SEPARATION, WEAK EXOGENEITY, AND P-T DECOMPOSITION IN COINTEGRATED VAR SYSTEMS WITH COMMON FEATURES

The aim of this paper is to study the concept of separability in multiple nonstationary time series displaying both common stochastic trends and common stochastic cycles. When modeling the dynamics of multiple time series for a panel of several entities such as countries, sectors, firms, imposing some form of separability and commonalities is often required to restrict the dimension of the parameter space. For this purpose we introduce the concept of common feature separation and investigate the relationships between separation in cointegration and separation in serial correlation common features. Loosely speaking we investigate whether a set of time series can be partitioned into subsets such that there are serial correlation common features within the sub-groups only. The paper investigates three issues. First, it provides conditions for separating joint cointegrating vectors into marginal cointegrating vectors as well as separating joint short-term dynamics into marginal short-term dynamics. Second, conditions for making permanent-transitory decompositions based on marginal systems are given. Third, issues of weak exogeneity are considered. Likelihood ratio type tests for the different hypotheses under study are proposed. An empirical analysis of the link between economic fluctuations in the United States and Canada shows the practical relevance of the approach proposed in this paper.     

Functional-coefficient cointegration models in the presence of deterministic trends

In this article, we extend the functional-coefficient cointegration model (FCCM) to the cases in which nonstationary regressors contain both stochastic and deterministic trends. A nondegenerate distributional theory on the local linear (LL) regression smoother of the FCCM is explored. It is demonstrated that even when integrated regressors are endogenous, the limiting distribution is the same as if they were exogenous. Finite-sample performance of the LL estimator is investigated via Monte Carlo simulations in comparison with an alternative estimation method. As an application of the FCCM, electricity demand analysis in Illinois is considered.     

Cointegration in VAR(1) process: Characterization and testing

In this paper we taken a rather different approach to the concept of cointegration (comparated to existing literature) by focusing on the distance norm of an appropriately defined stochastic process (the first differences of one series) and a closed linear subspace defined from the first differences of the other series. The main result contained in Theorem 2 states that, within a VAR(l) framework, two series are cointegrated if and only if this distance is smaller than the standard deviation of the former process. It links cointegration to the evaluation of the distance between two information sets concerning the short-run dynamic paths of the variables. Hence cointegration can be detected by the differenced series. We, also propose a test for cointegration     

中国费雪效应的门限协整检验

由于中国费雪效应的研究结果具有很大的不一致性,结合中国1991年1月至2008年12月之间的数据,应用可以刻画变量间非线性均衡关系的门限协整理论检验费雪效应,研究结果显示：第一,中国的名义利率与通货膨胀率均为单位根过程,二者之间不存在线性协整关系,而是存在两个门限值的门限协整关系;第二,当通货膨胀率小于-0．8％时,中国费雪效应不存在,而当通货膨胀率在-0．8％～12．03％2;间时,中国存在值为0．42的部分费雪效应;当通货膨胀率大于12．03％时,中国存在值为0．05的部分费雪效应。     

Tests for Parameter Instability in Regressions with 1(1) Processes

This article derives the large-sample distributions of Lagrange multiplier (LM) tests for parameter instability against several alternatives of interest in the context of cointegrated regression models. The fully modified estimator of Phillips and Hansen is extended to cover general models with stochastic and deterministic trends. The test statistics considered include the SupF test of Quandt, as well as the LM tests of Nyblom and of Nabeya and Tanaka. It is found that the asymptotic distributions depend on the nature of the regressor processes—that is, if the regressors are stochastic or deterministic trends. The distributions are noticeably different from the distributions when the data are weakly dependent. It is also found that the lack of cointegration is a special case of the alternative hypothesis considered (an unstable intercept), so the tests proposed here may also be viewed as a test of the null of cointegration against the alternative of no cointegration. The tests are applied to three data sets—an aggregate consumption function, a present value model of stock prices and dividends, and the term structure of interest rates.     

