Exploring the Impact of Multivariate GARCH Innovations on Hypothesis Testing for Cointegrating Vectors

This article investigates the impact of multivariate generalized autoregressive conditional heteroskedastic (GARCH) errors on hypothesis testing for cointegrating vectors. The study reviews a cointegrated vector autoregressive model incorporating multivariate GARCH innovations and a regularity condition required for valid asymptotic inferences. Monte Carlo experiments are then conducted on a test statistic for a hypothesis on the cointegrating vectors. The experiments demonstrate that the regularity condition plays a critical role in rendering the hypothesis testing operational. It is also shown that Bartlett-type correction and wild bootstrap are useful in improving the small-sample size and power performance of the test statistic of interest.     

Estimating systems of trending variables

A survey is given of some results on inference in cointegrated systems. We discuss some regression methods, and contrast them with the analysis of the vector autoregressive model. We discuss determination of cointegrating rank and estimation of parameters, as well as asymptotic inference. The problems are treated for 1(1) and for 1(2) variables.     

Estimating systems of trending variables

A survey is given of some results on inference in cointegrated systems. We discuss some regression methods, and contrast them with the analysis of the vector autoregressive model. We discuss determination of cointegrating rank and estimation of parameters, as well as asymptotic inference. The problems are treated for 1(1) and for 1(2) variables.     

Estimation of cointegrated models with exogenous variables

We consider a cointegrated vector autoregressive process of integrated order 1, where the process consists of endogenous variables and exogenous variables. Johansen [Cointegration in partial systems and the efficiency of single-equation analysis. J Econometrics. 1992;52:389–402], Harbo et al. [Asymptotic inference on cointegrating rank in partial systems. J Amer Statist Assoc. 1998;16:388–399], and Pesaran et al. [Structural analysis of vector error correction models with exogenous I(1) variables. J Econometrics. 2000;97:293–343] considered inference of such processes assuming that the non-stationary exogenous variables are not cointegrated, and thus they are weakly exogenous. We consider the case where exogenous variables are cointegrated. Parameterization and estimation of the model is considered, and the asymptotic properties of the estimators are presented. The method in this paper is also applicable for the models considered in Mosconi and Giannini [Non-causality in cointegrated systems: representation estimation and testing. Oxford Bull Econ Stat. 1992;54:399–417], Pradel and Rault [Exogeneity in vector error correction models with purely exogenous long-run paths. Oxford Bull Econ Stat. 2003;65:629–653], and Hunter [Cointegrating exogeneity. Econom Lett. 1990;34:33–35]. A real data example is provided to illustrate the methods. Finite sample properties of the estimators are also examined through a Monte Carlo simulation.     

Inference on cointegrating ranks using lr and lm tests based on pseudo-likelihoods

This paper considers Lagrange Multiplier (LM) and Likelihood Ratio (LR) tests for determining the cointegrating rank of a vector autoregressive system. n order to deal with outliers and possible fat-tailedness of the error process, non-Gaussian likelihoods are used to carry out the estimation. The limiting distributions of the tests based on these non-Gaussian pseudo-)likelihoods are derived. These distributions depend on nuisance parameters. An operational procedure is proposed to perform inference. It appears that the tests based on non-Gaussian pseudo-likelihoods are much more powerful than their Gaussian counterparts if the errors are fat-tailed. Moreover, the operational LM-type test has a better overall performance than the LR-type test. Copyright O 1998 by Marcel Dekker, Inc.     

Comments on specification of household expenditure functions and equivalence scales by nonparametric regression

This paper considers Lagrange Multiplier (LM) and Likelihood Ratio (LR) tests for determining the cointegrating rank of a vector autoregressive system. n order to deal with outliers and possible fat-tailedness of the error process, non-Gaussian likelihoods are used to carry out the estimation. The limiting distributions of the tests based on these non-Gaussian pseudo-)likelihoods are derived. These distributions depend on nuisance parameters. An operational procedure is proposed to perform inference. It appears that the tests based on non-Gaussian pseudo-likelihoods are much more powerful than their Gaussian counterparts if the errors are fat-tailed. Moreover, the operational LM-type test has a better overall performance than the LR-type test. Copyright O 1998 by Marcel Dekker, Inc.     

