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).     

Long memory versus structural breaks: An overview

We discuss the increasing literature on misspecifying structural breaks or more general trends as long-range dependence. We consider tests on structural breaks in the long-memory regression model as well as the behaviour of estimators of the memory parameter when structural breaks or trends are in the data but long memory is not. Methods for distinguishing both of these phenomena are proposed. The financial support of Volkswagenstiftung is gratefully acknowledged.     

A new nonlinear unit root test with Fourier function

Abstract

Traditional unit root tests display a tendency to be nonstationary in the case of structural breaks and nonlinearity. To eliminate this problem this paper proposes a new flexible Fourier form nonlinear unit root test. This test eliminates this problem to add structural breaks and nonlinearity together to the test procedure. In this test procedure, structural breaks are modeled by means of a Fourier function and nonlinear adjustment is modeled by means of an exponential smooth threshold autoregressive (ESTAR) model. The simulation results indicate that the proposed unit root test is more powerful than the Kruse and KSS tests.     

Bayesian tests for unit root and multiple breaks

A Bayesian approach is considered for identifying sources of nonstationarity for models with a unit root and breaks. Different types of multiple breaks are allowed through crash models, changing growth models, and mixed models. All possible nonstationary models are represented by combinations of zero or nonzero parameters associated with time trends, dummy for breaks, or previous levels, for which Bayesian posterior probabilities are computed. Multiple tests based on Markov chain Monte Carlo procedures are implemented. The proposed method is applied to a real data set, the Korean GDP data set, showing a strong evidence for two breaks rather than the usual unit root or one break.     

Testing for abrupt breaks in variance structures with smooth changes

Abstract

We investigate the problem of testing for variance breaks in the case where the variance structure is assumed to be smoothly time-varying under the null. Since the classical tests are aimed to detect any change in the variance, they are not able to distinguish between smooth non constant variance and abrupt breaks. In this paper a new procedure for detecting variance breaks taking into account for smooth changes in the variance under the null is proposed. The finite sample properties of the test we introduce are investigated by Monte Carlo experiments. The theoretical outputs are illustrated using U.S. macroeconomic data.     

Structural change and unit roots

Recent research (though sometimes disputed) has demonstrated that structural breaks can affect inference about unit roots. The present paper presents a selective survey of the literature on structural breaks, on tests for unit roots and on tests for unit roots under structural breaks. Both the classical and Bayesian literature are reviewed and some suggestions for further research are made.     

Powerful Parametric Tests Based on Sum‐Functions of Spacings

Assume that we have a sequence of n independent and identically distributed random variables with a continuous distribution function F, which is specified up to a few unknown parameters. In this paper, tests based on sum‐functions of sample spacings are proposed, and large sample theory of the tests are presented under simple null hypotheses as well as under close alternatives. Tests, which are optimal within this class, are constructed, and it is noted that these tests have properties that closely parallel those of the likelihood ratio test in regular parametric models. Some examples are given, which show that the proposed tests work also in situations where the likelihood ratio test breaks down. Extensions to more general hypotheses are discussed.     

Unit Roots,Level Shifts,and Trend Breaks in Per Capita Output: A Robust Evaluation

Determining whether per capita output can be characterized by a stochastic trend is complicated by the fact that infrequent breaks in trend can bias standard unit root tests towards nonrejection of the unit root hypothesis. The bulk of the existing literature has focused on the application of unit root tests allowing for structural breaks in the trend function under the trend stationary alternative but not under the unit root null. These tests, however, provide little information regarding the existence and number of trend breaks. Moreover, these tests suffer from serious power and size distortions due to the asymmetric treatment of breaks under the null and alternative hypotheses. This article estimates the number of breaks in trend employing procedures that are robust to the unit root/stationarity properties of the data. Our analysis of the per capita gross domestic product (GDP) for Organization for Economic Cooperation and Development (OECD) countries thereby permits a robust classification of countries according to the “growth shift,” “level shift,” and “linear trend” hypotheses. In contrast to the extant literature, unit root tests conditional on the presence or absence of breaks do not provide evidence against the unit root hypothesis.     

Unit roots and structural breaks: The case of India 1900-1988

SUMMARY This paper tests the hypothesis of difference stationarity of macro-economic time series against the alternative of trend stationarity, with and without allowing for possible structural breaks. The methodologies used are that of Dickey and Fuller familiarized by Nelson and Plosser, and that of dummy variables familiarized by Perron, including the Zivot and Andrews extension of Perron's tests. We have chosen 12 macro-economic variables in the Indian economy during the period 1900-1988 for this study. A study of this nature has not previously been undertaken for the Indian economy. The conventional Dickey-Fuller methodology without allowing for structural breaks cannot reject the unit root hypothesis (URH) for any series. Allowing for exogenous breaks in level and rate of growth in the years 1914, 1939 and 1951, Perron's tests reject the URH for three series after 1951, i.e. the year of introduction of economic planning in India. The Zivot and Andrews tests for endogenous breaks confirm the Perron tests and lead to the rejection of the URH for three more series.     

