Granger Causality Test in the Presence of Spillover Effects

In this article, we investigate the effect of spillover (i.e., causality in variance) on the reliability of Granger causality test based on ordinary least square estimates. We studied eight different versions of the test both, with and without Whites heteroskedasticity consistent covariance matrix (HCCME). The properties of the tests are investigated by means of a Monte Carlo experiment where 21 different data generating processes (DGP) are used and a number of factors that might affect the test are varied. The result shows that the best choice to test for Granger causality under the presence of spillover is the Lagrange Multiplier test with HCCME.     

Measuring Nonlinear Granger Causality in Mean

We propose model-free measures for Granger causality in mean between random variables. Unlike the existing measures, ours are able to detect and quantify nonlinear causal effects. The new measures are based on nonparametric regressions and defined as logarithmic functions of restricted and unrestricted mean square forecast errors. They are easily and consistently estimated by replacing the unknown mean square forecast errors by their nonparametric kernel estimates. We derive the asymptotic normality of nonparametric estimator of causality measures, which we use to build tests for their statistical significance. We establish the validity of smoothed local bootstrap that one can use in finite sample settings to perform statistical tests. Monte Carlo simulations reveal that the proposed test has good finite sample size and power properties for a variety of data-generating processes and different sample sizes. Finally, the empirical importance of measuring nonlinear causality in mean is also illustrated. We quantify the degree of nonlinear predictability of equity risk premium using variance risk premium. Our empirical results show that the variance risk premium is a very good predictor of risk premium at horizons less than 6 months. We also find that there is a high degree of predictability at the 1-month horizon, that can be attributed to a nonlinear causal effect. Supplementary materials for this article are available online.     

Adaptive estimation of vector autoregressive models with time-varying variance: Application to testing linear causality in mean

Linear vector autoregressive (VAR) models where the innovations could be unconditionally heteroscedastic are considered. The volatility structure is deterministic and quite general, including breaks or trending variances as special cases. In this framework we propose ordinary least squares (OLS), generalized least squares (GLS) and adaptive least squares (ALS) procedures. The GLS estimator requires the knowledge of the time-varying variance structure while in the ALS approach the unknown variance is estimated by kernel smoothing with the outer product of the OLS residual vectors. Different bandwidths for the different cells of the time-varying variance matrix are also allowed. We derive the asymptotic distribution of the proposed estimators for the VAR model coefficients and compare their properties. In particular we show that the ALS estimator is asymptotically equivalent to the infeasible GLS estimator. This asymptotic equivalence is obtained uniformly with respect to the bandwidth(s) in a given range and hence justifies data-driven bandwidth rules. Using these results we build Wald tests for the linear Granger causality in mean which are adapted to VAR processes driven by errors with a nonstationary volatility. It is also shown that the commonly used standard Wald test for the linear Granger causality in mean is potentially unreliable in our framework (incorrect level and lower asymptotic power). Monte Carlo experiments illustrate the use of the different estimation approaches for the analysis of VAR models with time-varying variance innovations.     

A Wavelet-Based Approach of Testing for Granger Causality in the Presence of GARCH Effects

The size and power of the most commonly used tests and a new wavelet-based approach of testing for Granger causality is evaluated by means of a Monte Carlo study in which the error term follows a generalized autoregressive conditional heteroscedasticity consistent (GARCH) process. In the simulation study it is shown that the commonly used causality tests tend to overreject the true null hypothesis in the presence of GARCH errors and that the new wavelet-based approach improves the size properties of the Granger causality test for all of the different situations evaluated.     

A simple investigation of the Granger-causality test in integrated-cointegrated VAR systems

The size and power of various generalization tests for the Granger-causality in integrated-cointegrated VAR systems are considered. By using Monte Carlo methods, properties of eight versions of the test are studied in two different forms, the standard form and the modified form by Dolado & Lütkepohl (1996) in a study confined to properties of the Wald test only. In their study as well as in ours, both the standard and the modified Wald tests are shown to perform badly especially in small samples. We find, however, that the corrected LR tests exhibit correct size even in small samples. The power of the test is higher when the true VAR(2) model is estimated, and the modified test loses information by estimating the extra coefficients. The same is true when considering the power results in the VAR(3) model, and the power of the tests is somewhat lower than those in the VAR(2).     

The effect of number of clusters and cluster size on statistical power and Type I error rates when testing random effects variance components in multilevel linear and logistic regression models

When using multilevel regression models that incorporate cluster-specific random effects, the Wald and the likelihood ratio (LR) tests are used for testing the null hypothesis that the variance of the random effects distribution is equal to zero. We conducted a series of Monte Carlo simulations to examine the effect of the number of clusters and the number of subjects per cluster on the statistical power to detect a non-null random effects variance and to compare the empirical type I error rates of the Wald and LR tests. Statistical power increased with increasing number of clusters and number of subjects per cluster. Statistical power was greater for the LR test than for the Wald test. These results applied to both the linear and logistic regressions, but were more pronounced for the latter. The use of the LR test is preferable to the use of the Wald test.     

