Testing for coefficient stability of AR(1) model when the null is an integrated or a stationary process

In this paper, we propose a new test for coefficient stability of an AR(1) model against the random coefficient autoregressive model of order 1 neither assuming a stationary nor a non-stationary process under the null hypothesis of a constant coefficient. The proposed test is obtained as a modification of the locally best invariant (LBI) test by Lee [(1998). Coefficient constancy test in a random coefficient autoregressive model. J. Statist. Plann. Inference 74, 93–101]. We examine finite sample properties of the proposed test by Monte Carlo experiments comparing with other existing tests, in particular, the LBI test by McCabe and Tremayne [(1995). Testing a time series for difference stationary. Ann. Statist. 23 (3), 1015–1028], which is for the null of a unit root process against the alternative of a stochastic unit root process.     

Bootstrap LR tests of stationarity,common trends and cointegration

This paper considers the likelihood ratio (LR) tests of stationarity, common trends and cointegration for multivariate time series. As the distribution of these tests is not known, a bootstrap version is proposed via a state- space representation. The bootstrap samples are obtained from the Kalman filter innovations under the null hypothesis. Monte Carlo simulations for the Gaussian univariate random walk plus noise model show that the bootstrap LR test achieves higher power for medium-sized deviations from the null hypothesis than a locally optimal and one-sided Lagrange Multiplier (LM) test that has a known asymptotic distribution. The power gains of the bootstrap LR test are significantly larger for testing the hypothesis of common trends and cointegration in multivariate time series, as the alternative asymptotic procedure – obtained as an extension of the LM test of stationarity – does not possess properties of optimality. Finally, it is shown that the (pseudo-)LR tests maintain good size and power properties also for the non-Gaussian series. An empirical illustration is provided.     

面板数据单位根似然比检验研究

本文采用似然比类检验统计量进行面板单位根检验（简称为LR检验）研究,在局部备择假设成立的条件下,推导了其在无确定项、仅含截距项以及存在线性时间趋势项三种模型下所对应的渐近分布与局部渐近势函数。Monte Carlo模拟结果显示,当面板数据中含确定项（截距项或时间趋势项）时,LR检验水平比LLC和IPS检验水平更接近于给定的显著性检验水平;此外,当面板数据中包含发散个体时,经水平修正后的LR检验势要远远高于经水平修正后的LLC与IPS检验势,其中,经水平修正后的LLC与IPS检验势接近于零。     

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.     

Asymptotic power comparison of T2-type test and likelihood ratio test for a mean vector based on two-step monotone missing data

Abstract

In a 2-step monotone missing dataset drawn from a multivariate normal population, T²-type test statistic (similar to Hotelling’s T² test statistic) and likelihood ratio (LR) are often used for the test for a mean vector. In complete data, Hotelling’s T² test and LR test are equivalent, however T²-type test and LR test are not equivalent in the 2-step monotone missing dataset. Then we interest which statistic is reasonable with relation to power. In this paper, we derive asymptotic power function of both statistics under a local alternative and obtain an explicit form for difference in asymptotic power function. Furthermore, under several parameter settings, we compare LR and T²-type test numerically by using difference in empirical power and in asymptotic power function. Summarizing obtained results, we recommend applying LR test for testing a mean vector.     

Likelihood Ratio Tests for the Non Equivalence of Means

We derive likelihood ratio (LR) tests for the null hypothesis of equivalence that the normal means fall into a practical indifference zone. The LR test can easily be constructed and applied to k ≥ 2 treatments. Simulation results indicate that the LR test might be slightly anticonservative statistically, but when the sample sizes are large, it always produces the nominal level for mean configurations under the null hypothesis. More powerful than the studentized range test, the LR test is a straightforward application that requires only current existing statistical tables, with no complicated computations.     

