ADF单位根检验中联合检验LM统计量研究

 本文研究了ADF单位根检验中参数联合约束的拉格朗日乘数检验。首先,本文构建了4个LM统计量并推导了它们的极限分布;然后,运用蒙特卡罗试验,模拟了有限样本容量常用检验水平下的临界值,拟合了临界值关于样本容量的响应面函数,并总结了LM统计量有限样本容量下的统计特性;比较分析了这4个LM统计量的检验功效及实际检验水平;最后,一个实例分析简要说明了这几个统计量在单位根检验中的应用。     

Detecting seasonal unit roots in a structural time series model

In this paper, we propose to detect seasonal unit roots within the context of a structural time series model. Such a model is often found to be useful in practice. Using Monte Carlo simulations, we show that our method works well. We illustrate our approach for several quarterly macroeconomic time series variables.     

检验式设定错误对时间序列单位根检验小样本性质的影响

首先对单位根检验的两类常见的数据生成系统进行比较，然后利用蒙特卡洛实验研究了时间序列单位根检验式的设定问题。研究发现在利用DF检验和DF-GLS检验进行时间序列的单位根检验时，检验式设定错误直接影响着检验结果，尤其在推断时间序列是趋势平稳过程还是有时间趋势项的随机游走过程或有二阶时间趋势多项式的随机游走过程时，检验式的错误设定很容易将趋势平稳过程误判为非平稳过程。     

Exact inference for a simple step-stress model with competing risks for failure from exponential distribution under Type-II censoring

In reliability analysis, accelerated life-testing allows for gradual increment of stress levels on test units during an experiment. In a special class of accelerated life tests known as step-stress tests, the stress levels increase discretely at pre-fixed time points, and this allows the experimenter to obtain information on the parameters of the lifetime distributions more quickly than under normal operating conditions. Moreover, when a test unit fails, there are often more than one fatal cause for the failure, such as mechanical or electrical. In this article, we consider the simple step-stress model under Type-II censoring when the lifetime distributions of the different risk factors are independently exponentially distributed. Under this setup, we derive the maximum likelihood estimators (MLEs) of the unknown mean parameters of the different causes under the assumption of a cumulative exposure model. The exact distributions of the MLEs of the parameters are then derived through the use of conditional moment generating functions. Using these exact distributions as well as the asymptotic distributions and the parametric bootstrap method, we discuss the construction of confidence intervals for the parameters and assess their performance through Monte Carlo simulations. Finally, we illustrate the methods of inference discussed here with an example.     

A Multivariate Unit Root Test Based on the Modified Weighted Symmetric Estimator for VAR(p)

Multivariate unit root tests for the VAR model have been commonly used in time series analysis. Several unit root tests were developed. Most of the estimators of coefficient matrices developed in the VAR model are obtained using ordinary least squares estimators. In this paper, we suggest a multivariate unit root test based on a modified weighted symmetric estimator. Using a limited Monte Carlo simulation, we compare the powers of the new test statistic and the test statistic suggested in Fuller (1996).     

Unit Root Tests in the Presence of Multi-Variance Break and Level Shifts That Have Power Against the Piecewise Stationary Alternative

Structural breaks in the level as well as in the volatility have often been exhibited in economic time series. In this paper, we propose new unit root tests when a time series has multiple shifts in its level and the corresponding volatility. The proposed tests are Lagrangian multiplier type tests based on the residual's marginal likelihood which is free from the nuisance mean parameters. The limiting null distributions of the proposed tests are the χ²distributions, and are affected not by the size and the location of breaks but only by the number of breaks.

We set the structural breaks under both the null and the alternative hypotheses to relieve a possible vagueness in interpreting test results in empirical work. The null hypothesis implies a unit root process with level shifts and the alternative connotes a stationary process with level shifts. The Monte Carlo simulation shows that our tests are locally more powerful than the OLSE-based tests, and that the powers of our tests, in a fixed time span, remain stable regardless the number of breaks. In our application, we employ the data which are analyzed by Perron (1990), and some results differ from those of Perron's (1990).     

