Short and long run causality measures: Theory and inference

The concept of causality introduced by Wiener [Wiener, N., 1956. The theory of prediction, In: E.F. Beckenback, ed., The Theory of Prediction, McGraw-Hill, New York (Chapter 8)] and Granger [Granger, C. W.J., 1969. Investigating causal relations by econometric models and cross-spectral methods, Econometrica 37, 424–459] is defined in terms of predictability one period ahead. This concept can be generalized by considering causality at any given horizon h$h$ as well as tests for the corresponding non-causality [Dufour, J.-M., Renault, E., 1998. Short-run and long-run causality in time series: Theory. Econometrica 66, 1099–1125; Dufour, J.-M., Pelletier, D., Renault, É., 2006. Short run and long run causality in time series: Inference, Journal of Econometrics 132 (2), 337–362]. Instead of tests for non-causality at a given horizon, we study the problem of measuring causality between two vector processes. Existing causality measures have been defined only for the horizon 1, and they fail to capture indirect causality. We propose generalizations to any horizon h$h$ of the measures introduced by Geweke [Geweke, J., 1982. Measurement of linear dependence and feedback between multiple time series. Journal of the American Statistical Association 77, 304–313]. Nonparametric and parametric measures of unidirectional causality and instantaneous effects are considered. On noting that the causality measures typically involve complex functions of model parameters in VAR and VARMA models, we propose a simple simulation-based method to evaluate these measures for any VARMA model. We also describe asymptotically valid nonparametric confidence intervals, based on a bootstrap technique. Finally, the proposed measures are applied to study causality relations at different horizons between macroeconomic, monetary and financial variables in the US.     

Another look at Swedish business cycles, 1861–1988

The linearity of nine long Swedish macroeconomic time series, whose business cycle properties were discussed by Englund, Persson, and Svensson (1992), is tested and rejected for all but two. Non-linear (STAR) models are estimated, and their properties are investigated. Business cycle frequency variation does not seem to be constant over time for all series; it is difficult to find a ‘Swedish business cycle’. Pairwise Granger non-causality tests are adapted to the STAR case, and non-causality is tested. The results point at strong temporal interactions and indicate that the functional form (linear or STAR) strongly affects the outcome of these tests. Copyright © 1999 John Wiley & Sons, Ltd.     

A new statistic and practical guidelines for nonparametric Granger causality testing

In this paper we introduce a new nonparametric test for Granger non-causality which avoids the over-rejection observed in the frequently used test proposed by Hiemstra and Jones [1994. Testing for linear and nonlinear Granger causality in the stock price-volume relation. Journal of Finance 49, 1639–1664]. After illustrating the problem by showing that rejection probabilities under the null hypothesis may tend to one as the sample size increases, we study the reason behind this phenomenon analytically. It turns out that the Hiemstra–Jones test for the null of Granger non-causality, which can be rephrased in terms of conditional independence of two vectors X and Z given a third vector Y, is sensitive to variations in the conditional distributions of X and Z that may be present under the null. To overcome this problem we replace the global test statistic by an average of local conditional dependence measures. By letting the bandwidth tend to zero at appropriate rates, the variations in the conditional distributions are accounted for automatically. Based on asymptotic theory we formulate practical guidelines for choosing the bandwidth depending on the sample size. We conclude with an application to historical returns and trading volumes of the Standard and Poor's index which indicates that the evidence for volume Granger-causing returns is weaker than suggested by the Hiemstra–Jones test.     

