Studying co-movements in large multivariate data prior to multivariate modelling

For non-stationary vector autoregressive models (VAR hereafter, or VAR with moving average, VARMA hereafter), we show that the presence of common cyclical features or cointegration leads to a reduction of the order of the implied univariate autoregressive-integrated-moving average (ARIMA hereafter) models. This finding can explain why we identify parsimonious univariate ARIMA models in applied research although VAR models of typical order and dimension used in macroeconometrics imply non-parsimonious univariate ARIMA representations.     

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.     

On the cointegration of international stock indices

This paper investigates the cointegration relationship among a group of international stock indices in light of new developments of econometric methods. Kasa (1992) first documented strong evidence for cointegration relations among five national stock indices, which suggests that there exists a common trend among those stock indices. Using Johansen multivariate cointegration test, we find that his findings are persistent in a sample of longer periods and more countries. In order to investigate whether these results are driven by statistical biases related to the sample size, we apply to our tests the Johansen’s small sample correction factor. The results still point toward the existence of a cointegration relationship but the evidence becomes much weaker. We next examine the empirical patterns emerged from different lag specifications and argue that Kasa’s findings are more likely due to the size distortion in extreme long lag VAR models. Indeed, when we employ a newly developed non-parametric test that does not require estimation VAR models, the null hypothesis of no cointegration cannot be rejected for the original sample of Kasa’s five-country stock indices from 1974 to 1990, nor for the extended period from 1970 to 2003.     

Stochastic trends and economic fluctuations in a small open economy

In this paper we describe how restricted vector autoregressions can be employed to examine the sources of macroeconomic fluctuations. We show how cointegration restrictions can be used to identify a VAR system with common stochastic trends subject to transitory and permanent changes in average growth, and how we may investigate the system's responses to permanent shocks, i.e. to innovations to the trends. Theoretical cointegration vectors are derived from a small open economy growth model for terms of trade, real GDP, real consumption, and real investments. Applying these methods to Swedish annual data (1875–1986) we find that permanent real (supply) shocks account for most of the fluctuations in GDP, even in the short run.     

Multicointegration in Stock-Flow Models

Multicointegration, in the sense of Granger and Lee (1990), frequently occurs in models of stock-flow adjustment and implies cointegration amongst I(2) variables and their differences (polynomial cointegration). The purpose of this article is two-fold. First, we demonstrate that based on a multicointegrated vector autoregression (VAR) two equivalent error correction model (ECM) representations can be derived; the first is expressed in terms of adjustments in the flows of the variables (the standard I(2) ECM), and the second is expressed in terms of adjustments in both the stocks and the flows. Secondly, we apply I(2) estimation and testing procedures for multicointegrated time series to analyze data for US housing construction. We find that stocks of housing units started and completed exhibit poly- nomial cointegration (and hence the flows are multicointegrated) and the associated ECM's are estimated. Lee (1992, 1996) also found multicointegration in this data set but without explicitly exploiting the I(2) property.     

Efficient tests of long‐run causation in trivariate VAR processes with a rolling window study of the money–income relationship

This paper develops a simple sequential multiple‐horizon non‐causation test strategy for trivariate VAR models (with one auxiliary variable). We apply the test strategy to a rolling window study of money supply and real income, with the price of oil, the unemployment rate and the spread between the Treasury bill and commercial paper rates as auxiliary processes. Ours is the first study to control simultaneously for common stochastic trends, sensitivity of test statistics to the chosen sample period, null hypothesis over‐rejection, sequential test size bounds, and the possibility of causal delays. Evidence suggests highly significant direct or indirect causality from M1 to real income, in particular through the unemployment rate and M2 once we control for cointegration. Copyright © 2007 John Wiley & Sons, Ltd.     

Forecasting with factor-augmented error correction models

As a generalization of the factor-augmented VAR (FAVAR) and of the Error Correction Model (ECM), Banerjee and Marcellino (2009) introduced the Factor-augmented Error Correction Model (FECM). The FECM combines error-correction, cointegration and dynamic factor models, and has several conceptual advantages over the standard ECM and FAVAR models. In particular, it uses a larger dataset than the ECM and incorporates the long-run information which the FAVAR is missing because of its specification in differences. In this paper, we examine the forecasting performance of the FECM by means of an analytical example, Monte Carlo simulations and several empirical applications. We show that FECM generally offers a higher forecasting precision relative to the FAVAR, and marks a useful step forward for forecasting with large datasets.     