Unobserved heterogeneity in Markovian analysis of the size distortion of unit root tests

We confirm that units root tests can exhibit substantial size distortion when breaks in mean are generated by a first-order Markov chain, but unlike previous literature, we find augmentation largely remedies this situation. However, considerable heterogeneity is evident in the size properties of the tests when faced with breaks in mean varying in duration, in number and in position within the sample. This heterogeneity will be hidden when a Markov chain is employed. For instance, when the transition probabilities generate single period outliers, rejection frequencies (RFs) rise substantially with the number of outliers, but augmentation results in approximately nominal RFs. Qualitatively similar results hold when a number of structural breaks are allocated randomly in the central section of the sample. Interestingly, very different behaviour is exposed by a design exploring the impact on the tests of two breaks imposed at a range of fixed intervals, RFs rising when break occur in the extremities of the sample, a situation unaffected by augmentation.     

Modified tests for change points in variance in the possible presence of mean breaks

This paper considers the detection problem of variance changes for the time series involving abrupt and/or smooth breaks in mean. Often, in these situations, the tests of choice are based on cumulative sum of squares statistics. We show that the test statistics are not robust in the presence of broken mean and their sizes suffer severe distortions. The adjusted residual-based method is then proposed to eliminate these deficiencies and makes a significant improvement. Finally, simulation results confirm the validity of these modified test statistics, and an empirical data analysis using some stock price series from the Shanghai Stock Exchange is reported.     

Quasi-likelihood ratio tests for cointegration,cobreaking, and cotrending

We consider a set of variables with two types of nonstationary features, stochastic trends and broken linear trends. We develop tests that can determine whether there is a linear combination of these variables under which the nonstationary features can be canceled out. The first test can determine whether stochastic trends can be eliminated and thus whether cointegration holds, regardless of whether structural breaks in linear trends are eliminated. The second test can determine whether both stochastic trends and breaks in linear trends are simultaneously removed and thus whether cointegration and cobreaking simultaneously hold. The third test can determine whether not only breaks in linear trends but also linear trends themselves are eliminated along with stochastic trends and thus whether both cointegration and cotrending hold.     

Embodied Technological Change and Tests of the Internal-Adjustment-Cost Hypothesis

Standard unit-root and cointegration tests are sensitive to atypical events such as outliers and structural breaks. In this article, we use outlier-robust estimation techniques to examine the impact of these events on cointegration analysis. Our outlier-robust cointegration test provides a new diagnostic tool for signaling when standard cointegration results might be driven by a few aberrant observations. A main feature of our approach is that the proposed robust estimator can be used to compute weights for all observations, which in turn can be used to identify the approximate dates of atypical events. We evaluate our method using simulated data and a Monte Carlo experiment. We also present an empirical example showing the usefulness of the proposed analysis.     

A new unit root test with two structural breaks in level and slope at unknown time

In this paper, we propose a new augmented Dickey–Fuller-type test for unit roots which accounts for two structural breaks. We consider two different specifications: (a) two breaks in the level of a trending data series and (b) two breaks in the level and slope of a trending data series. The breaks whose time of occurrence is assumed to be unknown are modeled as innovational outliers and thus take effect gradually. Using Monte Carlo simulations, we show that our proposed test has correct size, stable power, and identifies the structural breaks accurately.     

Detection of structural breaks in a time-varying heteroskedastic regression model

A Bayesian method for estimating a time-varying regression model subject to the presence of structural breaks is proposed. Heteroskedastic dynamics, via both GARCH and stochastic volatility specifications, and an autoregressive factor, subject to breaks, are added to generalize the standard return prediction model, in order to efficiently estimate and examine the relationship and how it changes over time. A Bayesian computational method is employed to identify the locations of structural breaks, and for estimation and inference, simultaneously accounting for heteroskedasticity and autocorrelation. The proposed methods are illustrated using simulated data. Then, an empirical study of the Taiwan and Hong Kong stock markets, using oil and gas price returns as a state variable, provides strong support for oil prices being an important explanatory variable for stock returns.     

