Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances

Quasi-maximum likelihood estimation in GARCH processes when some coefficients are equal to zero

The quasi-likelihood function proposed by Wedderburn [Quasi-likelihood functions, generalized linear models, and the Gauss–Newton method. Biometrika. 1974;61:439–447] broadened the application scope of generalized linear models (GLM) by specifying the mean and variance function instead of the entire distribution. However, in many situations, complete specification of variance function in the quasi-likelihood approach may not be realistic. Following Fahrmeir's [Maximum likelihood estimation in misspecified generalized linear models. Statistics. 1990;21:487–502] treating with misspecified GLM, we define a quasi-likelihood nonlinear models (QLNM) with misspecified variance function by replacing the unknown variance function with a known function. In this paper, we propose some mild regularity conditions, under which the existence and the asymptotic normality of the maximum quasi-likelihood estimator (MQLE) are obtained in QLNM with misspecified variance function. We suggest computing MQLE of unknown parameter in QLNM with misspecified variance function by the Gauss–Newton iteration procedure and show it to work well in a simulation study.

Strong consistency and (weak) distributional convergence to highly non-Gaussian limits are established for closed-form, two stage least squares (TSLS) estimators of linear and threshold ARCH (p) models, with special attention paid to the ARCH (1) and threshold ARCH (1) cases. Conditions supporting these results include (relatively) mild moment existence criteria that enjoy empirical support. These conditions are not shared by competing estimators like OLS. Identification of the TSLS estimators depends on asymmetry, either in the model’s rescaled errors or in the conditional variance function. Monte Carlo studies reveal TSLS estimation can sizably outperform quasi maximum likelihood (QML) in small samples and even best recently proposed two-step estimators specifically designed to enhance the efficiency of QML.

This paper performs a Monte Carlo study on efficient method of moments (EMM), generalized method of moments (GMM), quasi-maximum likelihood estimation (QMLE) and maximum likelihood estimation (MLE) for a continuous-time square-root model under two challenging scenarios – high persistence in mean and strong conditional volatility – that are commonly found in estimating the interest rate process. MLE turns out to be the most efficient of the four methods, but its finite sample inference and convergence rate suffer severely from approximating the likelihood function, especially in the scenario of highly persistent mean. QMLE comes second in terms of estimation efficiency, but it is the most reliable in generating inferences. GMM with lag-augmented moments has, overall, the lowest estimation efficiency, possibly due to the ad hoc choice of moment conditions. EMM shows an accelerated convergence rate in the high-volatility scenario, while its over-rejection bias in the mean persistence scenario is unacceptably large. Finally, under a stylized alternative model of the US interest rates, the overidentification test of EMM obtains the ultimate power for detecting misspecification, while the GMM J-test is increasingly biased downwards in finite samples.

Influence of News in Moscow and New York on Returns and Risks on Baltic State Stock Indices

To be utilized in the smart charging of plug-in electric vehicles, this paper proposes semiparametric conditional mean and variance models for the daily density forecasting of the electricity load. The mean is modeled by means of an autoregressive moving average model with exogenous inputs (ARMAX), whereas several options for the variance evolution are investigated, starting with modeling the variance as a power of the conditional mean, then as a piecewise constant function, and finally as a generalized autoregressive conditional heteroskedasticity (GARCH) model. Due to the possible non-Gaussianity of the distribution of the stochastic components, a quasi-maximum likelihood estimation (QMLE) with the Pearson type IV (P-IV) distribution is also considered, apart from the estimation with the Gaussianity assumption. Moreover, the daily density forecasts are generated in a non-parametric manner by propagating samples from the stochastic components iteratively through the available model. These strategies, involving different options for variance modeling and estimation, are compared in terms of their forecast performances on two representative phase currents from the low voltage cables of medium-to-low voltage transformers in the Netherlands. The results indicate that the QMLE with the Gaussianity assumption performs better due to the additional complexity of the P-IV distribution on estimation. Concerning the variance models, the piecewise constant and GARCH models are more preferable when processing phase currents exhibiting only daily seasonality, and the model as a power of the conditional mean outperforms the others if both daily and weekly seasonality and hence more complexity is present.