Cointegration and joint efficiency of international commodity markets

This paper investigates the semi-strong efficiency hypothesis in the international commodity markets of four industrialized countries, using vector autoregression (VAR) and cointegration techniques. Efficiency in these markets requires the corresponding real exchange rates to be martingales with respect to any information set available in the public domain. In the context of a VAR consisting only of real exchange rates, we show that necessary and sufficient conditions for joint efficiency of all the markets under consideration amount to the VAR being of order one (Markovness) and non-cointegrated. On the contrary, in a VAR extended by other potentially “relevant” variables, such as the corresponding real interest rates, non-cointegration and Markovness are only sufficient conditions for the same commodity markets to be characterized as jointly efficient. We also suggest methods for efficiency testing in each individual market within a cointegrated VAR and, finally, we discuss possible long-run linkages among the real exchange rates and real interest rates in association with efficiency in the commodity markets. JEL Classification Number: F31     

Steady‐state priors for vector autoregressions

Bayesian priors are often used to restrain the otherwise highly over‐parametrized vector autoregressive (VAR) models. The currently available Bayesian VAR methodology does not allow the user to specify prior beliefs about the unconditional mean, or steady state, of the system. This is unfortunate as the steady state is something that economists usually claim to know relatively well. This paper develops easily implemented methods for analyzing both stationary and cointegrated VARs, in reduced or structural form, with an informative prior on the steady state. We document that prior information on the steady state leads to substantial gains in forecasting accuracy on Swedish macro data. A second example illustrates the use of informative steady‐state priors in a cointegration model of the consumption‐wealth relationship in the USA. Copyright © 2009 John Wiley & Sons, Ltd.     

Monte Carlo Evidence on Cointegration and Causation

The small sample performance of Granger causality tests under different model dimensions, degree of cointegration, direction of causality, and system stability are presented. Two tests based on maximum likelihood estimation of error-correction models (LR and WALD) are compared to a Wald test based on multivariate least squares estimation of a modified VAR (MWALD). In large samples all test statistics perform well in terms of size and power. For smaller samples, the LR and WALD tests perform better than the MWALD test. Overall, the LR test outperforms the other two in terms of size and power in small samples.     

SOME REPARAMETERIZATIONS OF LAG POLYNOMIALS FOR DYNAMIC ANALYSIS

ABSTRACT Various reparameterizations of scalar polynomials are considered in the context of lag polynomials. These are used to explore possibilities of testing for stationary autoregressive roots, repeated roots, and polynomial factors of given form. Multivariate generalizations of these results are then applied to VAR models and to comovement between the component series of such systems. The link between the representation of unitroots in the univariate case and cointegration in multivariate systems is demonstrated.     

Common cyclical features analysis in VAR models with cointegration

The paper considers n-dimensional VAR models for variables exhibiting cointegration and common cyclical features. Two specific reduced rank vector error correction models are discussed. In one, named the “strong form” and denoted by SF, the collection of all coefficient matrices of a VECM has rank less than n, in the other, named the “weak form” and denoted by WF, the collection of all coefficient matrices except the matrix of coefficient of error correction terms has rank less than n. The paper explores the theoretical connections between these two forms, suggests asymptotic tests for each form and examines the small sample properties of these tests by Monte Carlo simulations.     

Discussion Paper Abstracts

Some Key Global Adjustment Scenarios and Their Effects on Major Developing Country Regions Forecasting Inflation from the Term Structure: A Cointegration Approach An International CAPM for Bonds and Equities Fiscal and Monetary Policy Under EMU: Credible inflation targets or unpleasant monetary arithmetic? Capital-Skill Complementarity and Relative Employment in West German Manufacturing Estimating Long-run Relationships from Dynamic Heterogeneous Panels Measuring and Forecasting Underlying Economic Activity Discussion Paper No.18–92 Recently, interest in the methodology of constructing coincident economic indicators has been revived by the work of Stock and Watson (1988,1991). They adopt the framework of the state space form and Kalman filter in which to construct an optimal estimate of an unobserved component. This is interpreted as corresponding to underlying economic activity derived from a set of observed indicator variables. In this paper we suggest a modification to the Stock and Watson approach which allows for cointegration between some of the variables. We also discuss the general relationship between cointegration and the appropriate specification of stochastic trend models. The technique is applied to the UK where the observed indicator variables used are those which make up the CSO coincident indicator, therefore constructing alternative measures of economic activity. Two of the calculated series are forecast using a systems VAR with error correction terms, where the VAR consists of the CSO longer leading indicator component variables plus a term structure variable. The derived forecasts represent an alternative longer leading economic indicator. Price and Quantity Responses to Cost and Demand Shocks     

Improved likelihood ratio tests for cointegration rank in the VAR model

We suggest improved tests for cointegration rank in the vector autoregressive (VAR) model and develop asymptotic distribution theory and local power results. The tests are (quasi-)likelihood ratio tests based on a Gaussian likelihood, but as usual the asymptotic results do not require normally distributed innovations. Our tests differ from existing tests in two respects. First, instead of basing our tests on the conditional (with respect to the initial observations) likelihood, we follow the recent unit root literature and base our tests on the full likelihood as in, e.g., Elliott et al. (1996). Second, our tests incorporate a “sign” restriction which generalizes the one-sided unit root test. We show that the asymptotic local power of the proposed tests dominates that of existing cointegration rank tests.     

