On forecasting cointegrated seasonal time series

We analyze periodic and seasonal cointegration models for bivariate quarterly observed time series in an empirical forecasting study. We include both single equation and multiple equation methods for those two classes of models. A VAR model in first differences, with and without cointegration restrictions, and a VAR model in annual differences are also included in the analysis, where they serve as benchmark models. Our empirical results indicate that the VAR model in first differences without cointegration is best if one-step ahead forecasts are considered. For longer forecast horizons however, the VAR model in annual differences is better. When comparing periodic versus seasonal cointegration models, we find that the seasonal cointegration models tend to yield better forecasts. Finally, there is no clear indication that multiple equations methods improve on single equation methods.     

Forecasting using a periodic transfer function: with an application to the UK price of ferrous scrap

The familiar concept of cointegration enables us to determine whether or not there is a long-run relationship between two integrated time series. However, this may not capture short-run effects such as seasonality. Two series which display different seasonal effects can still be cointegrated. Seasonality may arise independently of the long-run relationship between two time series or, indeed, the long-run relationship may itself be seasonal. The market for recycled ferrous scrap displays these features: the US and UK scrap prices are cointegrated, yet the local markets exhibit different forms of seasonality. The paper addresses the problem of using both cointegrating and seasonal relationships in forecasting time series through the use of periodic transfer function models. We consider the problems of testing for cointegration between series with differing seasonal patterns and develop a periodic transfer function model for the US and UK scrap markets. Forecast comparisons with other time series models suggest that forecasting efficiency may be improved by allowing for periodicity but that such improvement is by no means guaranteed. The correct specification of the periodic component of the model is critical for forecast accuracy.     

COMMON SEASONAL FEATURES: GLOBAL UNEMPLOYMENT

Seasonal patterns in economic time series are generally examined from a univariate point of view. Using extensions of the unit root literature, important classes of seasonal processes are deterministic, stationary stochastic or mean reverting, and unit root stochastic. Time series tests have been developed for each of these. This paper examines seasonality in a multivariate context. Systems of economic variables can have trends, cycles and unit roots as well as the various types of seasonality. Restrictions such as cointegration and common cycles are here applied also to examine multivariate seasonal behaviour of economic variables. If each of a collection of series has a certain type of seasonality but a linear combination of these series can be found without seasonality, then the seasonal is said to be ‘common’. New tests are developed to determine if seasonal characteristics are common to a set of time series. These tests can be employed in the presence of various other time series structures. The analysis is applied to OECD data on unemployment for the period 1975.1 to 1993.4, and it is found that four diverse countries (Australia, Canada, Japan and USA) not only have common trends in their unemployment, but also have common deterministic seasonal features and a common cycle/stochastic seasonal feature. Such a collection of characteristics were not found in other groups of OECD countries.     

Tests of seasonal integration and cointegration in multivariate unobserved component models

This paper considers tests of seasonal integration and cointegration for multivariate unobserved component models. First, the locally best invariant (LBI) test of the null hypothesis of a deterministic seasonal pattern against the alternative of seasonal integration is derived for a model with Gaussian i.i.d. disturbances and deterministic trend. Then the null hypothesis of seasonal cointegration is considered and a test for common nonstationary components at the seasonal frequencies is proposed. The tests are subsequently generalized to account for stochastic trends, weakly dependent errors and unattended unit roots. Asymptotic representations and critical values of the tests are provided, while the finite sample performance is evaluated by Monte Carlo simulation experiments. Finally, the tests are applied to the series of industrial production of the four largest countries of the European Monetary Union. It is found that Germany does not appear to cointegrate with the other countries at most seasonal frequencies, while there seems to exist a common nonstationary seasonal component between France, Italy and Spain. Copyright © 2006 John Wiley & Sons, Ltd.     

几类非线性协整模型研究综述

协整分析方法经过20多年的发展成为计量经济学界的一个前沿工具,在经济与金融领域得到了广泛的应用。线性协整分析已经成熟,而非线性协整的理论与方法仍在持续研究中。本文回顾了最近20年非线性协整的发展历史,其中包括结构变化、门限非线性、马尔可夫转换和平滑转换等几类非线性协整模型,强调了这些非线性机制的本质区别,总结了已取得的一些重要研究成果,最后对该问题的最新发展动向加以概括。     

A drunk and her dog: a spurious relation? Cointegration tests as instruments to detect spurious correlations between integrated time series

A significant correlation between integrated time series does not necessarily imply a meaningful relation. The relation can also be meaningless, i.e. spurious. Cointegration is sometimes illustrated by the metaphor of ‘a drunk and her dog’. The relation between integrated processes is meaningful, if they are cointegrated. To prevent spurious correlations, integrated series are usually transformed. This implies a loss of information. In case of cointegration, these transformations are no longer necessary. Moreover, it can be shown that cointegration tests are instruments to detect spurious correlations between integrated time series. This paper compares the Dickey–Fuller and the Johansen cointegration test. By means of Monte Carlo simulations, we found that these cointegration tests are a much more accurate alternative for the identification of spurious relations compared to the rather imprecise method of utilizing the R ²-and DW-statistics recommended by some authors. Furthermore, we demonstrate that cointegration techniques are precise methods of distinguishing between spurious and meaningful relations even if the dependency between the processes is very low. Using these tests, the researcher is not in danger of either neglecting a small but meaningful relation or regarding a relation as meaningful which is actually spurious.     