On the asymptotic distribution of a simple unit root test for trending and breaking series

In this paper, we derive the asymptotic distribution of Popp's (2008) innovational outlier unit root test for trending series with a break. The results of Zivot and Andrews (1992) are applied to provide the limiting results of these new test statistics. We tabulate their asymptotic and finite sample critical values, and illustrate the use of the new statistics with an application to the unemployment rate series for 23 OECD countries.     

The effect of temporal aggregation on the estimation accuracy of ARMA models

ABSTRACT

Autoregressive Moving Average (ARMA) time series model fitting is a procedure often based on aggregate data, where parameter estimation plays a key role. Therefore, we analyze the effect of temporal aggregation on the accuracy of parameter estimation of mixed ARMA and MA models. We derive the expressions required to compute the parameter values of the aggregate models as functions of the basic model parameters in order to compare their estimation accuracy. To this end, a simulation experiment shows that aggregation causes a severe accuracy loss that increases with the order of aggregation, leading to poor accuracy.     

The use of temporally aggregated data on detecting a mean change of a time series process

In this article we investigate the effects of temporal aggregation on testing for a mean change of time series through a likelihood ratio (LR) test. We derive the functional relationship between non aggregate-model parameters and aggregate-model parameters. Using the relationship, we propose a modified LR test when aggregate data are used. Through the theory, Monte Carlo simulations, and empirical examples, we show that aggregation leads the null distribution of the LR test statistic being shifted to the left. Hence, the test power increases as the order of aggregation increases.     

Integer autoregressive models with structural breaks

Even though integer-valued time series are common in practice, the methods for their analysis have been developed only in recent past. Several models for stationary processes with discrete marginal distributions have been proposed in the literature. Such processes assume the parameters of the model to remain constant throughout the time period. However, this need not be true in practice. In this paper, we introduce non-stationary integer-valued autoregressive (INAR) models with structural breaks to model a situation, where the parameters of the INAR process do not remain constant over time. Such models are useful while modelling count data time series with structural breaks. The Bayesian and Markov Chain Monte Carlo (MCMC) procedures for the estimation of the parameters and break points of such models are discussed. We illustrate the model and estimation procedure with the help of a simulation study. The proposed model is applied to the two real biometrical data sets.     

The effect of temporal aggregation on the estimation accuracy of time series models

In time series analysis, Autoregressive Moving Average (ARMA) models play a central role. Because of the importance of parameter estimation in ARMA modeling and since it is based on aggregate time series so often, we analyze the effect of temporal aggregation on estimation accuracy. We derive the relationships between the aggregate and the basic parameters and compute the actual values of the former from those of the latter in order to measure and compare their estimation accuracy. We run a simulation experiment that shows that aggregation seriously worsens estimation accuracy and that the impact increases with the order of aggregation.     

Heteroscedasticity in non-stationary time series,some Monte Carlo evidence

A frequent question raised by practitioners doing unit root tests is whether these tests are sensitive to the presence of heteroscedasticity. Theoretically this is not the case for a wide range of heteroscedastic models. However, for some limiting cases such as degenerate and integrated heteroscedastic processes it is not obvious whether this will have an effect. In this paper we report a Monte Carlo study analyzing the implications of various types of heteroscedasticity on three types of unit root tests: The usual Dickey-Fuller test, Phillips' (1987) semi-parametric test and finally a Dickey-Fuller type test using White's (1980) heteroscedasticity consistent standard errors. The sorts of heteroscedasticity we examine are the GARCH model of Bollerslev (1986) and the Exponential ARCH model of Nelson (1991). In particular, we call attention to situations where the conditional variances exhibit a high degree of persistence as is frequently observed for returns of financial time series, and the case where, in fact, the variance process for the first class of models becomes degenerate.     

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.     

A Detrended Range Unit Root (DRUR) Test

We extend the Range Unit Root test in two directions. First, we consider the process with time trend and prove that the modified standardized number of new records converges to a sum of two Rayleigh distributions. Second, more general structures of autocorrelated disturbances are also taken into account. Monte Carlo experiments show the good sample properties of this nonparametric unit root test.     

