A SCORE TEST FOR SEASONAL FRACTIONAL INTEGRATION AND COINTEGRATION

This paper develops a time domain score statistic for testing fractional integration at zero and seasonal frequencies in quarterly time series models. Further, it introduces the notion of fractional cointegration at different frequencies between two seasonally integrated, I(1) series. In testing problems involving seasonal fractional cointegration, it is argued that the alternative hypothesis is one-sided for which the usual score test may not be appropriate. Therefore, based on ideas in Silvapulle and Silvapulle (1995), a one-sided score statistic is constructed. A simulation study finds that the score statistic generally has desirable size and power properties in moderately sized samples. The score test is applied to the quarterly Australian consumption function. The income and consumption series are found to be I(1) at zero and seasonal frequencies and these two series are not cointegrated at any frequency.     

A bivariate fractionally cointegrated relationship in the context of cyclical structures

This paper deals with the analysis of cointegration in a bivariate system. However, we depart from the classic concept of cointegration in two aspects. First, we permit fractional degrees of integration in both the parent series and in their linear combination. Second, instead of assuming that the pole or singularity in the spectrum takes places at the zero frequency, we consider the case where the singularity occurs at a frequency λ in the interval (0, π]. We use a procedure that follows the same lines as the two-step testing strategy of R.F. Engle, and C.W.J. Granger, [Cointegration and error correction model. Representation, estimation and testing, Econometrica 55 (1987), pp. 251–276]. Thus, we test first the order of integration in the individual series, which are specified in terms of the Gegenbauer polynomials. Then, if the two series share the same degree of integration at a given frequency, we test the null hypothesis of no cointegration against the alternative of fractional cyclical cointegration, by testing the order of integration on the estimated residuals from the cointegrating regression. Finite sample critical values are obtained, and the power properties of the test are examined. An empirical application is also carried out at the end of the article.     

On the usability of the fluctuation test statistic to identify multiple cointegration break points

The fluctuation test suggested by Hansen and Johansen [Some tests for parameter constancy in cointegrated VAR models, Econometrics J. 2 (1999), pp. 306–333] intends to distinguish between the presence of zero and one break in cointegration relations. In this article, we provide evidence by Monte Carlo simulations that it also serves as a graphical device to detect even multiple break locations. It suffices to consider a simplified and easy-to-implement version of the original fluctuation test. Its break detection performance depends on the sign of change in cointegration parameters and the break height. The sign issue can be approached successfully by a backward application of the test statistic. If breaks are observable, the break locations are detected at the true location on average. We apply the graphical procedure to assess the cointegration of bond yields of Spain, Italy and Portugal with German yields for the period 1995–2013 which is surprisingly supported by the trace test. However, the recursive cointegration approach shows that a stable relationship with German yields is only present for sub-periods between the introduction of the Euro and the global financial crisis which is in line with expectations. The statistical robustness of these results is supported by a forward and backward application of the cointegration breakdown test by Andrews and Kim [Tests for cointegration breakdown over a short time period, J. Bus. Econom. Stat. 24 (2006), pp. 379–394].     

Panel Seasonal Unit Root Test: Further Simulation Results and An Application to Unemployment Data

Summary: In this paper the seasonal unit root test of Hylleberg et al. (1990) is generalized to cover a heterogenous panel. The procedure follows the work of Im, Pesaran and Shin (2002) and is independently proposed by Otero et al. (2004). Test statistics are given and critical values are obtained by simulation. Moreover, the properties of the tests are analyzed for different deterministic and dynamic specifications. Evidence is presented that for a small time series dimension the power is low even for increasing cross section dimension. Therefore, it seems necessary to have a higher time series dimension than cross section dimension. The test is applied to unemployment data in industrialized countries. In some cases seasonal unit roots are detected. However, the null hypotheses of panel seasonal unit roots are rejected. The null hypothesis of a unit root at the zero frequency is not rejected, thereby supporting the presence of hysteresis effects. * The research of this paper was supported by the Deutsche Forschungsgemeinschaft. The paper was presented at the workshop “Unit roots and cointegration in panel data” in Frankfurt, October 2004 and in the poster-session at the EC2 meeting in Marseille, December 2004. We are grateful to the participants of the workshops and an anonymous referee for their helpful comments.     

Cointegration analysis with state space models

This paper presents and exemplifies results developed for cointegration analysis with state space models by Bauer and Wagner in a series of papers. Unit root processes, cointegration, and polynomial cointegration are defined. Based upon these definitions, the major part of the paper discusses how state space models, which are equivalent to VARMA models, can be fruitfully employed for cointegration analysis. By detailing the cases most relevant for empirical applications, the I(1), multiple frequency I(1), and I(2) cases, a canonical representation is developed and thereafter some available statistical results are briefly discussed.     

