Quantile cointegrating regression

Quantile regression has important applications in risk management, portfolio optimization, and asset pricing. The current paper studies estimation, inference and financial applications of quantile regression with cointegrated time series. In addition, a new cointegration model with quantile-varying coefficients is proposed. In the proposed model, the value of cointegrating coefficients may be affected by the shocks and thus may vary over the innovation quantile. The proposed model may be viewed as a stochastic cointegration model which includes the conventional cointegration model as a special case. It also provides a useful complement to cointegration models with (G)ARCH effects. Asymptotic properties of the proposed model and limiting distribution of the cointegrating regression quantiles are derived. In the presence of endogenous regressors, fully-modified quantile regression estimators and augmented quantile cointegrating regression are proposed to remove the second order bias and nuisance parameters. Regression Wald tests are constructed based on the fully modified quantile regression estimators. An empirical application to stock index data highlights the potential of the proposed method.     

Nonlinear minimization estimators in the presence of cointegrating relations

In this paper, we consider estimation of a long-run and a short-run parameter jointly in the presence of nonlinearities. The theory developed establishes limit behavior of minimization estimators of the long- and short-run parameters jointly. Typically, if the long-run parameter that is present in a cointegrating relationship is estimated, its estimator will be superconsistent. Therefore, we may conjecture that the joint minimization estimation of both parameters jointly will result in the same limit distribution for the short-run parameter as if the long-run parameter was known. However, we show that unless a regularity condition holds, this intuition is false in general. This regularity condition, that clearly holds in the standard linear case, is identical to the condition for validity of a two-step Granger–Engle type procedure. Also, it is shown that if the cointegrated variables are measured in deviation from their averages, the standard asymptotic normality result (that one would obtain if the long-run parameter was known) holds.     

Two-sample rank tests with truncated populations

LetX ₁,…,X _m andY ₁,…,Y _n be two independent samples from continuous distributionsF andG respectively. Using a Hoeffding (1951) type theorem, we obtain the distributions of the vector S=(S ₍₁₎,…,S _(n)), whereS _(j)=# (X _i ’s≤Y _(j)) andY _(j) is thej-th order statistic ofY sample, under three truncation models: (a)G is a left truncation ofF orG is a right truncation ofF, (b)F is a right truncation ofH andG is a left truncation ofH, whereH is some continuous distribution function, (c)G is a two tail truncation ofF. Exploiting the relation between S and the vectorR of the ranks of the order statistics of theY-sample in the pooled sample, we can obtain exact distributions of many rank tests. We use these to compare powers of the Hajek test (Hajek 1967), the Sidak Vondracek test (1957) and the Mann-Whitney-Wilcoxon test. We derive some order relations between the values of the probagility-functions under each model. Hence find that the tests based onS ₍₁₎ andS _(n) are the UMP rank tests for the alternative (a). We also find LMP rank tests under the alternatives (b) and (c).     

Singular value decomposition in cointegration analysis: a note regarding the difference stationary series

Many methods and techniques have been developed gradually to compute cointegration vectors. We present here a comparatively simple method for computing the matrix of cointegrating vectors, by applying singular value decomposition. With this method, one can easily accommodate in the cointegrating vectors any deterministic factors, such as a dummy, apart from the constant term and the trend. Besides the errors corresponding to the finally selected cointegrating vector have the property of minimum variance. It is noted that this procedure is not mentioned in the relevant literature. Additionally, a sequential technique is introduced, for determining the order of integration for a given series. With this procedure one can directly detect whether the differencing process produces a stationary series or not, since it seems to be a common belief that differencing a variable (one or more times) we will always get a stationary series [Harris, R.: Using Cointegration Analysis in Econometric Modelling. Prentice Hall, London (1995)]. It will be seen that this is not necessarily the case.     

A generalization of Friedman's rank statistic

Abstract  In this paper a very natural generalization of the two-way analysis of variance rank statistic of F riedman is given. The general distribution-free test procedure based on this statistic for the effect of J treatments in a random block design can be applied in general two-way layouts without interactions and with different numbers of the continuous observations per cell provided the design scheme is connected. The asymptotic distribution under the null hypothesis of the test statistic is derived. A comparison with the method of m rankings of B enard and van E lteren is made. The disadvantage of B enard and van E lteren's test procedure is that the number of observations per block does influence the statistic twice, namely firstly by the number itself, as it should, and see ondly by the level of the ranks which will be different in different blocks if the numbers of observations per block are different. The proposed test statistic is not sensitive to differences in the levels of the ranks caused by the different numbers of observations per block. The test is derived from considerhg the K ruskal -W allis statistics per block.
Finally, the results of simulation experiments are given. The simulation is carried out for three designs and a number of normal location alternatives and gives some information about the power of the suggested test procedure. A comparison is made with B enard and van E lteren's test and with the classical analysis of variance technique. For some simple orthogonal designs the exact null distributions of B enard and van E lteren's test and the proposed test are compared.     