我国费雪效应的非参数检验

本文基于我国1990:01—2007:04期间的名义利率与通货膨胀率月度数据非线性变化的特征,应用非参数单位根和非参数协整理论检验我国是否存在费雪效应, 进而应用非参数局部线性变窗宽估计计算我国的费雪系数。由此产生的结论为：第一,非参数单位根检验发现我国名义利率与通货膨胀率都是非平稳的单位根过程;第二,非参数协整检验的结论为, 我国名义利率与通胀变化率之间存在长期的非线性协整关系, 这一结论表明我国至少存在弱的费雪效应;第三,非参数局部线性变窗宽估计计算的费雪效应(系数)的均值为0.4055,这一结果进一步支持我国存在弱的费雪效应,其隐含的意义为,当前加息对稳定通胀将产生正面效应,进一步, 如适时适度的调整利率, 很可能抑制当前较高的CPI向高通胀的转化。     

Linear and segmented trends in sea surface temperature data

This paper deals with the analysis of the MET Office Hadley Centre's sea surface temperature data set (HadSST3) by using long-range dependence techniques. We incorporate linear and segmented trends using fractional integration, and thus permitting long memory behavior in the detrended series. The results indicate the existence of warming trends in the three series examined (Northern and Southern Hemispheres along with global temperatures), with orders of integration which are in the range (0.5, 1) and thus implying nonstationary long memory and mean reverting behavior. This is innovative compared with other works that assume short memory behavior in the detrended series. Allowing for segmented trends two features are observed: increasing values in the degree of dependence of the series across time and significant warming trends from 1940 onwards.     

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.     

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.     

The role of information in nonstationary regression

The role of standard likelihood-based measures of information and efficiency is unclear when regressions involve nonstationary data. Typically the standardized score is not asymptotically Gaussian and the standardized Hessian has a stochastic, rather than deterministic limit. Here we consider a time series regression involving a deterministic covariate which can be evaporating, slowly evolving or nonstationary. It is shown that conditional information, or equivalently, profile Kullback–Leibler and Fisher information remain informative about both the accuracy, i.e. asymptotic variance, of profile maximum likelihood estimators, and the power of point optimal invariant tests for a unit root. Specifically, these information measures indicate fractional, rather than linear trends that may minimize inferential accuracy. Such is confirmed in a numerical experiment.     

Graphical Techniques for Selecting Explanatory Variables for Time Series Data

Bayesian model building techniques are developed for data with a strong time series structure and possibly exogenous explanatory variables that have strong explanatory and predictive power. The emphasis is on finding whether there are any explanatory variables that might be used for modelling if the data have a strong time series structure that should also be included. We use a time series model that is linear in past observations and that can capture both stochastic and deterministic trend, seasonality and serial correlation. We propose the plotting of absolute predictive error against predictive standard deviation. A series of such plots is utilized to determine which of several nested and non-nested models is optimal in terms of minimizing the dispersion of the predictive distribution and restricting predictive outliers. We apply the techniques to modelling monthly counts of fatal road crashes in Australia where economic, consumption and weather variables are available and we find that three such variables should be included in addition to the time series filter. The approach leads to graphical techniques to determine strengths of relationships between the dependent variable and covariates and to detect model inadequacy as well as determining useful numerical summaries.     

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。     

A bayesian predictive approach to the selection of variables in multiple regression

The squared error loss function applied to Bayesian predictive distributions is investigated as a variable selection criterion in linear regression equations. It is illustrated that “cost-free” variables may be eliminated if they are poor predictors. Regression models where the predictors are fixed and where they are stochastic are both considered. An empirical examination of the criterion and a comparison with other techniques are presented.     

