Analytical quasi maximum likelihood inference in multivariate volatility models

We study the properties of the quasi-maximum likelihood estimator (QMLE) and related test statistics in dynamic models that jointly parameterize conditional means and conditional covariances, when a normal log-likelihood os maximized but the assumption of normality is violated. Because the score of the normal log-likelihood has the martingale difference property when the forst two conditional moments are correctly specified, the QMLE is generally Consistent and has a limiting normal destribution. We provide easily computable formulas for asymptotic standard errors that are valid under nonnormality. Further, we show how robust LM tests for the adequacy of the jointly parameterized mean and variance can be computed from simple auxiliary regressions. An appealing feature of these robyst inference procedures is that only first derivatives of the conditional mean and variance functions are needed. A monte Carlo study indicates that the asymptotic results carry over to finite samples. Estimation of several AR and AR-GARCH time series models reveals that in most sotuations the robust test statistics compare favorably to the two standard (nonrobust) formulations of the Wald and IM tests. Also, for the GARCH models and the sample sizes analyzed here, the bias in the QMLE appears to be relatively small. An empirical application to stock return volatility illustrates the potential imprtance of computing robust statistics in practice.

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

GARCH models can contain outliers, such as Additive Level Outliers (ALO) and/or Additive Volatility Outliers (AVO), so a robust parameter estimation is used to overcome the presence of outliers. This study uses Quasi-Maximum Likelihood Estimation (QMLE) and M-estimation to estimate the parameters of the GARCH model containing outliers. Hierarchical and K-means clustering algorithms are then used to cluster time-series data based on distances calculated from these parameter estimates. The proposed approach applies hierarchical and K-means clustering using distances derived from QMLE and M-estimation results for GARCH (1,1) models containing outliers for clustering with simulation data and a case study using stock data listed in the Indonesia Stock Exchange. The results show that hierarchical and K-means clustering with distance between two GARCH (1,1) models based on M-estimation produces good clusters with smaller C index than QMLE estimations for simulation data and the case study of stock data in Indonesia.

BackgroundBayesian phylogenetic inference holds promise as an alternative to maximum likelihood, particularly for large molecular-sequence data sets. We have investigated the performance of Bayesian inference with empirical and simulated protein-sequence data under conditions of relative branch-length differences and model violation.ResultsWith empirical protein-sequence data, Bayesian posterior probabilities provide more-generous estimates of subtree reliability than does the nonparametric bootstrap combined with maximum likelihood inference, reaching 100% posterior probability at bootstrap proportions around 80%. With simulated 7-taxon protein-sequence datasets, Bayesian posterior probabilities are somewhat more generous than bootstrap proportions, but do not saturate. Compared with likelihood, Bayesian phylogenetic inference can be as or more robust to relative branch-length differences for datasets of this size, particularly when among-sites rate variation is modeled using a gamma distribution. When the (known) correct model was used to infer trees, Bayesian inference recovered the (known) correct tree in 100% of instances in which one or two branches were up to 20-fold longer than the others. At ratios more extreme than 20-fold, topological accuracy of reconstruction degraded only slowly when only one branch was of relatively greater length, but more rapidly when there were two such branches. Under an incorrect model of sequence change, inaccurate trees were sometimes observed at less extreme branch-length ratios, and (particularly for trees with single long branches) such trees tended to be more inaccurate. The effect of model violation on accuracy of reconstruction for trees with two long branches was more variable, but gamma-corrected Bayesian inference nonetheless yielded more-accurate trees than did either maximum likelihood or uncorrected Bayesian inference across the range of conditions we examined. Assuming an exponential Bayesian prior on branch lengths did not improve, and under certain extreme conditions significantly diminished, performance. The two topology-comparison metrics we employed, edit distance and Robinson-Foulds symmetric distance, yielded different but highly complementary measures of performance.ConclusionsOur results demonstrate that Bayesian inference can be relatively robust against biologically reasonable levels of relative branch-length differences and model violation, and thus may provide a promising alternative to maximum likelihood for inference of phylogenetic trees from protein-sequence data.

Matrix-variate time series data are largely available in applications. However, no attempt has been made to study their conditional heteroscedasticity that is often observed in economic and financial data. To address this gap, we propose a novel matrix generalized autoregressive conditional heteroscedasticity (GARCH) model to capture the dynamics of conditional row and column covariance matrices of matrix time series. The key innovation of the matrix GARCH model is the use of a univariate GARCH specification for the trace of conditional row or column covariance matrix, which allows for the model identification. Moreover, we introduce a quasi-maximum likelihood estimator (QMLE) for model estimation and develop a portmanteau test for model diagnostic checking. Simulation studies are conducted to assess the finite-sample performance of the QMLE and portmanteau test. To handle large dimensional matrix time series, we also propose a matrix factor GARCH model, and establish its theoretical properties. Finally, we demonstrate the superiority of the matrix GARCH and matrix factor GARCH models over existing multivariate GARCH-type models in volatility forecasting and portfolio allocations using three applications on credit default swap prices, global stock sector indices, and future prices. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

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.

The purpose of this paper is to incorporate semiparametric alternatives to maximum likelihood estimation and inference in the context of unordered multinomial response data when in practice there is often insufficient information to specify the parametric form of the function linking the observables to the unknown probabilities. We specify the function linking the observables to the unknown probabilities using a very general flexible class of functions belonging to the Pearson system of cumulative distribution equations. In this setting we consider the observations as arising from a multinomial distribution characterized by one of the CDFs in the Pearson system. Given this situation, it is possible to utilize the concept of unbiased estimating functions (EFs), combined with the concept of empirical likelihood (EL) to define an (empirical) likelihood function for the parameter vector based on a nonparametric representation of the sample’s PDF. This leads to the concept of maximum empirical likelihood (MEL) estimation and inference, which is analogous to parametric maximum likelihood methods in many respects.

