic are proposed. The asymptotic properties for the estimator and test statistic are established. Lastly, a residual-based empirical process test statistic marked by proper functions of the regressors is proposed for the model checking problem. We further suggest a bootstrap procedure to calculate critical values. Simulation studies demonstrate the performance of the proposed procedure and a real example is analysed to illustrate its practical usage.     

A bootstrap test for single index models

Single index models are frequently used in econometrics and biometrics. Logit and Probit models arc special cases with fixed link functions. In this paper we consider a bootstrap specification test that detects nonparametric deviations of the link function. The bootstrap is used with the aim to rind a more accurate distribution under the null than the normal approximation. We prove that the statistic and its bootstrapped version have the same asymptotic distribution. In a simulation study we show that the bootstrap is able to capture the negative bias and the skewness of the test statistic. It yields better approximations to the true critical values and consequently it has a more accurate level than the normal approximation.     

Bootstrapping a consistent nonparametric goodness-of-fit test

In this paper, we employ the parametric bootstrap to approximate the finite sample distribution of a goodness-of-fit test statistic in Fan (1994). We show that the proposed bootstrap procedure works in that the bootstrap distribution conditional on the random sample tends to the asymptotic distribution of the test statistic in probability. A simulation study demonstrates that the bootstrap approximation works extremely well in small samples with only 25 observations and is very robust to the value of the smoothing parameter in the kernel density estimation.     

Monitoring Change in Persistence Against the Null of Difference-Stationarity in Infinite Variance Observations

In this article, we propose a moving kernel-weighted variance ratio statistic to monitor persistence change in infinite variance observations. We focus on I(1) to I(0) persistence change for sequences in the domain of attraction of a stable law and local-to-finite variance sequences. The null distribution of the monitoring statistic and its consistency are proved. In particular, a bootstrap procedure is proposed to determine the critical values for the derived asymptotic distribution depends on unknown tail index. The small sample performances of proposed monitoring procedure are illustrated by both simulation and application to a high frequency financial data.     

Statistical inference on partial linear additive models with distortion measurement errors

We consider statistical inference for partial linear additive models (PLAMs) when the linear covariates are measured with errors and distorted by unknown functions of commonly observable confounding variables. A semiparametric profile least squares estimation procedure is proposed to estimate unknown parameter under unrestricted and restricted conditions. Asymptotic properties for the estimators are established. To test a hypothesis on the parametric components, a test statistic based on the difference between the residual sums of squares under the null and alternative hypotheses is proposed, and we further show that its limiting distribution is a weighted sum of independent standard chi-squared distributions. A bootstrap procedure is further proposed to calculate critical values. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analyzed for an illustration.     

Assessing goodness-of-fit of categorical regression models based on case-control data

Demonstrated equivalence between a categorical regression model based on case‐control data and an I‐sample semiparametric selection bias model leads to a new goodness‐of‐fit test. The proposed test statistic is an extension of an existing Kolmogorov–Smirnov‐type statistic and is the weighted average of the absolute differences between two estimated distribution functions in each response category. The paper establishes an optimal property for the maximum semiparametric likelihood estimator of the parameters in the I‐sample semiparametric selection bias model. It also presents a bootstrap procedure, some simulation results and an analysis of two real datasets.     

A NONPARAMETRIC TEST OF CHANGING CONDITIONAL VARIANCES IN AUTOREGRESSIVE TIME SERIES

A nonparametric test for detecting changing conditional variances in stationary AR(p) time series is proposed in this paper. For AR(1) models, the test statistic is a Kolmogorov-Smirnov type statistic and the asymptotic theory is developed under both the null and the alternative hypotheses. For AR(p) models (p ≥ 2), an approximate test procedure is proposed. The empirical upper percentage points for our test are tabulated for both p = 1 and p = 2 cases and a bootstrap procedure is suggested for the p ≥ 3 case. Monte Carlo simulations demonstrate that the test has very good powers for finite samples under both normal and non-normal errors.     