Testing inference in variable dispersion beta regressions

The class of beta regression models proposed by Ferrari and Cribari-Neto [Beta regression for modelling rates and proportions, Journal of Applied Statistics 31 (2004), pp. 799–815] is useful for modelling data that assume values in the standard unit interval (0, 1). The dependent variable relates to a linear predictor that includes regressors and unknown parameters through a link function. The model is also indexed by a precision parameter, which is typically taken to be constant for all observations. Some authors have used, however, variable dispersion beta regression models, i.e., models that include a regression submodel for the precision parameter. In this paper, we show how to perform testing inference on the parameters that index the mean submodel without having to model the data precision. This strategy is useful as it is typically harder to model dispersion effects than mean effects. The proposed inference procedure is accurate even under variable dispersion. We present the results of extensive Monte Carlo simulations where our testing strategy is contrasted to that in which the practitioner models the underlying dispersion and then performs testing inference. An empirical application that uses real (not simulated) data is also presented and discussed.     

Pitfalls in Estimating Cointegrating Vector when Cointegration Relationship has Nonlinear Adjustment

This article investigates the properties of the estimators of the cointegrating vector when the cointegration error has a nonlinear adjustment. We investigate the properties of three estimators, namely, ordinary least squares (OLS), dynamic OLS (DOLS), and autoregressive distributed lag (ADL) models. Monte Carlo simulation results demonstrate that although all the estimators have consistency under cointegration with a nonlinear adjustment, they suffer from severe size distortions for the t-statistics of the cointegrating vector when the cointegration error has a highly persistent nonlinear adjustment and endogeneity. The results imply that the use of DOLS and ADL for cointegration with nonlinear adjustment cannot sufficiently improve the estimates and size performances.     

The effect of non-normal disturbances and conditional heteroskedasticity on multiple cointegration tests

!t is well-known that Johansen's multiple cointegration tests' results and those of Johansen and Juselius' tests for restricrions on cointegrating vectors and their weights have far-reaching implications for economic modelling and analysis. Therefore, it is important to ensure that the tests have desirable finite sample properties. Although the statistics are derived under Gaussian distribution,the asympotic results are derived under a much wider class of distributions. Using simulation, this paper investigates the effect of non-normal disturbances on these tests in finite samples. Further, ARCH/GARCH type conditional heteroskedasticity is present in many economic and financial time series. This paper examines the finite properties of the tests when the error term follows ARCH/GARCH type processes. From the evidence, it appears that researchers should not be overly concerned by the possibility of small departures from non-normality when using Johansen's suggested techniques even in finite samples. ARCH and GARCH effects may be more problematic, however. In particular it becomes more important ro test whether the restriction implicit in the integrated (or near-integrated) ARCH-type Drocess actually holds in time series for the application of the cointegraiion rank tests and the test for restrictions on cointegrating weights. The tests for restrictions on cointegrating vectors apper to be robust for non-normal errors and for all ARCH and GARCH type processes considered.     

Testing normalization and overidentification of cointegrating vectors in vector autoregressive processes

This paper develops test procedures for testing the validity of general linear identifying restrictions imposed on cointegrating vectors in the context of a vector autoregressive model. In addition to overidentifying restrictions the considered restrictions may also involve normalizing restrictions. Tests for both types of restrictions are developed and their asymptotic properties are obtained. Under the null hypothesis tests for normalizing restrictions have an asymptotic "multivariate unit root distribution", similar to that obtained for the likelihood ratio test for cointegration, while tests for overidentifying restrictions have a standard chi-square limiting distribution. Since these two types of tests are asymptotically independent they are easy to cotnbine to an overall test for the spccifed identifying restrictions. An overall test of this kind can consistently reveal the failure of the identifying restrictions in a wider class of cases than previous tests which only test for overidentifying restrictions.     