Tests for balanced incomplete block ranked data with ties

We consider balanced incomplete block data when ties occur and propose new statistics for testing (a) differences in mean ranks, (b) differences in distributions of ranks, (c) differences in nonlinear effects of ranks and (d) linear contrasts. A sensory evaluation example where the data are ranks is given.     

Locally Most Powerful and Other Rank Tests for Independence – with a Contaminated Weighted Alternative

Several authors in the literature have attempted the quantification of the concept of stochastic dependence for bivariate distribution. Two weighted rank tests for testing independence against a weighted contamination alternative is proposed and their distributional properties are studied. We also derived a locally most powerful rank test for the alternative setting. The rank tests proposed are shown to be asymptotic locally most powerful for specific distributions.     

Determining the cointegrating rank in nonstationary fractional systems by the exact local Whittle approach

We propose to extend the cointegration rank determination procedure of Robinson and Yajima [2002. Determination of cointegrating rank in fractional systems. Journal of Econometrics 106, 217–242] to accommodate both (asymptotically) stationary and nonstationary fractionally integrated processes as the common stochastic trends and cointegrating errors by applying the exact local Whittle analysis of Shimotsu and Phillips [2005. Exact local Whittle estimation of fractional integration. Annals of Statistics 33, 1890–1933]. The proposed method estimates the cointegrating rank by examining the rank of the spectral density matrix of the d$d$th differenced process around the origin, where the fractional integration order, d$d$, is estimated by the exact local Whittle estimator. Similar to other semiparametric methods, the approach advocated here only requires information about the behavior of the spectral density matrix around the origin, but it relies on a choice of (multiple) bandwidth(s) and threshold parameters. It does not require estimating the cointegrating vector(s) and is easier to implement than regression-based approaches, but it only provides a consistent estimate of the cointegration rank, and formal tests of the cointegration rank or levels of confidence are not available except for the special case of no cointegration. We apply the proposed methodology to the analysis of exchange rate dynamics among a system of seven exchange rates. Contrary to both fractional and integer-based parametric approaches, which indicate at most one cointegrating relation, our results suggest three or possibly four cointegrating relations in the data.     

面板协整检验有限样本性质的模拟比较

面板协整检验是基于渐近分布的检验,有限样本下统计量的检验水平和检验功效的表现涉及检验的可靠性。本文针对目前实证研究中应用最广的一类基于残差的统计量及文献中最新提出的基于准残差的统计量进行蒙特卡罗模拟,比较10个检验统计量在不同DGP设定下的检验水平和检验功效,尤其是在DGP误设时的表现。模拟结果表明:基于准残差的面板协整检验大多数情况下有着更好的检验水平和检验功效表现。这一研究为解决实证中面临的统计量可靠性甄别与选择问题提供了依据。     

协整向量误设对相关检验的影响

常用于检验既定协整关系的统计量有tDF和tECM两种，但由于真实数据生成过程未知，估计模型中可能存在一定程度的协整向量误设，从而使统计量的分布特征受到影响。本文首先探讨tDF检验的隐含系数约束α=γ，即短期弹性等于先验长期弹性；其次分析零假设下两种统计量的分布特征，以及先验设定γ对信号噪声比q进而对tECM分布特征的影响；最后在局部备择假设下，给出两种统计量的渐近分布，并表明向量误设会降低协整检验的势，其程度与设定误差d正相关。     

Local asymptotic normality for randomly censored models with applications to rank tests

For randomly right censored models we study the asymptotic behaviour of linear (rank) statistics under local alternatives. The results can be used to evaluate the asymptotic power of the corresponding tests. For instance we treat the question how to choose the best scores in order to derive asymptotically optimal (rank) tests under certain alternatives.     

Forecasting performance of multivariate time series models with full and reduced rank: an empirical examination

Since Quenouille's influential work on multiple time series, much progress has been made towards the goal of parameter reduction and model fit. Relatively less attention has been paid to the systematic evaluation of out-of-sample forecast performance of multivariate time series models. In this paper, we update the hog data set studied by Quenouille (and other researchers who followed him). We re-estimate his model with extended observations (1867–1966), and generate recursive one- to four-steps-ahead forecasts for the period of 1967 through 2000. These forecasts are compared to forecasts from an unrestricted vector autoregression, a reduced rank regression model, an index model and a cointegration-based error correction model. The error correction model that takes into account both nonstationarity of the data and rank reduction performs best at all four forecasting horizons. However, differences among competing models are statistically insignificant in most cases. No model consistently encompasses the others at all four horizons.     

Rearrangements and sequential rank order dominance

Distributive analysis typically involves comparisons of distributions where individuals differ in more than just one attribute. In the particular case where there are two attributes and where the distribution of one of these two attributes is fixed, one can appeal to sequential rank order dominance for comparing distributions. We show that sequential rank order domination of one distribution over another implies that the dominating distribution can be obtained from the dominated one by means of a finite sequence of favourable permutations, and conversely. We provide two examples where favourable permutations prove to have interesting implications from a normative point of view.     