Some thoughts on the development of cointegration

This paper describes how the notion of cointegration came about, and discusses some generalizations to indicate where the topic may go next. In particular, some issues in the analysis of possibly cointegrated quantile time series are discussed.     

Testing for cointegration using partially linear models

A partially linear model of cointegration is developed where stationary covariates enter nonparametrically. We propose tests for cointegration using singular values of the estimated autoregressive matrix. The tests are based on eigenvalues of standardized matrices and are relatively simple to compute. Asymptotic theory of the proposed test is developed. It is shown that the limiting distribution of the proposed test is similar to that of several tests in the recent literature. A Gamma approximation of the distribution is discussed to facilitate inference. Finite sample properties of the proposed procedure are illustrated in some limited Monte Carlo experiments. An empirical application to US macroeconomic time series is conducted to highlight the approach.     

线性和非线性视角下的费雪效应实证检验

基于2000年1月~2009年12月我国名义利率和通货膨胀率均为非平稳时间序列的事实,采用Johansen协整检验和门限协整(threshold cointegration)两种方法对我国是否存在费雪效应进行检验;实证发现,两种方法均支持我国存在弱费雪效应,但得出来的弱费雪效应程度却存在差别,前者认为我国通货膨胀率的变化只有6%反应到名义利率上面,而后者认为这个比例达到42.4%。     

Advances in the Analysis of Non-stationary Time Series: An Illustration of Cointegration and Error Correction Methods in Research on Crime and Immigration

Time series data of interest to social scientists often have the property of random walks in which the statistical properties of the series including means and variances vary over time. Such non-stationary series are by definition unpredictable. Failure to meet the assumption of stationarity in the process of analyzing time series variables may result in spurious and unreliable statistical inferences. This paper outlines the problems of using non-stationary data in regression analysis and identifies innovative solutions developed recently in econometrics. Cointegration and error-correction models have recently received positive attention as remedies to the problems of ``spurious regression' arising from non-stationary series. In this paper, we illustrate the relevant statistical concepts concerning these methods by referring to similar concepts used in cross-sectional analysis. An historical example is used to demonstrate how such techniques are applied. It illustrates that ``foreign' immigrants to Canada (1896–1940) experienced elevated levels of social control in areas of high police discretion. ``Foreign' immigration was unrelated to trends in serious crimes but closely related to vagrancy and drunkenness. The merits of cointegration are compared to traditional approaches to the regression analysis of time series.     

Applied cointegration analysis in the mirror of macroeconomic theory

Cointegration analyses of macroeconomic time series are often not based on fully specified theoretical models. We use a theoretical model to scrutinize common procedures in applied cointegration analysis. Monte Carlo experiments show that (1) some tests of the cointegration vectors do not work well on series generated by an equilibrium business cycle model; (2) cointegration restrictions add little to forecasting; (3) structural VAR models based on weak long-run restrictions seem to work well. The main disadvantages of cointegration analysis without strong links to economic theory are that it makes it hard to estimate and interpret the cointegration vectors.     

On the Role of Seasonal Intercepts in Seasonal Cointegration

In the paper we consider the role of seasonal intercepts in seasonal cointegration analysis. For the nonseasonal unit root, such intercepts can generate a stochastic trend with a drift common to all observations. For the seasonal unit roots, however, we show that unrestricted seasonal intercepts generate trends that are different across the seasons. Since such seasonal trends may not appear in economic data, we propose a modified empirical method to test for seasonal cointegration. We evaluate our method using Monte Carlo simulations and using a four-dimensional data set of Austrian macroeconomic variables.     