Testing for the Cointegrating Rank of a VAR Process With Structural Shifts

Tests for the cointegrating rank of a vector autoregressive process are considered that allow for possible exogenous shifts in the mean of the data-generation process. The break points are assumed to be known a priori. It is proposed to estimate and remove the deterministic terms such as mean, linear-trend term, and a shift in a first step. Then systems cointegration tests are applied to the adjusted series. The resulting tests are shown to have known limiting null distributions that are free of nuisance parameters and do not depend on the break point. The tests are applied for analyzing the number of cointegrating relations in two German money-demand systems.     

Structural Instability and the Production-Smoothing Model of Inventories

The production-smoothing model of inventories implies that inventories, labor inputs, sales, and factor input prices are cointegrated if sales and factor prices are I(1) with one cointegrating vector for each state variable held. These propositions are tested in six nondurable-goods industries. All industries provide evidence of cointegration. Fewer quasi-fixed factors are found than previous research often assumed. Estimates of cointegrating vectors provide implausible parameter estimates. Rank stability tests, with fixed or seqentially chosen breakpoints, indicate that the cointegrating matrix has unstable rank. Parameter estimates of cointegrating vectors do not provide much support for the production-smoothing model of inventories.     

A directional look at F‐tests

Directional testing of vector parameters, based on higher order approximations of likelihood theory, can ensure extremely accurate inference, even in high‐dimensional settings where standard first order likelihood results can perform poorly. Here we explore examples of directional inference where the calculations can be simplified, and prove that in several classical situations, the directional test reproduces exact results based on F‐tests. These findings give a new interpretation of some classical results and support the use of directional testing in general models, where exact solutions are typically not available. The Canadian Journal of Statistics 47: 619–627; 2019 © 2019 Statistical Society of Canada     

Efficient Estimation of the Seemingly Unrelated Regression Cointegration Model and Testing for Purchasing Power Parity

This paper studies the efficient estimation of seemingly unrelated linear models with integrated regressors and stationary errors. We consider two cases. The first one has no common regressor among the equations. In this case, we show that by adding leads and lags of the first differences of the regressors and estimating this augmented dynamic regression model by generalized least squares using the long-run covariance matrix, we obtain an efficient estimator of the cointegrating vector that has a limiting mixed normal distribution. In the second case we consider, there is a common regressor to all equations, and we discuss efficient minimum distance estimation in this context. Simulation results suggests that our new estimator compares favorably with others already proposed in the literature. We apply these new estimators to the testing of the proportionality and symmetry conditions implied by purchasing power parity (PPP) among the G-7 countries. The tests based on the efficient estimates easily reject the joint hypotheses of proportionality and symmetry for all countries with either the United States or Germany as numeraire. Based on individual tests, our results suggest that Canada and Germany are the most likely countries for which the proportionality condition holds, and that Italy and Japan for the symmetry condition relative to the United States.     

Bayesian Inference in Cointegrated I (2) Systems: A Generalization of the Triangular Model

This paper generalizes the cointegrating model of Phillips (1991) to allow for I (0), I (1) and I (2) processes. The model has a simple form that permits a wider range of I (2) processes than are usually considered, including a more flexible form of polynomial cointegration. Further, the specification relaxes restrictions identified by Phillips (1991) on the I (1) and I (2) cointegrating vectors and restrictions on how the stochastic trends enter the system. To date there has been little work on Bayesian I (2) analysis and so this paper attempts to address this gap in the literature. A method of Bayesian inference in potentially I (2) processes is presented with application to Australian money demand using a Jeffreys prior and a shrinkage prior.     

The power of bootstrap based tests for parameters in cointegrating regressions

Several asymptotic procedures have been suggested for inference on cointegrating parameters. But the tests based on asymptotic theory have been found to have substantial size distortions. The present paper shows that the bootstrap method gives the proper test sizes and that the power of the bootstrap based tests is satisfactory.     