Testing for linear vector autoregressive dynamics under multivariate generalized autoregressive heteroskedasticity

In this paper, we propose a fixed design wild bootstrap procedure to test parameter restrictions in vector autoregressive models, which is robust in cases of conditionally heteroskedastic error terms. The wild bootstrap does not require any parametric specification of the volatility process and takes contemporaneous error correlation implicitly into account. Via a Monte Carlo investigation, empirical size and power properties of the method are illustrated for the case of white noise under the null hypothesis. We compare the bootstrap approach with standard ordinary least squares (OLS)-based, weighted least squares (WLS) and quasi-maximum likelihood (QML) approaches. In terms of empirical size, the proposed method outperforms competing approaches and achieves size-adjusted power close to WLS or QML inference. A White correction of standard OLS inference is satisfactory only in large samples. We investigate the case of Granger causality in a bivariate system of inflation expectations in France and the United Kingdom. Our evidence suggests that the former are Granger causal for the latter while for the reverse relation Granger non-causality cannot be rejected.     

Monotone dependence in graphical models for multivariate Markov chains

We show that a deeper insight into the relations among marginal processes of a multivariate Markov chain can be gained by testing hypotheses of Granger noncausality, contemporaneous independence and monotone dependence. Granger noncausality and contemporaneous independence conditions are read off a mixed graph, and the dependence of an univariate component of the chain on its parents—according to the graph terminology—is described in terms of stochastic dominance criteria. The examined hypotheses are proven to be equivalent to equality and inequality constraints on some parameters of a multivariate logistic model for the transition probabilities. The introduced hypotheses are tested on real categorical time series.     

Testing conditional independence via empirical likelihood

We construct two classes of smoothed empirical likelihood ratio tests for the conditional independence hypothesis by writing the null hypothesis as an infinite collection of conditional moment restrictions indexed by a nuisance parameter. One class is based on the CDF; another is based on smoother functions. We show that the test statistics are asymptotically normal under the null hypothesis and a sequence of Pitman local alternatives. We also show that the tests possess an asymptotic optimality property in terms of average power. Simulations suggest that the tests are well behaved in finite samples. Applications to some economic and financial time series indicate that our tests reveal some interesting nonlinear causal relations which the traditional linear Granger causality test fails to detect.     

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.     

Keeping track of global trade in real time

This paper builds an innovative composite world trade-cycle index by means of a dynamic factor model for short-term forecasts of world trade growth of both goods and (usually neglected) services. Trade indicators are selected using a multidimensional approach, including Bayesian model averaging techniques, dynamic correlations, and Granger non-causality tests in a linear vector autoregression framework. To overcome real-time forecasting challenges, the dynamic factor model is extended to account for mixed frequencies, to deal with asynchronous data publication, and to include hard and survey data along with leading indicators. Nonlinearities are addressed with a Markov switching model. Pseudo-real-time empirical simulations suggest that: (i) the global trade index is a useful tool for tracking and forecasting world trade in real time; (ii) the model is able to infer global trade cycles very precisely and better than several competing alternatives; and (iii) global trade finance conditions seem to lead the trade cycle, a conclusion that is in line with the theoretical literature.     

Generalized spectral testing for multivariate continuous-time models

We develop an omnibus specification test for multivariate continuous-time models using the conditional characteristic function, which often has a convenient closed-form or can be accurately approximated for many multivariate continuous-time models in finance and economics. The proposed test fully exploits the information in the joint conditional distribution of underlying economic processes and hence is expected to have good power in a multivariate context. A class of easy-to-interpret diagnostic procedures is supplemented to gauge possible sources of model misspecification. Our tests are also applicable to discrete-time distribution models. Simulation studies show that the tests provide reliable inference in finite samples.     

Extended Neyman smooth goodness-of-fit tests,applied to competing heavy-tailed distributions

A simplified version of the Neyman (1937) “Smooth” goodness-of-fit test is extended to account for the presence of estimated model parameters, thereby removing overfitting bias. Using a Lagrange Multiplier approach rather than the Likelihood Ratio statistic proposed by Neyman greatly simplifies the calculations. Polynomials, splines, and the step function of Pearson’s test are compared as alternative perturbations to the theoretical uniform distribution. The extended tests have negligible size distortion and more power than standard tests. The tests are applied to competing symmetric leptokurtic distributions with US stock return data. These are generally rejected, primarily because of the presence of skewness.     