A residual-based test of the null of cointegration in panel data

This paper proposes a residual-based Lagrange Multiplier (LM) test for the null of cointegration in panel data. The test is analogous to the locally best unbiased invariant (LBUI) for a moving average (MA) unit root. The asymptotic distribution of the test is derived under the null. Monte Carlo simulations are performed to study the size and power properties of the proposed test.

overall, the empirical sizes of the LM-FM and LM-DOLs are close to the true size even in small samples. The power is quite good for the panels where T ≥ 50, and decent with panels for fewer observation in T. In our fixed sample of N = 50 and T = 50, the presence of a moving average and correlation between the LM-DOLS test seems to be better at correcting these effects, although in some cases the LM-FM test is more powerful.

Although much of the non-stationary time series econometrics has been criticized for having more to do with the specific properties of the data set rather than underlying economic models, the recent development of the cointegration literature has allowed for a concrete bridge between economic long run theory and time series methods. Our test now allows for the testing of the null of cointegration in a panel setting and should be of considerable interest to economists in a wide variety of fields.     

Testing for structural change in cointegrated regression models: some comparisons and generalizations

This paper compares and generalizes some testing procedures for structural change in the context of cointegrated regression models. The Lagrange Multiplier (LM) tests proposod by Hansen (1992) are generalized to testing for partial structural change. An exponential average LM test is also suggested following the idea of Andrews and Ploberger (1992). In particular, an optimal test for cointegration is developed. We also propose a new cointegration test which is robust to a possible one-time discrete jump in the intercept. We tabulate the asymptotic critical values for the above tests and conduct a small Monte Carlo simulation to investigate their finite sample performance.     

Testing for structural change in cointegrated regression models: some comparisons and generalizations

This paper compares and generalizes some testing procedures for structural change in the context of cointegrated regression models. The Lagrange Multiplier (LM) tests proposod by Hansen (1992) are generalized to testing for partial structural change. An exponential average LM test is also suggested following the idea of Andrews and Ploberger (1992). In particular, an optimal test for cointegration is developed. We also propose a new cointegration test which is robust to a possible one-time discrete jump in the intercept. We tabulate the asymptotic critical values for the above tests and conduct a small Monte Carlo simulation to investigate their finite sample performance.     

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。     

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     

基于半参数混频误差修正模型的中国CPI预测研究

考虑宏观经济变量具有明显的非线性特征,将非线性误差修正项引入存在协整关系的非平稳混频数据抽样（MIDAS）模型中,构建半参数混频数据抽样误差修正（SEMI-ECM-MIDAS）模型。使用广义似然比（GLR）检验,拓展了混频数据下模型函数形式的一致性检验问题。模拟结果表明SEMI-ECM-MIDAS模型对存在非线性误差修正机制的数据具有显著的预测优势。最后使用该模型研究中国股票市场周度数据、广义货币发行量月度数据和国际原油市场月度数据对中国CPI的短期预测效果。基于AIC准则,对包含半参数模型在内的4种混频数据抽样模型和2种同频模型的连续预测效果进行了全面的比较。研究结果发现：GLR检验表明误差修正项具有明显的非线性特征且在回归中具有显著的反向修正机制,无论采用递归样本、滚动样本还是固定样本,本文提出的SEMI-ECM-MIDAS模型在进行连续预测时均具有最优的预测精度,且预测结果不受混频动态协整关系选择的影响。     

The effects of additive outliers on the seasonal KPSS test: a Monte Carlo analysis

This article builds on the test proposed by Lyhagen [The seasonal KPSS statistic, Econom. Bull. 3 (2006), pp. 1–9] for seasonal time series and having the null hypothesis of level stationarity against the alternative of unit root behaviour at some or all of the zero and seasonal frequencies. This new test is qualified as seasonal-frequency Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test and it is not originally supported by a regression framework.

The purpose of this paper is twofold. Firstly, we propose a model-based regression method and provide a clear illustration of Lyhagen's test and we establish its asymptotic theory in the time domain. Secondly, we use the Monte Carlo method to study the finite-sample performance of the seasonal KPSS test in the presence of additive outliers. Our simulation analysis shows that this test is robust to the magnitude and the number of outliers and the statistical results obtained cast an overall good performance of the test finite-sample properties.     