Composite likelihood functions are often used for inference in applications where the data have a complex structure. While inference based on the composite likelihood can be more robust than inference based on the full likelihood, the inference is not valid if the associated conditional or marginal models are misspecified. In this paper, we propose a general class of specification tests for composite likelihood inference. The test statistics are motivated by the fact that the second Bartlett identity holds for each component of the composite likelihood function when these components are correctly specified. We construct the test statistics based on the discrepancy between the so-called composite information matrix and the sensitivity matrix. As an illustration, we study three important cases of the proposed tests and establish their limiting distributions under both null and local alternative hypotheses. Finally, we evaluate the finite-sample performance of the proposed tests in several examples.

Estimating time-varying conditional covariance matrices of financial returns play important role in portfolio analysis, risk management, and financial econometrics research. The availability of high-frequency financial data can provide an additional data source for dynamic covariance modeling. In this paper, we propose to use the information of asset return vector and realized covariance measures simultaneously to develop a new conditional covariance matrix model. We derive the stationary condition of the new model. We use the normal and Wishart distributions to construct the quasi-log-likelihood function. We also consider the variance targeting (VT) method, which plugs in the weighted average of the sample covariance matrix of returns and the sample mean of realized covariance measure for the unconditional covariance matrix, in order to maximize the quasi-log-likelihood function. We show the consistency and asymptotic normality of the quasi-maximum likelihood (QML) and VT estimators. We investigate the finite sample property of these estimators via Monte Carlo experiments. The empirical example for the bivariate data of the Nikkei 225 index and its futures indicates that the first-step VT estimation could have non-negligible effects on the standard errors of the second-step VT estimates.

This paper examines the estimation of the Stochastic Conditional Duration model by the empirical characteristic function and the generalized method of moments when maximum likelihood is unavailable. The joint characteristic function for the durations along with general expressions for the moments are derived, leading naturally to estimation via the empirical characteristic function and generalized method of moments. In a Monte Carlo study as well as an empirical application, these alternative methods are compared with quasi maximum likelihood. These experiments reveal that the empirical characteristic function approach outperforms the quasi maximum likelihood and generalized method of moments in terms of both bias and root mean square error. Copyright The Author(s). Journal compilation Royal Economic Society 2008

Quasi maximum likelihood estimation and inference in multivariate volatility models remains a challenging computational task if, for example, the dimension of the parameter space is high. One of the reasons is that typically numerical procedures are used to compute the score and the Hessian, and often they are numerically unstable. We provide analytical formulae for the score and the Hessian for a variety of multivariate GARCH models including the Vec and BEKK specifications as well as the recent dynamic conditional correlation model. By means of a Monte Carlo investigation of the BEKK–GARCH model we illustrate that employing analytical derivatives for inference is clearly preferable to numerical methods.

We consider a model for a multivariate time series where the conditional covariance matrix is a function of a finite-dimensional parameter and the innovation distribution is nonparametric. The semiparametric lower bound for the estimation of the euclidean parameter is characterized, and it is shown that adaptive estimation without reparametrization is not possible. Based on a consistent first-stage estimator (such as quasi maximum likelihood), we propose a semiparametric estimator that estimates the efficient influence function using kernel estimators. We state conditions under which the estimator attains the semiparametric lower bound. For particular models such as the constant conditional correlation model, adaptive estimation of the dynamic part of the model is shown to be possible. To avoid the curse of dimensionality one can, e.g., restrict the multivariate density to the class of spherical distributions, for which we also derive the semiparametric efficiency bound and an estimator that attains this bound. A simulation experiment demonstrates the efficiency gain of the proposed estimator compared with quasi maximum likelihood estimation.Rombouts work was supported by the Centre for Research on e-Finance, HEC Montreal. Hafner gratefully acknowledges financial support by the Fonds Sp ciaux de Recherche (FSR 05) of the Universit catholique de Louvain. The authors thank three anonymous referees for valuable comments and suggestions and Luc Bauwens, Geert Dhaene, Feico Drost, Wolfgang H rdle, Douglas Hodgson, Jens Peter Kreiss, Oliver Linton, and Bas Werker for helpful discussions. We also thank participants of the CORE econometrics seminar, the York annual meeting in econometrics, the annual econometric study group meeting 2002 in Bristol, the 2003 workshop The Art of Semiparametrics in Berlin, and the statistics seminar of the Stockholm School of Economics for valuable comments.