Short run and long run causality in time series: inference

We propose methods for testing hypothesis of non-causality at various horizons, as defined in Dufour and Renault (Econometrica 66, (1998) 1099–1125). We study in detail the case of VAR models and we propose linear methods based on running vector autoregressions at different horizons. While the hypotheses considered are nonlinear, the proposed methods only require linear regression techniques as well as standard Gaussian asymptotic distributional theory. Bootstrap procedures are also considered. For the case of integrated processes, we propose extended regression methods that avoid nonstandard asymptotics. The methods are applied to a VAR model of the US economy.     

Testing of Fractional Cointegration in Macroeconomic Time Series*

We propose in this article a two‐step testing procedure of fractional cointegration in macroeconomic time series. It is based on Robinson's (Journal of the American Statistical Association, Vol. 89, p. 1420) univariate tests and is similar in spirit to the one proposed by Engle & Granger (Econometrica, Vol. 55, p. 251), testing initially the order of integration of the individual series and then, testing the degree of integration of the residuals from the cointegrating relationship. Finite‐sample critical values of the new tests are computed and Monte Carlo experiments are conducted to examine the size and the power properties of the tests in finite samples. An empirical application, using the same datasets as in Engle & Granger (Econometrica, Vol. 55, p. 251) and Campbell & Shiller (Journal of Political Economy, Vol. 95, p. 1062), is also carried out at the end of the article.     

Partial maximum likelihood estimation of spatial probit models

This paper analyzes spatial Probit models for cross sectional dependent data in a binary choice context. Observations are divided by pairwise groups and bivariate normal distributions are specified within each group. Partial maximum likelihood estimators are introduced and they are shown to be consistent and asymptotically normal under some regularity conditions. Consistent covariance matrix estimators are also provided. Estimates of average partial effects can also be obtained once we characterize the conditional distribution of the latent error. Finally, a simulation study shows the advantages of our new estimation procedure in this setting. Our proposed partial maximum likelihood estimators are shown to be more efficient than the generalized method of moments counterparts.     

A consistent nonparametric test for nonlinear causality—Specification in time series regression

Since the pioneering work by Granger (1969), many authors have proposed tests of causality between economic time series. Most of them are concerned only with “linear causality in mean”, or if a series linearly affects the (conditional) mean of the other series. It is no doubt of primary interest, but dependence between series may be nonlinear, and/or not only through the conditional mean. Indeed conditional heteroskedastic models are widely studied recently. The purpose of this paper is to propose a nonparametric test for possibly nonlinear causality. Taking into account that dependence in higher order moments are becoming an important issue especially in financial time series, we also consider a test for causality up to the Kth conditional moment. Statistically, we can also view this test as a nonparametric omitted variable test in time series regression. A desirable property of the test is that it has nontrivial power against T^1/2-local alternatives, where T is the sample size. Also, we can form a test statistic accordingly if we have some knowledge on the alternative hypothesis. Furthermore, we show that the test statistic includes most of the omitted variable test statistics as special cases asymptotically. The null asymptotic distribution is not normal, but we can easily calculate the critical regions by simulation. Monte Carlo experiments show that the proposed test has good size and power properties.     

Modelling security market events in continuous time: Intensity based,multivariate point process models

A continuous time econometric modelling framework for multivariate financial market event (or ‘transactions’) data is developed in which the model is specified via the vector conditional intensity. Generalised Hawkes models are introduced that incorporate inhibitory events and dependence between trading days. Novel omnibus specification tests based on a multivariate random time change theorem are proposed. A bivariate point process model of the timing of trades and mid-quote changes is then presented for a New York Stock Exchange stock and related to the market microstructure literature. The two-way interaction of trades and quote changes in continuous time is found to be important empirically.     