Structural changes during a century of the world’s most popular sport

The analysis in this paper employs a methodology for dating structural breaks in tests with non-standard asymptotic distributions. The application examines whether changes in the rules of a game and major social and political events during the past century had significant effects upon various outcomes of this game. The statistical methodology first applied here proves successful as most breaks can be traced to specific events and rule changes. Dating these breaks allows us to obtain useful insights into production and competition processes in this industry. As such, using empirical tests we illustrate the utility of a valuable statistical technique not applied previously.Ignacio Palacios-Huerta: I am grateful to Gary S. Becker, Tony Lancaster, Robin Lumsdaine, Kevin M. Murphy, Gabriel Perez-Quiros, Ana I. Saracho, Amy Serrano, an associate editor and a referee for useful suggestions. I am also indebted to Barry Blake, Vicki Bogan, Salwa Hammami and Karen Wong for able research assistance, Tony Brown at the Association of Football Statisticians for the data the Hoover Institution for its hospitality, and the Spanish Ministerio de Ciencia y Tecnologia for financial support (grant BEC 2003-08182). The Gauss programs used in this paper were kindly provided by Bruce E. Hansen. Any errors are mine alone.     

Estimation of structural mean breaks for long-memory data sets

In long-memory data sets such as the realized volatilities of financial assets, a sequential test is developed for the detection of structural mean breaks. The long memory, if any, is adjusted by fitting an HAR (heterogeneous autoregressive) model to the data sets and taking the residuals. Our test consists of applying the sequential test of Bai and Perron [Estimating and testing linear models with multiple structural changes. Econometrica. 1998;66:47–78] to the residuals. The large-sample validity of the proposed test is investigated in terms of the consistency of the estimated number of breaks and the asymptotic null distribution of the proposed test. A finite-sample Monte-Carlo experiment reveals that the proposed test tends to produce an unbiased break time estimate, while the usual sequential test of Bai and Perron tends to produce biased break times in the case of long memory. The experiment also reveals that the proposed test has a more stable size than the Bai and Perron test. The proposed test is applied to two realized volatility data sets of the S&P index and the Korea won-US dollar exchange rate for the past 7 years and finds 2 or 3 breaks, while the Bai and Perron test finds 8 or more breaks.     

我国通货膨胀结构突变及不确定性检验

 我们利用GARCH (1, 1) 模型对我国通货膨胀率动态过程中的结构转变点进行了样本内及样本外检验,进而对通货膨胀不确定性进行测度。研究发现,我国通货膨胀率序列在1983年1月至2008年5月之间存在一个显著的结构转变,结构转变点发生在1996年1月,这与我国在1996年成功实现经济“软着陆”的事实相一致。基于两个基准模型和五个比较模型在不同预测水平下对样本外数据进行预测所得结果表明,五个比较模型在大多数情况下能够获得小于两个基准模型的均值损失。此外,我们使用多个模型进行联合预测,发现联合预测的结果具有一定的代表性。     

基于傅里叶变换的含确定性趋势结构突变的协整回归模型和不等方差检验

一般来说,数据结构突变点的位置是未知的或突变点的存在性无法准确预知。Enders和Lee（2009,2011）[1][2]证明低频的傅里叶变换（Fourier transformation）就能较精确地处理单位根检验中的数据结构突变（异质结构突变）问题。本文在协整模型框架下,使用傅里叶变换处理协整模型确定性趋势项下的结构突变,考察了协整模型参数的收敛速度,并重新推导了不等方差检验。傅里叶近似项参数的收敛速度为： 。使用蒙特卡洛模拟表明：在缺乏结构突变的先验知识的情况下,使用低频的傅里叶变换能较好地处理协整回归中的确定性趋势的结构突变的问题,显著提高协整向量的估计效率。使用改进后的方法,重新研究了中国股市和国际股市联动关系的密切程度,实证结果更为强烈地支持：中国投资者投资于澳大利亚市场分散风险的收益显著弱于投资其他国际市场。     

Seasonality Tests

This article modifies and extends the test against nonstationary stochastic seasonality proposed by Canova and Hansen. A simplified form of the test statistic in which the nonparametric correction for serial correlation is based on estimates of the spectrum at the seasonal frequencies is considered and shown to have the same asymptotic distribution as the original formulation. Under the null hypothesis, the distribution of the seasonality test statistics is not affected by the inclusion of trends, even when modified to allow for structural breaks, or by the inclusion of regressors with nonseasonal unit roots. A parametric version of the test is proposed, and its performance is compared with that of the nonparametric test using Monte Carlo experiments. A test that allows for breaks in the seasonal pattern is then derived. It is shown that its asymptotic distribution is independent of the break point, and its use is illustrated with a series on U.K. marriages. A general test against any form of permanent seasonality, deterministic or stochastic, is suggested and compared with a Wald test for the significance of fixed seasonal dummies. It is noted that tests constructed in a similar way can be used to detect trading-day effects. An appealing feature of the proposed test statistics is that under the null hypothesis, they all have asymptotic distributions belonging to the Cramér–von Mises family.