Offer Lieberman

Asymptotic theory for multivariate GARCH processes

We provide in this paper asymptotic theory for the multivariate GARCH(p, q) process. Strong consistency of the quasi-maximum likelihood estimator (MLE) is established by appealing to conditions given by Jeantheau (Econometric Theory 14 (1998), 70) in conjunction with a result given by Boussama (Ergodicity, mixing and estimation in GARCH models, Ph.D. Dissertation, University of Paris 7, 1998) concerning the existence of a stationary and ergodic solution to the multivariate GARCH(p, q) process. We prove asymptotic normality of the quasi-MLE when the initial state is either stationary or fixed.

GENERALIZED AUTOREGRESSIVE CONDITIONAL CORRELATION

This paper develops a generalized autoregressive conditional correlation (GARCC) model when the standardized residuals follow a random coefficient vector autoregressive process. As a multivariate generalization of the Tsay (1987, Journal of the American Statistical Association 82, 590–604) random coefficient autoregressive (RCA) model, the GARCC model provides a motivation for the conditional correlations to be time varying. GARCC is also more general than the Engle (2002, Journal of Business & Economic Statistics 20, 339–350) dynamic conditional correlation (DCC) and the Tse and Tsui (2002, Journal of Business & Economic Statistics 20, 351–362) varying conditional correlation (VCC) models and does not impose unduly restrictive conditions on the parameters of the DCC model. The structural properties of the GARCC model, specifically, the analytical forms of the regularity conditions, are derived, and the asymptotic theory is established. The Baba, Engle, Kraft, and Kroner (BEKK) model of Engle and Kroner (1995, Econometric Theory 11, 122–150) is demonstrated to be a special case of a multivariate RCA process. A likelihood ratio test is proposed for several special cases of GARCC. The empirical usefulness of GARCC and the practicality of the likelihood ratio test are demonstrated for the daily returns of the Standard and Poors 500, Nikkei, and Hang Seng indexes.

A characterization of the price behavior of international dual stocks: an error correction approach

This paper deals with the interrelations between stocks listed and traded in two international unsynchronized markets. The data exhibits first order nonstationarity and the series across markets are cointegrated. This gives a justification for an error correction model which incorporates a short run adjustment mechanism. The model is applied for different day-groups. The main findings are: (1) The domestic country emerges as the dominant market and the foreign market as the satellite one; (2) The adjustment mechanism coefficient is highly significant for most shares; (3) Different behavioural patterns emerge for middle-of-the- week days as compared with beginning/end-of-week days; (4) The model fits better for the more heavily traded shares.

Second‐Order Noncausality in Multivariate GARCH Processes

Typical multivariate economic time series may exhibit co-behavior patterns not only in the conditional means, but also in the conditional variances. In this paper we give two new definitions of variance noncausality in a multivariate setting a Granger-type noncausality and a linear Granger noncausality through projections on Hilbert spaces. Both definitions are related to a previous second-order noncausality concept defined by Granger et al. in a bivariate setting. The implications of second-order noncausality on multivariate ARMA processes with GARCH-type errors are investigated. We derive exact testable restrictions on the parameters of the processes considered, implied by this type of noncausality. Conditions for the finiteness of the fourth-order moment of the multivariate GARCH process are derived and related to earlier results in the univariate framework. We include an illustration of second-order noncausality in a trivariate model of daily financial returns.

Saddlepoint Approximation for the Distribution of a Ratio of Quadratic Forms in Normal Variables

Abstract In this article, the saddlepoint approximations to the density and tail probability of a ratio of quadratic forms in normal variables are derived. A numerical exposition via the Durbin-Watson test statistic reveals several desirable features. The approximations, which involve only a limited number of computable functions, provide the practitioner with an accessible and a very powerful tool.