Tests for cointegration rank and choice of the alternative

This paper discusses likelihood-ratio (LR) tests on the cointegrating (CI) rank which consider any possible dimension of the CI rank under the alternative. The trace test and lambda-max test are obtained as special cases. Limit quantiles for all the tests in the class are derived. It is found that any of these tests can be used to construct an estimator of the CI rank, with no differences in asymptotic properties when the alternative is fixed. The properties of the class of tests are investigated by local asymptotic analysis, a simulation study and an empirical illustration. It is found that all the tests in the class have comparable power, which deteriorates substantially as the number of random walks increases. Tests constructed for a specific class of alternatives present minor power gains for alternatives in the class, and require the alternative to be far from the null. No test in this class is found to be asymptotically (in-)admissible. Some of the new tests in the class can also be arranged to give a constrained estimator of the CI rank, that restricts the minimum number of common trends. The power gains that these tests can obtain by constraining the minimum number of common trends appears to be limited and outweighted by the risk of inconsistency induced by the constrains. As a consequence, no value of the CI rank should be left untested, unless it can be excluded beyond any reasonable doubt.     

A Parametric Bootstrap Test for Two-Way ANOVA Model Without Interaction Under Heteroscedasticity

In this article we consider the two-way ANOVA model without interaction under heteroscedasticity. For the problem of testing equal effects of factors, we propose a parametric bootstrap (PB) approach and compare it with existing the generalized F (GF) test. The Type I error rates and powers of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test performs better than the GF test. The PB test performs very satisfactorily even for small samples while the GF test exhibits poor Type I error properties when the number of factorial combinations or treatments goes up. It is also noted that the same tests can be used to test the significance of random effect variance component in a two-way mixed-effects model under unequal error variances.     

Testing for normality in linear regression models

The importance of the normal distribution for fitting continuous data is well known. However, in many practical situations data distribution departs from normality. For example, the sample skewness and the sample kurtosis are far away from 0 and 3, respectively, which are nice properties of normal distributions. So, it is important to have formal tests of normality against any alternative. D'Agostino et al. [A suggestion for using powerful and informative tests of normality, Am. Statist. 44 (1990), pp. 316–321] review four procedures Z ²(g ₁), Z ²(g ₂), D and K ² for testing departure from normality. The first two of these procedures are tests of normality against departure due to skewness and kurtosis, respectively. The other two tests are omnibus tests. An alternative to the normal distribution is a class of skew-normal distributions (see [A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), pp. 171–178]). In this paper, we obtain a score test (W) and a likelihood ratio test (LR) of goodness of fit of the normal regression model against the skew-normal family of regression models. It turns out that the score test is based on the sample skewness and is of very simple form. The performance of these six procedures, in terms of size and power, are compared using simulations. The level properties of the three statistics LR, W and Z ²(g ₁) are similar and close to the nominal level for moderate to large sample sizes. Also, their power properties are similar for small departure from normality due to skewness (γ₁≤0.4). Of these, the score test statistic has a very simple form and computationally much simpler than the other two statistics. The LR statistic, in general, has highest power, although it is computationally much complex as it requires estimates of the parameters under the normal model as well as those under the skew-normal model. So, the score test may be used to test for normality against small departure from normality due to skewness. Otherwise, the likelihood ratio statistic LR should be used as it detects general departure from normality (due to both skewness and kurtosis) with, in general, largest power.     

Unit Root Tests under Time-Varying Variances

The paper provides a general framework for investigating the effects of permanent changes in the variance of the errors of an autoregressive process on unit root tests. Such a framework - which is based on a novel asymptotic theory for integrated and near integrated processes with heteroskedastic errors - allows to evaluate how the variance dynamics affect the size and the power function of unit root tests. Contrary to previous studies, it is shown that non-constant variances can both inflate and deflate the rejection frequency of the commonly used unit root tests, both under the null and under the alternative, with early negative and late positive variance changes having the strongest impact on size and power. It is also shown that shifts smoothed across the sample have smaller impacts than shifts occurring as a single abrupt jump, while periodic variances have a negligible effect even when a small number of cycles take place over a given sample. Finally, it is proved that the locally best invariant (LBI) test of a unit root against level stationarity is robust to heteroskedasticity of any form under the null hypothesis.     