A resampling‐based test for two crossing survival curves

The area between two survival curves is an intuitive test statistic for the classical two‐sample testing problem. We propose a bootstrap version of it for assessing the overall homogeneity of these curves. Our approach allows ties in the data as well as independent right censoring, which may differ between the groups. The asymptotic distribution of the test statistic as well as of its bootstrap counterpart are derived under the null hypothesis, and their consistency is proven for general alternatives. We demonstrate the finite sample superiority of the proposed test over some existing methods in a simulation study and illustrate its application by a real‐data example.     

Block bootstrap testing for changes in persistence with heavy-tailed innovations

This article considers the detection of changes in persistence in heavy-tailed series. We adopt a Dickey–Fuller-type ratio statistic and derive its null asymptotic distribution of test statistic. We find that the asymptotic distribution depends on the stable index, which is often typically unknown and difficult to estimate. Therefore, the block bootstrap method is proposed to detect changes without estimating κ. The empirical sizes and power values are investigated to show that the block bootstrap test is valid. Finally, the validity of the method is demonstrated by analyzing the exchange rate of RMB and US dollars.     

Testing Conditional Mean Independence Under Symmetry

Conditional mean independence (CMI) is one of the most widely used assumptions in the treatment effect literature to achieve model identification. We propose a Kolmogorov–Smirnov-type statistic to test CMI under a specific symmetry condition. We also propose a bootstrap procedure to obtain the p-values and critical values that are required to carry out the test. Results from a simulation study suggest that our test can work very well even in small to moderately sized samples. As an empirical illustration, we apply our test to a dataset that has been used in the literature to estimate the return on college education in China, to check whether the assumption of CMI is supported by the dataset and show the plausibility of the extra symmetry condition that is necessary for this new test.     

Tests for noncorrelation of two multivariate ARMA time series

In many situations, we want to verify the existence of a relationship between multivariate time series. In this paper, we generalize the procedure developed by Haugh (1976) for univariate time series in order to test the hypothesis of noncorrelation between two multivariate stationary ARMA series. The test statistics are based on residual cross-correlation matrices. Under the null hypothesis of noncorrelation, we show that an arbitrary vector of residual cross-correlations asymptotically follows the same distribution as the corresponding vector of cross-correlations between the two innovation series. From this result, it follows that the test statistics considered are asymptotically distributed as chi-square random variables. Two test procedures are described. The first one is based on the residual cross-correlation matrix at a particular lag, whilst the second one is based on a portmanteau type statistic that generalizes Haugh's statistic. We also discuss how the procedures for testing noncorrelation can be adapted to determine the directions of causality in the sense of Granger (1969) between the two series. An advantage of the proposed procedures is that their application does not require the estimation of a global model for the two series. The finite-sample properties of the statistics introduced were studied by simulation under the null hypothesis. It led to modified statistics whose upper quantiles are much better approximated by those of the corresponding chi-square distribution. Finally, the procedures developed are applied to two different sets of economic data.     

Robust unit root tests with autoregressive errors

ABSTRACT

This article presents a new test for unit roots based on least absolute deviation estimation specially designed to work for time series with autoregressive errors. The methodology used is a bootstrap scheme based on estimating a model and then the innovations. The resampling part is performed under the null hypothesis and, as it is customary in bootstrap procedures, is automatic and does not rely on the calculation of any nuisance parameter. The validity of the procedure is established and the asymptotic distribution of the statistic proposed is proved to converge to the correct distribution. To analyze the performance of the test for finite samples, a Monte Carlo study is conducted showing a very good behavior in many different situations.     

Homogeneity test based on ranked set samples

Abstract

The homogeneity hypothesis is investigated in a location family of distributions. A moment-based test is introduced based on data collected from a ranked set sampling scheme. The asymptotic distribution of the proposed test statistic is determined and the performance of the test is studied via simulation. Furthermore, for small sample sizes, the bootstrap procedure is used to distinguish the homogeneity of data. An illustrative example is also presented to explain the proposed procedures in this paper.