Improved small sample inference in the mixed linear model: Bartlett correction and adjusted likelihood

The mixed linear model is a popular method for analysing unbalanced repeated measurement data. The classical statistical tests for parameters in this model are based on asymptotic theory that is unreliable in the small samples that are often encountered in practice. For testing a given fixed effect parameter with a small sample, we develop and investigate refined likelihood ratio (LR) tests. The refinements considered are the Bartlett correction and use of the Cox–Reid adjusted likelihood; these are examined separately and in combination. We illustrate the various LR tests on an actual data set and compare them in two simulation studies. The conventional LR test yields type I error rates that are higher than nominal. The adjusted LR test yields rates that are lower than nominal, with absolute accuracy similar to that of the conventional LR test in the first simulation study and better in the second. The Bartlett correction substantially improves the accuracy of the type I error rates with either the conventional or the adjusted LR test. In many cases, error rates that are very close to nominal are achieved with the refined methods.

Refined Inference on Long Memory in Realized Volatility

There is an emerging consensus in empirical finance that realized volatility series typically display long range dependence with a memory parameter (d) around 0.4 (Andersen et al., 2001; Martens et al., 2004). The present article provides some illustrative analysis of how long memory may arise from the accumulative process underlying realized volatility. The article also uses results in Lieberman and Phillips (2004, 2005) to refine statistical inference about d by higher order theory. Standard asymptotic theory has an O(n −1/2) error rate for error rejection probabilities, and the theory used here refines the approximation to an error rate of o(n −1/2). The new formula is independent of unknown parameters, is simple to calculate and user-friendly. The method is applied to test whether the reported long memory parameter estimates of Andersen et al. (2001) and Martens et al. (2004) differ significantly from the lower boundary (d = 0.5) of nonstationary long memory, and generally confirms earlier findings.

The Optimal Size of a Preliminary Test of Linear Restrictions in a Misspecified Regression Model

Abstract When the choice of estimator for the coefficients in a linear regression model is determined by the outcome of a prior test of the validity of restrictions on the model, it is well known that a minimax (risk) regret criterion leads to the simple rule that the optimal critical value for the preliminary test is approximately two in value, regardless of the degrees of freedom. We show that this result no longer holds in the (likely) event that relevant regressors are excluded from the model at the outset.

ASYMPTOTIC THEORY FOR EMPIRICAL SIMILARITY MODELS

We consider the stochastic process null , t = 2, …, n , where s ( x , x ) is a similarity function between the t th and the i th observations and { e } is a random disturbance term. This process was originally axiomatized by Gilboa, Lieberman, and Schmeidler (2006, Review of Economics and Statistics 88, 433–444) as a way by which agents, or even nature, reason. In the present paper, consistency and the asymptotic distribution of the quasi-maximum likelihood estimator of the parameters of the model are established. Connections to other models and techniques are drawn. In its general form, the model does not fall within any class of nonstationary econometric models for which asymptotic theory is available. For this reason, the developments in this paper are new and nonstandard.

Valid Edgeworth Expansions for the Whittle Maximum Likelihood Estimator for Stationary Long-memory Gaussian Time Series

In this paper, we prove the validity of an Edgeworth expansion to the distribution of the Whittle maximum likelihood estimator for stationary long-memory Gaussian models with unknown parameter theta in Theta subset R^{d_{theta}} . The error of the (s-2)-order expansion is shown to be o(n^{(s-2)/2}) -- the usual iid rate -- for a wide range of models, including the popular ARFIMA(p,d,q) models. The expansion is valid under mild assumptions on the behavior of spectral density and its derivatives in the neighborhood of the origin. As a by-product, we generalize a Theorem by Fox and Taqqu (1987) concerning the asymptotic behavior of Toeplitz matrices. Lieberman, Rousseau, and Zucker (2002) (LRZ) establish a valid Edgeworth expansion for the maximum likelihood estimator for stationary long-memory Gaussian models. For a significant class of models, their expansion is shown to have an error of o(n-1). The results given here improve upon those of LRZ in that the results provide an Edgeworth expansion for an asymptotically efficient estimator, as LRZ do, but the error of the expansion is shown to be o(n^{-(s-2)/2}), not o(n^{-1}), for a broad range of models.