Granger-causal analysis of GARCH models: A Bayesian approach

A multivariate GARCH model is used to investigate Granger causality in the conditional variance of time series. Parametric restrictions for the hypothesis of noncausality in conditional variances between two groups of variables, when there are other variables in the system as well, are derived. These novel conditions are convenient for the analysis of potentially large systems of economic variables. To evaluate hypotheses of noncausality, a Bayesian testing procedure is proposed. It avoids the singularity problem that may appear in the Wald test, and it relaxes the assumption of the existence of higher-order moments of the residuals required in classical tests.     

Linear and nonlinear causality tests in an LSTAR model: wavelet decomposition in a nonlinear environment

In this paper, we use simulated data to investigate the power of different causality tests in a two-dimensional vector autoregressive (VAR) model. The data are presented in a nonlinear environment that is modelled using a logistic smooth transition autoregressive function. We use both linear and nonlinear causality tests to investigate the unidirection causality relationship and compare the power of these tests. The linear test is the commonly used Granger causality F test. The nonlinear test is a non-parametric test based on Baek and Brock [A general test for non-linear Granger causality: Bivariate model. Tech. Rep., Iowa State University and University of Wisconsin, Madison, WI, 1992] and Hiemstra and Jones [Testing for linear and non-linear Granger causality in the stock price–volume relation, J. Finance 49(5) (1994), pp. 1639–1664]. When implementing the nonlinear test, we use separately the original data, the linear VAR filtered residuals, and the wavelet decomposed series based on wavelet multiresolution analysis. The VAR filtered residuals and the wavelet decomposition series are used to extract the nonlinear structure of the original data. The simulation results show that the non-parametric test based on the wavelet decomposition series (which is a model-free approach) has the highest power to explore the causality relationship in nonlinear models.     

Testing for multivariate heteroscedasticity

In this article, we propose a testing technique for multivariate heteroscedasticity, which is expressed as a test of linear restrictions in a multivariate regression model. Four test statistics with known asymptotical null distributions are suggested, namely the Wald, Lagrange multiplier (LM), likelihood ratio (LR) and the multivariate Rao F-test. The critical values for the statistics are determined by their asymptotic null distributions, but bootstrapped critical values are also used. The size, power and robustness of the tests are examined in a Monte Carlo experiment. Our main finding is that all the tests limit their nominal sizes asymptotically, but some of them have superior small sample properties. These are the F, LM and bootstrapped versions of Wald and LR tests.     

The small sample properties of the reset test as applied to systems of equations

The RESET test for functional misspecification is generalised to cover systems of equations, and the properties of 7 versions are studied using Monte Carlo methods. The Rao F -test clearly exhibits the best performance as regards correct size, whilst the commonly used LRT (uncorrected for degrees-of-freedom), and LM and Wald tests (both corrected and uncorrected) behave badly even in single equations. The Rao test exhibits correct size even in ten equation systems, which is better than previous research concerning autocorrelation tests. The power of the test is low, however, when the number of equations grows and the correlation between the omitted variables and the RESET proxies is small.     

Testing for panel cointegration in an error-correction framework with an application to the Fisher hypothesis

In this article, three innovative panel error-correction model (PECM) tests are proposed. These tests are based on the multivariate versions of the Wald (W), likelihood ratio (LR), and Lagrange multiplier (LM) tests. Using Monte Carlo simulations, the size and power of the tests are investigated when the error terms exhibit both cross-sectional dependence and independence. We find that the LM test is the best option when the error terms follow independent white-noise processes. However, in the more empirically relevant case of cross-sectional dependence, we conclude that the W test is the optimal choice. In contrast to previous studies, our method is general and does not rely on the strict assumption that a common factor causes the cross-sectional dependency. In an empirical application, our method is also demonstrated in terms of the Fisher effect—a hypothesis about the existence of which there is still no clear consensus. Based on our sample of the five Nordic countries we utilize our powerful test and discover evidence which, in contrast to most previous research, confirms the Fisher effect.     

A permutation alternative and other test procedures for spatially correlated data in one-way anova

Research in many disciplines involves data with spatially correlated observations. Spatial dependence violates the independent errors assumption required for techniques such as the standard one-way analysis of variance for a completely randomized design. The testing methodology within the correlated errors approach has not been investigated within a spatial context. For one-way fixed effects analysis of variance, a permutation test and tests associated with the correlated errors approach are investigated through simulation. No single test was superior with respect to both power and size but the standard Wald F test and a simple adjustment to it performed well overall.     

Making wald tests work for cointegrated VAR systems

Wald tests of restrictions on the coefficients of vector autoregressive (VAR) processes are known to have nonstandard asymptotic properties for 1(1) and cointegrated systems of variables. A simple device is proposed which guarantees that Wald tests have asymptotic X2-distributions under general conditions. If the true generation process is a VAR(p) it is proposed to fit a VAR(p+l) to the data and perform a Wald test on the coefficients of the first p lags only. The power properties of the modified tests are studied both analytically and numerically by means of simple illustrative examples.     

