On the invertibility of EGARCH(p,q)

Of the two most widely estimated univariate asymmetric conditional volatility models, the exponential GARCH (or EGARCH) specification is said to be able to capture asymmetry, which refers to the different effects on conditional volatility of positive and negative effects of equal magnitude, and leverage, which refers to the negative correlation between the returns shocks and subsequent shocks to volatility. However, the statistical properties of the (quasi-)maximum likelihood estimator (QMLE) of the EGARCH(p, q) parameters are not available under general conditions, but only for special cases under highly restrictive and unverifiable sufficient conditions, such as EGARCH(1,0) or EGARCH(1,1), and possibly only under simulation. A limitation in the development of asymptotic properties of the QMLE for the EGARCH(p, q) model is the lack of an invertibility condition for the returns shocks underlying the model. It is shown in this article that the EGARCH(p, q) model can be derived from a stochastic process, for which sufficient invertibility conditions can be stated simply and explicitly when the parameters respect a simple condition.¹¹Using the notation introduced in part 2, this refers to the cases where α ≥ |γ| or α ≤ ? |γ|. The first inequality is generally assumed in the literature related to the invertibility of EGARCH. This article provides (in the Appendix) an argument for the possible lack of invertibility when these conditions are not met. This will be useful in reinterpreting the existing properties of the QMLE of the EGARCH(p, q) parameters.     

Inference for Box–Cox Transformed Threshold GARCH Models with Nuisance Parameters

Abstract. Generalized autoregressive conditional heteroscedastic (GARCH) models have been widely used for analyzing financial time series with time‐varying volatilities. To overcome the defect of the Gaussian quasi‐maximum likelihood estimator (QMLE) when the innovations follow either heavy‐tailed or skewed distributions, Berkes & Horváth (Ann. Statist., 32, 633, 2004) and Lee & Lee (Scand. J. Statist. 36, 157, 2009) considered likelihood methods that use two‐sided exponential, Cauchy and normal mixture distributions. In this paper, we extend their methods for Box–Cox transformed threshold GARCH model by allowing distributions used in the construction of likelihood functions to include parameters and employing the estimated quasi‐likelihood estimators (QELE) to handle those parameters. We also demonstrate that the proposed QMLE and QELE are consistent and asymptotically normal under regularity conditions. Simulation results are provided for illustration.     

Estimating the variance for heterogeneity in arm‐based network meta‐analysis

Network meta‐analysis can be implemented by using arm‐based or contrast‐based models. Here we focus on arm‐based models and fit them using generalized linear mixed model procedures. Full maximum likelihood (ML) estimation leads to biased trial‐by‐treatment interaction variance estimates for heterogeneity. Thus, our objective is to investigate alternative approaches to variance estimation that reduce bias compared with full ML. Specifically, we use penalized quasi‐likelihood/pseudo‐likelihood and hierarchical (h) likelihood approaches. In addition, we consider a novel model modification that yields estimators akin to the residual maximum likelihood estimator for linear mixed models. The proposed methods are compared by simulation, and 2 real datasets are used for illustration. Simulations show that penalized quasi‐likelihood/pseudo‐likelihood and h‐likelihood reduce bias and yield satisfactory coverage rates. Sum‐to‐zero restriction and baseline contrasts for random trial‐by‐treatment interaction effects, as well as a residual ML‐like adjustment, also reduce bias compared with an unconstrained model when ML is used, but coverage rates are not quite as good. Penalized quasi‐likelihood/pseudo‐likelihood and h‐likelihood are therefore recommended.     

Finite sample properties of maximum likelihood and quasi-maximum likelihood estimators of egarch models

In this paper I examine finite sample properties of the maximum likelihood and quasi-maximum likelihood estimators of EGARCH(1,1) processes using Monte Carlo methods. I use response surface methodology to summarize the results of a wide array of experiments which suggest that the maximum likelihood estimator has reasonable finite sample properties. The Gaussian quasi-maximum likelihood estimator has poor finite sample properties when the data generating process has conditional excess kurtosis. Some of these poor properties appear to be asymptotic in nature.     