Unit root econometrics and economic nonlinearities

This paper concentrates on some shortcomings of contemporary unit root econometric methodology (testing for cointegration, common roots and stationarity) where the dynamics of an economy are described by a nonlinear process. It is shown that, in such circumstances, traditionally applied unit root econometrics may not lead to interpretable or statistically significant results. Two cases of such nonlinearities are discussed: (i) a stochastically nonlinear data generating process and (ii) a time-varying parameters cointegrating relation, typical of an economic reform process. It is shown that case (i) consists of a wide family of economic processes and in most such cases the results of standard unit root tests are not directly interpretable. Case (ii) does not result in a (conventionally understood) error-correction representation of a cointegrated process. Some Monte Carlo experiments evaluate the validity of cointegration tests in situations where there is a change in the cointegration parameter and from cointegration regime to noncointegration and vice versa. A simple method of estimation through simulation is proposed and its finite-sample properties examined.     

Exact Inference in the Proportional Hazard Model: Possibilities and Limitations

It is suggested that inference under the proportional hazard model can be carried out by programs for exact inference under the logistic regression model. Advantages of such inference is that software is available and that multivariate models can be addressed. The method has been evaluated by means of coverage and power calculations in certain situations. In all situations coverage was above the nominal level, but on the other hand rather conservative. A different type of exact inference is developed under Type II censoring. Inference was then less conservative, however there are limitations with respect to censoring mechanism, multivariate generalizations and software is not available. This method also requires extensive computational power. Performance of large sample Wald, score and likelihood inference was also considered. Large sample methods works remarkably well with small data sets, but inference by score statistics seems to be the best choice. There seems to be some problems with likelihood ratio inference that may originate from how this method works with infinite estimates of the regression parameter. Inference by Wald statistics can be quite conservative with very small data sets.     

Efficient Estimation of the Seemingly Unrelated Regression Cointegration Model and Testing for Purchasing Power Parity

Abstract

This paper studies the efficient estimation of seemingly unrelated linear models with integrated regressors and stationary errors. We consider two cases. The first one has no common regressor among the equations. In this case, we show that by adding leads and lags of the first differences of the regressors and estimating this augmented dynamic regression model by generalized least squares using the long-run covariance matrix, we obtain an efficient estimator of the cointegrating vector that has a limiting mixed normal distribution. In the second case we consider, there is a common regressor to all equations, and we discuss efficient minimum distance estimation in this context. Simulation results suggests that our new estimator compares favorably with others already proposed in the literature. We apply these new estimators to the testing of the proportionality and symmetry conditions implied by purchasing power parity (PPP) among the G-7 countries. The tests based on the efficient estimates easily reject the joint hypotheses of proportionality and symmetry for all countries with either the United States or Germany as numeraire. Based on individual tests, our results suggest that Canada and Germany are the most likely countries for which the proportionality condition holds, and that Italy and Japan for the symmetry condition relative to the United States.     

Efficient network meta-analysis: A confidence distribution approach

Network meta-analysis synthesizes several studies of multiple treatment comparisons to simultaneously provide inference for all treatments in the network. It can often strengthen inference on pairwise comparisons by borrowing evidence from other comparisons in the network. Current network meta-analysis approaches are derived from either conventional pairwise meta-analysis or hierarchical Bayesian methods. This paper introduces a new approach for network meta-analysis by combining confidence distributions (CDs). Instead of combining point estimators from individual studies in the conventional approach, the new approach combines CDs, which contain richer information than point estimators, and thus achieves greater efficiency in its inference. The proposed CD approach can efficiently integrate all studies in the network and provide inference for all treatments, even when individual studies contain only comparisons of subsets of the treatments. Through numerical studies with real and simulated data sets, the proposed approach is shown to outperform or at least equal the traditional pairwise meta-analysis and a commonly used Bayesian hierarchical model. Although the Bayesian approach may yield comparable results with a suitably chosen prior, it is highly sensitive to the choice of priors (especially for the between-trial covariance structure), which is often subjective. The CD approach is a general frequentist approach and is prior-free. Moreover, it can always provide a proper inference for all the treatment effects regardless of the between-trial covariance structure.