Lagrance-multiplier tersts for weak exogeneity: a synthesis

This paper unifies two seemingly separate approaches to test weak exogeneity in dynamic regression models with Lagrange-mulptiplier statistics. The first class of tests focuses on the orthogonality between innovations and conditioning variables, and thus is related to the Durbin-Wu-Hausman specification test. The second approach has been developed more recently in the context of context of cointegration and error correction models, ad concentrates on the question whether the conditioning variables display error correction behaviour. It is shown that the vital difference between the two approaches stems from the choice of the parmeters of interest. A new test is derived, which encompasses both its predecessors. The test is applied to an error correction model of the demand for money in Switzerland.     

Lagrance-multiplier tersts for weak exogeneity: a synthesis

This paper unifies two seemingly separate approaches to test weak exogeneity in dynamic regression models with Lagrange-mulptiplier statistics. The first class of tests focuses on the orthogonality between innovations and conditioning variables, and thus is related to the Durbin-Wu-Hausman specification test. The second approach has been developed more recently in the context of context of cointegration and error correction models, ad concentrates on the question whether the conditioning variables display error correction behaviour. It is shown that the vital difference between the two approaches stems from the choice of the parmeters of interest. A new test is derived, which encompasses both its predecessors. The test is applied to an error correction model of the demand for money in Switzerland.     

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.     

非线性协整检验与“费雪效应”机制分析

本文分别在线性Engle-Granger协整模型和非线性指数平滑迁移自回归误差修正模型 (ESTAR-ECM) 的框架下,对我国名义利率与通货膨胀率序列进行了长期均衡关系的检验。发现线性协整模型不能捕捉到我国名义利率与通货膨胀率的长期均衡关系,而对于ESTAR-ECM模型,无论利用商业银行1年期贷款利率还是7天期银行间同业拆借利率作为名义利率的代理变量,均证实名义利率与通货膨胀率具有长期稳定的均衡关系,表明“费雪效应”在我国是成立的。但由于“费雪效应”系数小于1,表明名义利率与通货膨胀率之间仅存在弱的“费雪效应”。其意义在于,我国利率政策对稳定通胀预期、抑制通货膨胀具有一定的正面效应,但由于利率对通货膨胀反应不足,导致完全依靠利率政策控制目前较高的通货膨胀有一定的困难。     

Cointegration Detection Using Dynamic Factor Models

A method of information-criterion-based cointegration detection using dynamic factor models is proposed. The results of the data-based and non data-based Monte Carlo simulations suggest that this method is as effective as conventional hypothesis-testing methods. In the proposed method, an observed multivariate time series is described in terms of common stochastic trends plus stationary autoregressive cycles. Then the best model is selected from among alternative models obtained by changing the number of common stochastic trends, on the basis of information criteria. Consequently, the cointegration rank is determined on the basis of the selected model. Two advantages of the proposed method are also discussed.     

Deterministic versus stochastic seasonal fractional integration and?structural breaks

This paper considers a general model which allows for both deterministic and stochastic forms of seasonality, including fractional (stationary and nonstationary) seasonal orders of integration, and also incorporating endogenously determined structural breaks. Monte Carlo analysis shows that, in the case of a single break, the suggested procedure performs well even in small samples, accurately capturing the seasonal properties of the series, and correctly detecting the break date. As an illustration, the model is estimated using four US series (output, consumption, imports and exports). The results suggest that the seasonal patterns of these variables have changed over time: specifically, in the second subsample the systematic component of seasonality becomes insignificant, whilst the degree of persistence increases.     

基于LSTAR模型的非线性协整检验

大量的经济理论和实践都表明,宏观经济时间序列经常会出现非平稳和非线性特征,因而在统计分析时,需要进行非线性协整检验。基于逻辑平滑转换自回归(LSTAR)模型将传统的线性协整表述方法拓展为非线性形式,构造实用的检验程序及合适的统计量,利用软件R进行蒙特卡洛模拟给出非线性协整检验统计量的临界值,并通过实际数据分析购买力平价动态系统的非线性协整关系,说明方法的有效性。