Linear rank tests for comparing treatments with a control when data are subject to unequal patterns of censorship

A class of asymptotically distribution-free tests is considered for comparing several treatments with a control when the, data are subject to unequal right-censorship. A particular member of this class is proposed for use in practice and an illustrative numerical example is given. A general result for the Pitman efficacy of a test based on an asymptotically normal test statistic is proved for the multiparameter case and using this result the efficacy of the proposed class of tests is obtained under sequences of translation and proportional hazards alternatives. Simulation studies are conducted to compare the performance of the proposed test, in terms of power, with some other tests.     

Cointegration in a historical perspective

We analyse the impact of the Engle and Granger (1987) article by means of its citations over time, and find evidence of a second life starting in the new millennium. Next, we propose a possible explanation of the success of this citation classic. We argue that the conditions for its success were just right at the time of its appearance, because of the growing emphasis on time series properties in econometric modelling, the empirical importance of stochastic trends, the availability of sufficiently long macroeconomic time series, and the availability of personal computers and econometric software for carrying out the new techniques.     

A survey of two-sample location-scale problem, asymptotic relative efficiencies of some rank tests

Here the parametric as well as the non parametric approach to the two-sample location-scale problem are reviewed. Special attention is paid to the quadratic form of the linear rank statistics for the location and the scale suggested by Lepage for testing this problem. Several such quadratic forms are compared through ARE computations.     

Asymptotic distribution of the cointegrating vector estimator in error correction models with conditional heteroskedasticity

This paper explores the asymptotic distribution of the cointegrating vector estimator in error correction models with conditionally heteroskedastic errors. Asymptotic properties of the maximum likelihood estimator (MLE) of the cointegrating vector, which estimates the cointegrating vector and the multivariate GARCH process jointly, are provided. The MLE of the cointegrating vector follows mixture normal, and its asymptotic distribution depends on the conditional heteroskedasticity and the kurtosis of standardized innovations. The reduced rank regression (RRR) estimator and the regression-based cointegrating vector estimators do not consider conditional heteroskedasticity, and thus the efficiency gain of the MLE emerges as the magnitude of conditional heteroskedasticity increases. The simulation results indicate that the relative power of the t-statistics based on the MLE improves significantly as the GARCH effect increases.     

Racial harassment and job satisfaction in South Africa: the moderating effects of career orientations and managerial rank

Employee job satisfaction is a strong predictor of pivotal individual and organizational outcomes, e.g. commitment, productivity, retention. This study examines the mediating role of workplace racial harassment in the association of employee racioethnicity and job satisfaction in the South African context. Furthermore, this study investigates the moderating effects of career-related variables, i.e. career orientations and managerial rank. Results of a survey of 154 employees in South Africa indicate that black South Africans experience more workplace racial harassment than white employees and thus show lower job satisfaction. Moreover, the detrimental effects of workplace racial harassment on job satisfaction are more pronounced among highly career-oriented individuals and/or among employees with no or low managerial rank. The implications of these findings for theory and practice are discussed.     

Asymptotic properties of the efficient estimators for cointegrating regression models with serially dependent errors

In this paper, we analytically investigate three efficient estimators for cointegrating regression models: Phillips and Hansen’s [Phillips, P.C.B., Hansen, B.E., 1990. Statistical inference in instrumental variables regression with I(1) processes. Review of Economic Studies 57, 99–125] fully modified OLS estimator, Park’s [Park, J.Y., 1992. Canonical cointegrating regressions. Econometrica 60, 119–143] canonical cointegrating regression estimator, and Saikkonen’s [Saikkonen, P., 1991. Asymptotically efficient estimation of cointegration regressions. Econometric Theory 7, 1–21] dynamic OLS estimator. We consider the case where the regression errors are moderately serially correlated and the AR coefficient in the regression errors approaches 1 at a rate slower than 1/T$1/T$, where T$T$ represents the sample size. We derive the limiting distributions of the efficient estimators under this system and find that they depend on the approaching rate of the AR coefficient. If the rate is slow enough, efficiency is established for the three estimators; however, if the approaching rate is relatively faster, the estimators will have the same limiting distribution as the OLS estimator. For the intermediate case, the second-order bias of the OLS estimator is partially eliminated by the efficient methods. This result explains why, in finite samples, the effect of the efficient methods diminishes as the serial correlation in the regression errors becomes stronger. We also propose to modify the existing efficient estimators in order to eliminate the second-order bias, which possibly remains in the efficient estimators. Using Monte Carlo simulations, we demonstrate that our modification is effective when the regression errors are moderately serially correlated and the simultaneous correlation is relatively strong.     

协整向量、调节参数同为非线性的阈值协整检验

针对经济变量的长期均衡和短期调节关系可能同时存在非线性的事实,本文扩展现有阈值协整模型,提出了协整向量、调节参数都为非线性的阈值协整模型,并着重探讨了该模型的检验方法。研究表明,在协整关系的检验中,Wald统计量有较好的有限样本性质。在协整关系的非线性检验中,LMW和LMG统计量的水平扭曲和检验势都较好。在调节参数的非线性检验中,当调节参数具有显著的非线性时,LMH统计量表现出较好的有限样本性质。