Semi-nonparametric cointegration testing

This paper considers a semi-nonparametric cointegration test. The test uses the LM-testing principle. The score function needed for the LM-test is estimated from the data using an expansion of the density around a Student t distribution. In this way, we capture both the possible fat-tailedness and the skewness of the innovation process. Using a Monte Carlo experiment, we show that the semi-nonparametric cointegration test has good size and power properties over a broad class of distributions for the innovation process. We also investigate the effect of order selection of the underlying VAR on inference. The complete methodology is illustrated using an interest rate example.     

股指期货的最优套期保值率实证研究——基于沪深300指数期货仿真交易视角

在分析和比较常用的几种股指期货最优套期保值比率确定模型的基础上,基于风险最小化模型框架,利用沪深300指数期货合约模拟运行以来的样本数据,通过最小二乘回归模型、向量自回归模型、误差修正模型以及广义自回归条件异方差模型四种估计方法,对其最优套期保值比率进行了实证测算和绩效比较,提出了相应的政策建议和投资策略。     

Long-run Causality, with an Application to International Links Between Long-term Interest Rates

In this paper we give a precise definition of long-run causality in a multivariate non-stationary, possibly cointegrated, framework. A variable is said to be causal for another in the long-run if knowledge of the past of the former improves long-run predictions of the latter. In a VAR framework, we show that long-run non-causality can be easily tested with a Wald statistics, conditionally on the cointegration rank. The methodology is used to study long-run causal links between US, German, and French long-term interest rates from January 1990 to June 1997.     

Tests of Predictive Ability for Vector Autoregressions Used for Conditional Forecasting

Many forecasts are conditional in nature. For example, a number of central banks routinely report forecasts conditional on particular paths of policy instruments. Even though conditional forecasting is common, there has been little work on methods for evaluating conditional forecasts. This paper provides analytical, Monte Carlo and empirical evidence on tests of predictive ability for conditional forecasts from estimated models. In the empirical analysis, we examine conditional forecasts obtained with a VAR in the variables included in the DSGE model of Smets and Wouters (American Economic Review 2007; 97 : 586–606). Throughout the analysis, we focus on tests of bias, efficiency and equal accuracy applied to conditional forecasts from VAR models. Copyright © 2016 John Wiley & Sons, Ltd.     

The linkage between insurance activity and banking credit: Some evidence from dynamic analysis

This paper investigates the long-run and short-run linkages between insurance activity and banking credit for G-7 countries. To minimize the pretest bias and overcome the structural changes, we adopt the bootstrap Granger causality test applied to full sample and subsamples with a fixed window size. The Johansen cointegration test with GMM-IV estimator finds a long-run positive relation between the series. The full sample results of bootstrap Granger causality test show that there is predictive power from life insurance activity to banking credit only for France and Japan, while the short-run causal relationships between nonlife insurance activity and banking credit are country-specific. However, parameter stability test results suggest that the short-run results in full sample are unreliable. The results of rolling VAR models report that the causal linkages between them are time-varying across various subsamples. These findings offer some useful insights for achieving the co-evolution between insurance and banking credit markets.     

Detecting Co‐Movements in Non‐Causal Time Series

This paper introduces the notion of common non‐causal features and proposes tools to detect them in multivariate time series models. We argue that the existence of co‐movements might not be detected using the conventional stationary vector autoregressive (VAR) model as the common dynamics are present in the non‐causal (i.e. forward‐looking) component of the series. We show that the presence of a reduced rank structure allows to identify purely causal and non‐causal VAR processes of order P>1 even in the Gaussian likelihood framework. Hence, usual test statistics and canonical correlation analysis can be applied, where either lags or leads are used as instruments to determine whether the common features are present in either the backward‐ or forward‐looking dynamics of the series. The proposed definitions of co‐movements are also valid for the mixed causal—non‐causal VAR, with the exception that a non‐Gaussian maximum likelihood estimator is necessary. This means however that one loses the benefits of the simple tools proposed. An empirical analysis on Brent and West Texas Intermediate oil prices illustrates the findings. No short run co‐movements are found in a conventional causal VAR, but they are detected when considering a purely non‐causal VAR.