In this study, we introduce a novel approach, called the quasi maximum likelihood estimation (Q-MLE), for estimating large-dimensional matrix factor models. In contrast to the principal component analysis based approach, Q-MLE considers the heteroscedasticity of the idiosyncratic error term, the heteroscedasticity of which is simultaneously estimated with other parameters. Interestingly, under the homoscedasticity assumption of the idiosyncratic error, the Q-MLE estimator encompassed the projected estimator (PE) as a special case. We provide the convergence rates and asymptotic distributions of the Q-MLE estimators under mild conditions. Extensive numerical experiments demonstrate that the Q-MLE method performs better, especially when heteroscedasticity exists. Furthermore, two real examples in finance and macroeconomics reveal factor patterns across rows and columns, which coincide with financial, economic, or geographical interpretations.

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.

Spatial dynamic panel data (SDPD) models have received great attention in economics in recent 10 years. Existing approaches for the estimation and test of SDPD models are quasi-maximum likelihood (QML) approach and generalized method of moments (GMM). In this article, we introduce the empirical likelihood (EL) method to the statistical inference for SDPD models. The EL ratio statistics are constructed for the parameters of spatial dynamic panel data models. It is shown that the limiting distributions of the empirical likelihood ratio statistics are chi-squared distributions, which are used to construct confidence regions for the parameters of the models. Simulation results show that the EL based confidence regions outperform the normal approximation based confidence regions.

A spatial generalized ordered-response model with skew normal kernel error terms with an application to bicycling frequency

For inference in complex models, composite likelihood combines genuine likelihoods based on low‐dimensional portions of the data, with weights to be chosen. Optimal weights in composite likelihood may be searched following different routes, leading to a solution only in scalar parameter models. Here, after briefly reviewing the main approaches, we show how to obtain the first‐order optimal weights when using composite likelihood for inference on a scalar parameter in the presence of nuisance parameters. These weights depend on the true parameter value and need to be estimated. Under regularity conditions, the resulting likelihood ratio statistic has the standard asymptotic null distribution and improved local power. Simulation results in multivariate normal models show that estimation of optimal weights maintains the standard approximate null distribution and produces a visible gain in power with respect to constant weights.

PhyloNet was released in 2008 as a software package for representing and analyzing phylogenetic networks. At the time of its release, the main functionalities in PhyloNet consisted of measures for comparing network topologies and a single heuristic for reconciling gene trees with a species tree. Since then, PhyloNet has grown significantly. The software package now includes a wide array of methods for inferring phylogenetic networks from data sets of unlinked loci while accounting for both reticulation (e.g., hybridization) and incomplete lineage sorting. In particular, PhyloNet now allows for maximum parsimony, maximum likelihood, and Bayesian inference of phylogenetic networks from gene tree estimates. Furthermore, Bayesian inference directly from sequence data (sequence alignments or biallelic markers) is implemented. Maximum parsimony is based on an extension of the "minimizing deep coalescences" criterion to phylogenetic networks, whereas maximum likelihood and Bayesian inference are based on the multispecies network coalescent. All methods allow for multiple individuals per species. As computing the likelihood of a phylogenetic network is computationally hard, PhyloNet allows for evaluation and inference of networks using a pseudolikelihood measure. PhyloNet summarizes the results of the various analyzes and generates phylogenetic networks in the extended Newick format that is readily viewable by existing visualization software.

The commonality in idiosyncratic volatility cannot be fully explained by time-varying volatility; correlated idiosyncratic volatility shocks are an important contributing factor. We empirically document this fact for various characteristic-sorted portfolios and industry portfolios. To capture the commonality in idiosyncratic volatility, we propose a novel multivariate generalized autoregressive conditional heteroscedasticity (GARCH) model called dynamic factor correlation (DFC). The DFC model has a closed-form likelihood function that allows computationally cheap estimation, even for a large number of assets. The DFC model improves statistical fit compared with existing multivariate GARCH models such as the dynamic conditional correlation model and the dynamic equicorrelation (DECO) model, achieving the lowest root mean square error and mean absolute error in simulations. Empirical tests also corroborate simulations in demonstrating the improved statistical fit of the DFC model. Mean--variance portfolio optimization using the DFC model outperforms alternative volatility models such as the historical covariance matrix and the DECO model. Out-of-sample mean--variance-efficient portfolios using the DFC model have the lowest volatility and the highest Sharpe ratios, thereby improving the investor's opportunity set. Under parametric restrictions, the DFC model reduces to the constant conditional correlation model or the DECO model.

Two competitive analytical approaches, namely, the generalized method of moments (GMM) and quasi-maximum likelihood (QML) are widely used in statistics and econometrics literature for inferences in stochastic volatility models (SVMs). Alternative numerical approaches such as monte carlo markov chain (MCMC), simulated maximum likelihood (SML) and Bayesian approaches are also available. All these later approaches are, however, based on simulations. In this paper, we revisit the analytical estimation approaches and briefly demonstrate that the existing GMM approach is unnecessarily complicated. Also, the asymptotic properties of the likelihood approximation based QML approach are unknown and the finite sample based QML estimators can be inefficient. We then develop a precise set of moment estimating equations and demonstrate that the proposed method of moments (MM) estimators are easy to compute and they perform well in estimating the parameters of the SVMs in both small and large time series set up. A ‘working’ generalized quasi-likelihood (WGQL) estimation approach is also considered. Estimation methods are illustrated by reanalyzing a part of the Swiss-Franc and U.S. dollar exchange rates data.