固定效应模型基于个体LM的新可混合性检验

对固定效应模型,本文基于拉格朗日乘数（LM）原理提出了一种新的可混合性检验。不同于已有的LM型可混合性检验,这里使用每个截面个体的LM统计量构建可混合性检验统计量。数理分析表明,本文所提的方法有着渐进正态性,对于扰动项的异方差和非正态均稳健,且与PY检验（Pesaran&Yamagata,2008）渐近等价。Monte Carlo模拟实验表明,相对于PY检验及另外两种LM型的可混合性检验,对于不同大小的 ,本文提出的方法有着良好的水平表现和更优越的检验势。     

Likelihood ratio tests based on subglobal optimization: A power comparison in exponential mixture models

The paper compares several versions of the likelihood ratio test for exponential homogeneity against mixtures of two exponentials. They are based on different implementations of the likelihood maximization algorithm. We show that global maximization of the likelihood is not appropriate to obtain a good power of the LR test. A simple starting strategy for the EM algorithm, which under the null hypothesis often fails to find the global maximum, results in a rather powerful test. On the other hand, a multiple starting strategy that comes close to global maximization under both the null and the alternative hypotheses leads to inferior power.     

Likelihood ratio test for testing equality of location parameters of two exponential distributions from doubly censored samples

The Likelihood Ratio (LR) test for testing equality of two exponential distributions with common unknown scale parameter is obtained. Samples are assumed to be drawn under a type II doubly censored sampling scheme. Effects of left and right censoring on the power of the test are studied. Further, the performance of the LR test is compared with the Tiku(1981) test.     

A permutation test for randomness with power against smooth variation

A permutation test for the white noise hypothesis is described, offering power against a general class of smooth alternatives. Simulation results show that it performs well, as compared with similar tests available in the literature, in terms of power. An example demonstrates its use in a particular problem in which a test for randomness was sought without any specific alternative.     

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.     

Tests of additivity in mixed and fixed effect two-way ANOVA models with single sub-class numbers

In variety testing as well as in psychological assessment, the situation occurs that in a two-way ANOVA-type model with only one replication per cell, analysis is done under the assumption of no interaction between the two factors. Tests for this situation are known only for fixed factors and normally distributed outcomes. In the following we will present five additivity tests and apply them to fixed and mixed models and to quantitative as well as to Bernoulli distributed data. We consider their performance via simulation studies with respect to the type-I-risk and power. Furthermore, two new approaches will be presented, one being a modification of Tukey’s test and the other being a new experimental design to test for interactions.     

Assessing additivity in nonparametric models —A kernel‐based method

In this article, we address the testing problem for additivity in nonparametric regression models. We develop a kernel‐based consistent test of a hypothesis of additivity in nonparametric regression, and establish its asymptotic distribution under a sequence of local alternatives. Compared to other existing kernel‐based tests, the proposed test is shown to effectively ameliorate the influence from estimation bias of the additive component of the nonparametric regression, and hence increase its efficiency. Most importantly, it avoids the tuning difficulties by using estimation‐based optimal criteria, while there is no direct tuning strategy for other existing kernel‐based testing methods. We discuss the usage of the new test and give numerical examples to demonstrate the practical performance of the test. The Canadian Journal of Statistics 39: 632–655; 2011. © 2011 Statistical Society of Canada     

Robust test for independence in high dimensions

This article develops a new test based on Spearman’s rank correlation coefficients for total independence in high dimensions. The test is robust to the non normality and heavy tails of the data, which is a merit that is not shared by the existing tests in literature. Simulation results suggest that the new test performs well under several typical null and alternative hypotheses. Besides, we employ a real data set to illustrate the use of the new test.     

On the LBI criterion for the multivariate one-way random effects model under non-normality

The asymptotic null distribution of the locally best invariant (LBI) test criterion for testing the random effect in the one-way multivariable analysis of variance model is derived under normality and non-normality. The error of the approximation is characterized as O(1/n). The non-null asymptotic distribution is also discussed. In addition to providing a way of obtaining percentage points and p-values, the results of this paper are useful in assessing the robustness of the LBI criterion. Numerical results are presented to illustrate the accuracy of the approximation.     

Tests for bivariate exponentiality against BHNBUE alternatives

We present statistical procedures to test that a life distribution is bivariate exponential (BVE) against the alternative that it is bivariate harmonic new better than used in expectation (BHNBUE). We present a simulation study to compare the power the proposed test with tests proposed by Basu and Ebrahimi (1984) and Sen and Jain (1990) and we observe that the proposed test performs better than the other two tests.