Finite sample properties of maximum likelihood and quasi-maximum likelihood estimators of egarch models

In this paper I examine finite sample properties of the maximum likelihood and quasi-maximum likelihood estimators of EGARCH(1,1) processes using Monte Carlo methods. I use response surface methodology to summarize the results of a wide array of experiments which suggest that the maximum likelihood estimator has reasonable finite sample properties. The Gaussian quasi-maximum likelihood estimator has poor finite sample properties when the data generating process has conditional excess kurtosis. Some of these poor properties appear to be asymptotic in nature.     

Comment on “recent developments in bootstrapping time series”

The paper develops a general framework for identification, estimation, and hypothesis testing in cointegrated systems when the cointegrating coefficients are subject to (possibly) non-linear and cross-equation restrictions, obtained from economic theory or other relevant a priori information. It provides a proof of the consistency of the quasi maximum likelihood estimators (QMLE), establishes the relative rates of convergence of the QMLE of the short-run and the long-run parameters, and derives their asymptotic distributions; thus generalizing the results already available in the literature for the linear case. The paper also develops tests of the over-identifying (possibly) non-linear restrictions on the cointegrating vectors. The estimation and hypothesis testing procedures are applied to an Almost Ideal Demand System estimated on U.K. quarterly observations. Unlike many other studies of consumer demand this application does not treat relative prices and real per capita expenditures as exogenously given.     

LONG-RUN STRUCTURAL MODELLING

The paper develops a general framework for identification, estimation, and hypothesis testing in cointegrated systems when the cointegrating coefficients are subject to (possibly) non-linear and cross-equation restrictions, obtained from economic theory or other relevant a priori information. It provides a proof of the consistency of the quasi maximum likelihood estimators (QMLE), establishes the relative rates of convergence of the QMLE of the short-run and the long-run parameters, and derives their asymptotic distributions; thus generalizing the results already available in the literature for the linear case. The paper also develops tests of the over-identifying (possibly) non-linear restrictions on the cointegrating vectors. The estimation and hypothesis testing procedures are applied to an Almost Ideal Demand System estimated on U.K. quarterly observations. Unlike many other studies of consumer demand this application does not treat relative prices and real per capita expenditures as exogenously given.     

A Monte Carlo comparison of GMM and QMLE estimators for short dynamic panel data models with spatial errors

We suggest a generalized spatial system GMM (SGMM) estimation for short dynamic panel data models with spatial errors and fixed effects when n is large and T is fixed (usually small). Monte Carlo studies are conducted to evaluate the finite sample properties with the quasi-maximum likelihood estimation (QMLE). The results show that, QMLE, with a proper approximation for initial observation, performs better than SGMM in general cases. However, it performs poorly when spatial dependence is large. QMLE and SGMM perform better for different parameters when there is unknown heteroscedasticity in the disturbances and the data are highly persistent. Both estimates are not sensitive to the treatment of initial values. Estimation of the spatial autoregressive parameter is generally biased when either the data are highly persistent or spatial dependence is large. Choices of spatial weights matrices and the sign of spatial dependence do affect the performance of the estimates, especially in the case of the heteroscedastic disturbance. We also give empirical guidelines for the model.     

M-estimators of some GARCH-type models; computation and application

In this paper, we consider robust M-estimation of time series models with both symmetric and asymmetric forms of heteroscedasticity related to the GARCH and GJR models. The class of estimators includes least absolute deviation (LAD), Huber’s, Cauchy and B-estimator as well as the well-known quasi maximum likelihood estimator (QMLE). Extensive simulations are used to check the relative performance of these estimators in both models and the weighted resampling methods are used to approximate the sampling distribution of M-estimators. Our study indicates that there are estimators that can perform better than QMLE and even outperform robust estimator such as LAD when the error distribution is heavy-tailed. These estimators are also applied to real data sets.     