Which Wald statistic? Choosing a parameterization of the Wald statistic to maximize power in k-sample generalized estimating equations

The Wald statistic is known to vary under reparameterization. This raises the question: which parameterization should be chosen, in order to optimize power of the Wald statistic? We specifically consider k-sample tests of generalized linear models (GLMs) and generalized estimating equations (GEEs) in which the alternative hypothesis contains only two parameters. An example is presented in which such an alternative hypothesis is of interest. Amongst a general class of parameterizations, we find the parameterization that maximizes power via analysis of the non-centrality parameter, and show how the effect on power of reparameterization depends on sampling design and the differences in variance across samples. There is no single parameterization with optimal power across all alternatives. The Wald statistic commonly used under the canonical parameterization is optimal in some instances but it performs very poorly in others. We demonstrate results by example and by simulation, and describe their implications for likelihood ratio statistics and score statistics. We conclude that due to poor power properties, the routine use of score statistics and Wald statistics under the canonical parameterization for GEEs is a questionable practice.     

Finite Sample Modifications of the Granger Non Causality Test in Cointegrated Vector Autoregressions

This article deals with the Granger non causality test in cointegrated vector autoregressive processes. We propose a new testing procedure that yields an asymptotically standard distribution and performs well in small samples by combining the standard Wald test and the generalized inverse procedure. We also propose a few simple modifications to the test statistics in order to help our procedure perform better in finite samples. Monte Carlo simulations show that our procedure works better than the conventional approach.     

Vector autoregression and causality: a theoretical overview and simulation study

This paper provides a theoretical overview of Wald tests for Granger causality in levels vector autoregressions (VAR's) and Johansen-type error correction models (ECM's). The theory is based on results in Toda and Phillips (1991a) and allows for stochastic and deterministic trends as well as arbitrary degrees of cointegration. We recommend some operational procedures for conducting Granger causality tests that are based on the Gaussian maximum likelihood estimation of ECM's. These procedures are applicable in the important practical case of testing the causal effects of one variable on another group of variables and vice versa. This paper also investigates the sampling properties of these testing procedures through simulation exercises. Three sequential causality tests in ECM's are compared with conventional causality tests in levels and differences VAR's.     

Likelihood Ratio Tests for Fixed Model Terms using Residual Maximum Likelihood

Likelihood ratio tests for fixed model terms are proposed for the analysis of linear mixed models when using residual maximum likelihood estimation. Bartlett-type adjustments, using an approximate decomposition of the data, are developed for the test statistics. A simulation study is used to compare properties of the test statistics proposed, with or without adjustment, with a Wald test. A proposed test statistic constructed by dropping fixed terms from the full fixed model is shown to give a better approximation to the asymptotic χ²-distribution than the Wald test for small data sets. Bartlett adjustment is shown to improve the χ²-approximation for the proposed tests substantially.     

Simple Tests for Exogeneity of a Binary Explanatory Variable in Count Data Regression Models

This article investigates power and size of some tests for exogeneity of a binary explanatory variable in count models by conducting extensive Monte Carlo simulations. The tests under consideration are Hausman contrast tests as well as univariate Wald tests, including a new test of notably easy implementation. Performance of the tests is explored under misspecification of the underlying model and under different conditions regarding the instruments. The results indicate that often the tests that are simpler to estimate outperform tests that are more demanding. This is especially the case for the new test.     

Empirical Study of Six Tests for Equality of Populations with Zero-Inflated Continuous Distributions

We evaluated the properties of six statistical methods for testing equality among populations with zero-inflated continuous distributions. These tests are based on likelihood ratio (LR), Wald, central limit theorem (CLT), modified CLT (MCLT), parametric jackknife (PJ), and nonparametric jackknife (NPJ) statistics. We investigated their statistical properties using simulated data from mixed distributions with an unknown portion of non zero observations that have an underlying gamma, exponential, or log-normal density function and the remaining portion that are excessive zeros. The 6 statistical tests are compared in terms of their empirical Type I errors and powers estimated through 10,000 repeated simulated samples for carefully selected configurations of parameters. The LR, Wald, and PJ tests are preferred tests since their empirical Type I errors were close to the preset nominal 0.05 level and each demonstrated good power for rejecting null hypotheses when the sample sizes are at least 125 in each group. The NPJ test had unacceptable empirical Type I errors because it rejected far too often while the CLT and MCLT tests had low testing powers in some cases. Therefore, these three tests are not recommended for general use but the LR, Wald, and PJ tests all performed well in large sample applications.     

Vector autoregression and causality: a theoretical overview and simulation study

This paper provides a theoretical overview of Wald tests for Granger causality in levels vector autoregressions (VAR's) and Johansen-type error correction models (ECM's). The theory is based on results in Toda and Phillips (1991a) and allows for stochastic and deterministic trends as well as arbitrary degrees of cointegration. We recommend some operational procedures for conducting Granger causality tests that are based on the Gaussian maximum likelihood estimation of ECM's. These procedures are applicable in the important practical case of testing the causal effects of one variable on another group of variables and vice versa. This paper also investigates the sampling properties of these testing procedures through simulation exercises. Three sequential causality tests in ECM's are compared with conventional causality tests in levels and differences VAR's.