Generalized fixed‐T panel unit root tests

Panel data unit root tests, which can be applied to data that do not have many time series observations, are based on very restrictive error and deterministic component specification assumptions. In this paper, we develop a new, doubly modified estimator, based on which we propose a panel unit root test that allows for multiple structural breaks, linear and nonlinear trends, heteroscedasticity, serial correlation, and error cross‐section heterogeneity, when the number of time series observations is finite. The test has the additional perk that it is invariant to the initial condition.     

Testing for unit roots in panel data using a GMM approach

For aggregated time series unit root tests are routinely applied to choose among trend and difference stationary models. Recent work demonstrates that such test can also be applied for testing panel data. However, it is well known that disaggregated data often exhibit a considerable amount of heterogeneity so that standard tests may perform poorly. To account for the heterogeneity in the data we allow for individual specific deterministics, that is, we let the time trends vary across the cross section units. It is shown that standard GMM estimators suggested for the dynamic panel data model may fail to give a valid test procedure. To overcome this difficulty, a modified GMM estimator is suggested. In a Monte Carlo study the finite sample properties of the alternative tests are compared.     

The Joy of Copulas: Bivariate Distributions with Uniform Marginals

We describe a class of bivariate distributions whose marginals are uniform on the unit interval. Such distributions are often called “copulas.” The particular copulas we present are especially well suited for use in undergraduate mathematical statistics courses, as many of their basic properties can be derived using elementary calculus. In particular, we show how these copulas can be used to illustrate the existence of distributions with singular components and to give a geometric interpretation to Kendall's tau.     

Robust Stationarity Tests in Seasonal Time Series Processes

This article builds on the existing literature on (stationarity) tests of the null hypothesis of deterministic seasonality in a univariate time series process against the alternative of unit root behavior at some or all of the zero and seasonal frequencies. This article considers the case where, in testing for unit roots at some proper subset of the zero and seasonal frequencies, there are unattended unit roots among the remaining frequencies. Monte Carlo results are presented that demonstrate that in this case, the stationarity tests tend to distort below nominal size under the null and display an associated (often very large) loss of power under the alternative. A modification to the existing tests, based on data prefiltering, that eliminates the problem asymptotically is suggested. Monte Carlo evidence suggests that this procedure works well in practice, even at relatively small sample sizes. Applications of the robustified statistics to various seasonally unadjusted time series measures of U.K. consumers' expenditure are considered; these yield considerably more evidence of seasonal unit roots than do the existing stationarity tests.     

趋势平稳过程与单位根过程的趋势特征研究

为了深入研究具有高次趋势特征序列的单位根(平稳性)检验问题,研究了高次趋势平稳过程和带高次趋势的单位根过程的概念及其时间趋势特征。结果表明,带漂移的单位根过程实际具有线性趋势,带k(k≥1)次趋势的单位根过程实际具有k+1次趋势;而k(k≥0)次(趋势)平稳过程则具有k次趋势。无论是趋势平稳过程,还是单位根过程,都可以通过差分变换确定其时间趋势特征。     

Testing stationarity and trend stationarity against the unit root hypothesis

In this paper we propose a family of relativel simple nonparametrics tests for a unit root in a univariate time series. Almost all the tests proposed in the literature test the unit root hypothesis against the alternative that the time series involved is stationarity or trend stationary. In this paper we take the (trend) stationarity hypothesis as the null and the unit root hypothesis as the alternative. The order differnce with most of the tests proposed in the literature is that in all four cases the asymptotic null distribution is of a well-known type, namely standard Cauchy. In the first instance we propose four Cauchy tests of the stationarity hypothesis against the unit root hypothesis. Under H1 these four test statistics involved, divided by the sample size n, converge weakly to a non-central Cauchy distribution, to one, and to the product of two normal variates, respectively. Hence, the absolute values of these test statistics converge in probability to infinity 9at order n). The tests involved are therefore consistent against the unit root hypothesis. Moreover, the small sample performance of these test are compared by Monte Carlo simulations. Furthermore, we propose two additional Cauchy tests of the trend stationarity hypothesis against the alternative of a unit root with drift.