Inference in a synchronization game with social interactions

This paper studies inference in a continuous time game where an agent’s decision to quit an activity depends on the participation of other players. In equilibrium, similar actions can be explained not only by direct influences but also by correlated factors. Our model can be seen as a simultaneous duration model with multiple decision makers and interdependent durations. We study the problem of determining the existence and uniqueness of equilibrium stopping strategies in this setting. This paper provides results and conditions for the detection of these endogenous effects. First, we show that the presence of such effects is a necessary and sufficient condition for simultaneous exits. This allows us to set up a nonparametric test for the presence of such influences, which is robust to multiple equilibria. Second, we provide conditions under which parameters in the game are identified. Finally, we apply the model to data on desertion in the Union Army during the American Civil War, and find evidence of endogenous influences.     

Semiparametric estimation of a bivariate Tobit model

The existing semiparametric estimation literature has mainly focused on univariate Tobit models and no semiparametric estimation has been considered for bivariate Tobit models. In this paper, we consider semiparametric estimation of the bivariate Tobit model proposed by Amemiya (1974), under the independence condition without imposing any parametric restriction on the error distribution. Our estimator is shown to be consistent and asymptotically normal, and simulation results show that our estimator performs well in finite samples. It is also worth noting that while Amemiya’s (1974) instrumental variables estimator (IV) requires the normality assumption, our semiparametric estimator actually outperforms his IV estimator even when normality holds. Our approach can be extended to higher dimensional multivariate Tobit models.     

Searching for the Causal Structure of a Vector Autoregression*

We provide an accessible introduction to graph‐theoretic methods for causal analysis. Building on the work of Swanson and Granger (Journal of the American Statistical Association, Vol. 92, pp. 357–367, 1997), and generalizing to a larger class of models, we show how to apply graph‐theoretic methods to selecting the causal order for a structural vector autoregression (SVAR). We evaluate the PC (causal search) algorithm in a Monte Carlo study. The PC algorithm uses tests of conditional independence to select among the possible causal orders – or at least to reduce the admissible causal orders to a narrow equivalence class. Our findings suggest that graph‐theoretic methods may prove to be a useful tool in the analysis of SVARs.     

Exogeneity in Vector Error Correction Models with Purely Exogenous Long-Run Paths

Existing exogeneity conditions of literature are only sufficient and imply 'overly strong' constraints on long-run parameters. This paper presents some new results on exogeneity in vector error correction models. A key concept of the analysis is the 'purely exogenous long-run path', i.e. a cointegrating vector only including 'exogenous' variables. Extending earlier results of Johansen, S. (1992). 'Cointegration in partial systems and the efficiency of single-equation analysis', Journal of Econometrics , Vol. 52, pp. 389–402 and of Toda and Phillips (1991) . Vector Autoregressions and Causality , Cowles Foundation Discussion Paper, No. 977 among others, we propose a framework based on two canonical representations of the long-run matrix, which can constitute a suitable basis to formulate a necessary and sufficient condition for non-causality as well as a condition for strong exogeneity. An interesting property is that the statistics involved in the sequential procedures for testing these conditions are distributed as χ ² variables and can, therefore, easily be calculated with the usual statistical computer packages, which makes our approach fully operational, empirically. Finally, the power and size distortions of the sequential test procedures are analysed using Monte Carlo experiments.     

The conditional autoregressive Wishart model for multivariate stock market volatility

We propose a Conditional Autoregressive Wishart (CAW) model for the analysis of realized covariance matrices of asset returns. Our model assumes an autoregressive moving average structure for the scale matrix of the Wishart distribution. It accounts for positive definiteness of covariance matrices without imposing parametric restrictions, and can be estimated by Maximum Likelihood. We also propose extensions of the CAW model obtained by including a Mixed Data Sampling (MIDAS) component and Heterogeneous Autoregressive (HAR) dynamics for long-run fluctuations. The CAW models are applied to realized variances and covariances for five New York Stock Exchange stocks.