GMM estimation of spatial autoregressive models with moving average disturbances

Test statistics for checking the independence between the innovations of several time series are developed. The time series models considered allow for general specifications for the conditional mean and variance functions that could depend on common explanatory variables. In testing for independence between more than two time series, checking pairwise independence does not lead to consistent procedures. Thus a finite family of empirical processes relying on multivariate lagged residuals are constructed, and we derive their asymptotic distributions. In order to obtain simple asymptotic covariance structures, Möbius transformations of the empirical processes are studied, and simplifications occur. Under the null hypothesis of independence, we show that these transformed processes are asymptotically Gaussian, independent, and with tractable covariance functions not depending on the estimated parameters. Various procedures are discussed, including Cramér–von Mises test statistics and tests based on non‐parametric measures. The ranks of the residuals are considered in the new methods, giving test statistics which are asymptotically margin‐free. Generalized cross‐correlations are introduced, extending the concept of cross‐correlation to an arbitrary number of time series; portmanteau procedures based on them are discussed. In order to detect the dependence visually, graphical devices are proposed. Simulations are conducted to explore the finite sample properties of the methodology, which is found to be powerful against various types of alternatives when the independence is tested between two and three time series. An application is considered, using the daily log‐returns of Apple, Intel and Hewlett‐Packard traded on the Nasdaq financial market.The Canadian Journal of Statistics40: 447–479; 2012 © 2012 Statistical Society of Canada

In this article, we propose a new joint portmanteau test for checking the specification of parametric conditional mean and variance functions of linear and nonlinear time‐series models. The use of a joint test is motivated for complete control of the asymptotic size since marginal tests for the conditional variance may lead to misleading conclusions when the conditional mean is misspecified. The new test is based on an asymptotically distribution‐free transformation on the sample autocorrelations of both normalized residuals and squared normalized residuals. This makes it unnecessary to full detail the asymptotic properties of the estimates used to obtain residuals, which could be inefficient two‐step ones, avoiding also choices of maximum lag parameters increasing with sample length to control asymptotic size. The robust versions of the new test also properly account for higher‐order moment dependence at a reduced cost. The finite‐sample performance of the new test is compared with that of well‐known tests through simulations.

This paper investigates the asymptotic theory for a vector autoregressive moving average–generalized autoregressive conditional heteroskedasticity (ARMA-GARCH) model. The conditions for the strict stationarity, the ergodicity, and the higher order moments of the model are established. Consistency of the quasi-maximum-likelihood estimator (QMLE) is proved under only the second-order moment condition. This consistency result is new, even for the univariate autoregressive conditional heteroskedasticity (ARCH) and GARCH models. Moreover, the asymptotic normality of the QMLE for the vector ARCH model is obtained under only the second-order moment of the unconditional errors and the finite fourth-order moment of the conditional errors. Under additional moment conditions, the asymptotic normality of the QMLE is also obtained for the vector ARMA-ARCH and ARMA-GARCH models and also a consistent estimator of the asymptotic covariance.The authors thank the co-Editor, Bruce Hansen, and two referees for very helpful comments and suggestions and acknowledge the financial support of the Australian Research Council.