The Effects of Ignoring Level Shifts on Systems Cointegration Tests

Summary: In this paper I analyse the effects of ignoring level shifts in the data generating process on systems cointegration tests that do not accommodate level shifts. I consider two groups of Likelihood Ratio tests based on procedures suggested by Johansen (1988) and Saikkonen and Lütkepohl (2000b). The Monte Carlo analysis reveals that ignoring level shifts reduces the tests’ sizes to zero and causes an important drop in the small sample power for increasing shift magnitudes. This suggests that one should apply test procedures, which take account of level shifts. * This paper is a revised and summarized version of Chapter 3 of my PhD thesis (Trenkler, 2002). I would like to thank two anonymous referees for helpful comments on the submitted paper. Furthermore, I am grateful to Christian Müller, Ralf Brüggemann, and Helmut Lütkepohl for many useful suggestions and comments on an earlier version of the paper and the corresponding chapter of my thesis. The research was supported by the Deutsche Forschungsgemeinschaft (DFG) through the SFB 373 “Quantification and Simulation of Economic Processes” and the SFB 649 “Economic Risk”.     

Comparisons of tests for multivariate cointegration

This paper compares the small sample properties of different tests for multivariate cointegration like Johansen's trace test, stock &; Watson's common trend test, Phillips &; Ouliaris' principal component test, as well as cointegration rank decisions based on order selection criteria. Under the null hypothesis of non-cointegration we find a slow convergence rate of the test statistics. In bivariate models the Phillips &; Ouliaris test is extremely dependent on the specification and is outperformed by the other procedures. For trivariate processes we find dependence of the power results on the dynamic specification. The lag order is successfully estimated by order selection criteria.     

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。     

Size and power of some cointegration tests under structural breaks and heteroskedastic noise

In this paper we examine the power and size distortions of a number of representative cointegration tests in some large-scale Monte Carlo simulations, when the underlying system is subjected to regime shifts and conditional (ARCH-type) heteroskedasticity. Following the suggestion by Gregory and Hansen, we select the minimum (maximum for the Johansen test) value for the statistics evaluated over a set of tentative break points for the regime shifts. The performance of these statistics is compared to the corresponding ordinary statistics in the presence of regime shifts. We demonstrate that the size of the cointegration tests is severely distorted by conditional heteroskedasticity.     

A test for fractional cointegration using the sieve bootstrap

The concept of fractional cointegration (Cheung and Lai in J Bus Econ Stat 11:103–112, 1993) has been introduced to generalize traditional cointegration (Engle and Granger in Econometrica 55:251–276, 1987) to the long memory framework. In this work we propose a test for fractional cointegration with the sieve bootstrap and compare by simulations the performance of our proposal with other semiparametric methods existing in literature: the three steps technique of Marinucci and Robinson (J Econom 105:225–247, 2001) and the procedure to determine the fractional cointegration rank of Robinson and Yajima (J Econom 106:217–241, 2002).     

Modification of autoregressive fractionally integrated moving average models for the estimation of persistence

In this paper, it is proposed to modify autoregressive fractionally integrated moving average (ARFIMA) processes by introducing an additional parameter to comply with the criticism of Hauser et al . (1999) that ARFIMA processes are not appropriate for the estimation of persistence, because of the degenerate behavior of their spectral densities at frequency zero. When fitting these modified ARFIMA processes to the US GNP, it turns out that the estimated spectra are very similar to those obtained with conventional ARFIMA models, indicating that, in this special case, the disadvantage of ARFIMA models cited by Hauser et al. (1999) does not seriously aff ect the estimation of persistence. However, according to the results of a goodness-of-fit test applied to the estimated spectra, both the ARFIMA models and the modified ARFIMA models seem to overfit the data in the neighborhood of frequency zero.     

No-cointegration test based on fractional differencing: Some Monte Carlo results

This paper examines the use of the t-statistic in the Geweke–Porter-Hudak regression for the estimation of the fractional differencing parameter as a test for cointegration. The critical values of the test statistic are estimated using Monte Carlo methods. The results confirm that the test will over-reject the null hypothesis of no cointegration if the standard-normal critical values are used. The estimated critical values are generally robust to the nuisance parameters in the autoregressive or moving average specification of the error process of the component time series. Exceptions occur when the dependent variable in the cointegration regression follows an autoregressive process with a large positive parameter or a moving average process with a large negative parameter.     

北京市能源消费与经济增长关系研究

 本文利用面板协整理论和基于面板误差修正模型的Granger因果关系检验分析了北京市能源消费和经济增长的关系。面板协整检验表明北京市能源消费和经济增长之间存在长期协整关系,进一步基于面板误差修正模型的Granger因果关系检验表明北京市短期存在能源消费到经济增长的单向因果关系,长期能源消费和经济增长之间存在双向的因果关系。因此北京市在进行节能减排的工作时,必须考虑到能源消费减少对经济增长的负作用,尽可能采取提高能源利用效率的措施,包括利用财政税收优惠政策鼓励节能技术的研发,在政府采购时要求产品在生产过程中采用节能技术,更关键也是更重要的是积极探索能源价格机制改革,通过价格手段促进企业真正具备节能意识,主动节约能源,提高利用效率。