向量分整序列非线性变换的研究

自Granger提出整数阶单整向量序列的协整概念以来,协整理论在国内外学者的共同努力下不断得到完善。关于向量时间序列的分数维单整的研究主要集中在分整的线性协整的存在性条件及性质研究。文中通过引入交换条件数学期望算法(ACE),研究分整时间序列的非线性变换的协整性,对最优非线性变换函数进行估计。     

基于Sieve Bootstrap方法的季节时间序列模型检验问题研究

研究目标：完善季节时间序列模型建模理论,解决建模过程烦琐、各类检验方法的结论差异大以及模型误设定问题。研究方法：基于对各季节时间序列模型的数理分析及比较,提出合理的模型检验程序;再运用Sieve Bootstrap方法,给出季节性单位根检验及确定性季节过程检验的统计量的临界值,并比较基于Sieve Bootstrap的检验方法与HEGY检验、BT检验的异同。研究发现：本文提出的检验程序能有效识别模型,检验统计量有限样本性质优良;实证分析表明,本文提出的检验程序及方法能更有效地识别中国宏观经济数据中的季节性。研究创新：将Sieve Bootstrap方法应用于季节时间序列的平稳性检验及趋势性检验中。研究价值：提出季节时间序列模型检验程序及检验方法,促进其在季节性经济数据中的应用。     

A fractional cointegration analysis of Fisher hypothesis: evidence from Turkey

This paper examines the validity of Fisher hypothesis in Turkey over the period from 1990:01 through 2010:03 by using cointegration and fractional cointegration approaches. The findings from Engle and Granger cointegration test indicate that inflation and nominal interest rate series are cointegrated. Since the conventional cointegration tests do not provide strong evidence on the long run relationship, we also use fractional cointegration definition suggested by Cheung and Lai (J Bus Econ Stat 11:103–112, 1993) which requires only a mean reverting (d < 1) relationship between the series. The results from fractional cointegration tests based on GPH and Robinson methods show that inflation and nominal interest rate series are fractionally cointegrated. These findings support the validity of the Fisher hypothesis in Turkey.     

Bayesian point estimation of the cointegration space

A neglected aspect of the otherwise fairly well developed Bayesian analysis of cointegration is point estimation of the cointegration space. It is pointed out here that, due to the well known non-identification of the cointegration vectors, the parameter space is not Euclidean and the loss functions underlying the conventional Bayes estimators are therefore questionable. We present a Bayes estimator of the cointegration space which takes the curved geometry of the parameter space into account. This estimate has the interpretation of being the posterior mean cointegration space and is invariant to the order of the time series, a property not shared with many of the Bayes estimators in the cointegration literature. An overall measure of cointegration space uncertainty is also proposed. Australian interest rate data are used for illustration. A small simulation study shows that the new Bayes estimator compares favorably to the maximum likelihood estimator.     

Inference of Seasonal Cointegration: Gaussian Reduced Rank Estimation and Tests for Various Types of Cointegration*

An extension of Gaussian reduced rank estimation of Ahn and Reinsel (Journal of Econometrics, Vol. 62, pp. 317–350, 1994) to seasonal periods other than four is presented. Simple adjustments for estimation that are necessary because of complex‐valued seasonal unit roots are presented in detail and the asymptotic distribution of the estimators that takes the same form as that in Ahn and Reinsel (1994) is derived. Tests for contemporaneous cointegration and common polynomial cointegrating vectors (PCIVs) for different seasonal unit roots are presented. Finite sample properties are briefly examined through a small Monte Carlo simulation study and a numerical example is presented to illustrate the methods.     

An Econometric Analysis of I(2) Variables

This paper provides a selective survey of the recent literature dealing with I(2) variables in economic time series, that is, processes that require to be differenced twice in order to become stationary. With reference to particular economic models intuition is provided of why I(2)-and polynomial cointegration are features likely to occur in economics. The properties of I(2) series are discussed and I review topics such as: Testing for double unit roots, representations of I(2) cointegrated systems, and hypothesis testing in single equations as well as in systems of equations. Different data sets are used to illustrate the various econometric and statistical techniques.     

Seasonal cointegration and cross-equation restrictions on a forward-looking buffer stock model of money demand

The present study derives a set of cross-equation restrictions imposed on a forward-looking buffer stock model of money demand. Since typically data are seasonally unadjusted for many countries, a seasonal difference rather than the conventional first difference is employed here to compute the growth rate. This seemingly innocuous change in the computation of the growth rate nevertheless makes the multi-period forward-looking money demand equilibrium model substantially different from previous studies. In addition, the related cointegration analysis and implied cross-equation restrictions are also considerably changed. The existence of up to three seasonal unit roots derived from a seasonal difference supports the need for seasonal cointegration, suggesting a new test for the forward-looking equilibrium model. An application of testing such derived cross-equation restrictions to the equilibrium model is illustrated through use of macroeconomic data on Taiwan.     

Common cycles in seasonal non-stationary time series

This paper extends the notion of common cycles to quarterly time series having unit roots both at the zero and seasonal frequencies. It is shown that common cycles are present in the Hylleberg–Engle–Granger–Yoo decomposition of these series when there exists a linear combination of their seasonal differences which follows an MA process of order, at most, three. The pitfalls of seasonal adjustment for common cycles analysis are also documented. Inference on common cycles in seasonally cointegrated series is derived from existing statistical methods for codependence. Concepts and methods are illustrated with an empirical analysis of the comovements between consumption and output using Italian data. Copyright © 1999 John Wiley & Sons, Ltd.