Normal Mixture Quasi-maximum Likelihood Estimator for GARCH Models

Abstract.  The generalized autoregressive conditional heteroscedastic (GARCH) model has been popular in the analysis of financial time series data with high volatility. Conventionally, the parameter estimation in GARCH models has been performed based on the Gaussian quasi-maximum likelihood. However, when the innovation terms have either heavy-tailed or skewed distributions, the quasi-maximum likelihood estimator (QMLE) does not function well. In order to remedy this defect, we propose the normal mixture QMLE (NM-QMLE), which is obtained from the normal mixture quasi-likelihood, and demonstrate that the NM-QMLE is consistent and asymptotically normal. Finally, we present simulation results and a real data analysis in order to illustrate our findings.     

Robust inference in generalized linear models for longitudinal data

The author develops a robust quasi‐likelihood method, which appears to be useful for down‐weighting any influential data points when estimating the model parameters. He illustrates the computational issues of the method in an example. He uses simulations to study the behaviour of the robust estimates when data are contaminated with outliers, and he compares these estimates to those obtained by the ordinary quasi‐likelihood method.     

Asymptotic properties of a quast-maximum likelihood estimator in truncated regression model with serial correlation

Robinson (1982a) presented a general approach to serial correlation in limited dependent variable models and proved the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for the Tobit model with serial correlation, obtained under the assumption of independent errors. This paper proves the strong consistency and asymptotic normality of the QMLE based on independent errors for the truncated regression model with serial correlation and gives consistent estimators for the limiting covariance matrix of the QMLE.     

Asymptotic properties of a quast-maximum likelihood estimator in truncated regression model with serial correlation

Robinson (1982a) presented a general approach to serial correlation in limited dependent variable models and proved the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for the Tobit model with serial correlation, obtained under the assumption of independent errors. This paper proves the strong consistency and asymptotic normality of the QMLE based on independent errors for the truncated regression model with serial correlation and gives consistent estimators for the limiting covariance matrix of the QMLE.     

Quasi‐Symmetric Graphical Log‐Linear Models

Abstract. We propose an extension of graphical log‐linear models to allow for symmetry constraints on some interaction parameters that represent homologous factors. The conditional independence structure of such quasi‐symmetric (QS) graphical models is described by an undirected graph with coloured edges, in which a particular colour corresponds to a set of equality constraints on a set of parameters. Unlike standard QS models, the proposed models apply with contingency tables for which only some variables or sets of the variables have the same categories. We study the graphical properties of such models, including conditions for decomposition of model parameters and of maximum likelihood estimates.     

Adaptive quasi-maximum likelihood estimation of GARCH models with Student’s t likelihood

ABSTRACT

This paper proposes an adaptive quasi-maximum likelihood estimation (QMLE) when forecasting the volatility of financial data with the generalized autoregressive conditional heteroscedasticity (GARCH) model. When the distribution of volatility data is unspecified or heavy-tailed, we worked out adaptive QMLE based on data by using the scale parameter η_f to identify the discrepancy between wrongly specified innovation density and the true innovation density. With only a few assumptions, this adaptive approach is consistent and asymptotically normal. Moreover, it gains better efficiency under the condition that innovation error is heavy-tailed. Finally, simulation studies and an application show its advantage.     

Generalised quasi‐likelihood inference in a semi‐parametric binary dynamic mixed logit model

There exists a recent study where dynamic mixed‐effects regression models for count data have been extended to a semi‐parametric context. However, when one deals with other discrete data such as binary responses, the results based on count data models are not directly applicable. In this paper, we therefore begin with existing binary dynamic mixed models and generalise them to the semi‐parametric context. For inference, we use a new semi‐parametric conditional quasi‐likelihood (SCQL) approach for the estimation of the non‐parametric function involved in the semi‐parametric model, and a semi‐parametric generalised quasi‐likelihood (SGQL) approach for the estimation of the main regression, dynamic dependence and random effects variance parameters. A semi‐parametric maximum likelihood (SML) approach is also used as a comparison to the SGQL approach. The properties of the estimators are examined both asymptotically and empirically. More specifically, the consistency of the estimators is established and finite sample performances of the estimators are examined through an intensive simulation study.     

Double AR model without intercept: An alternative to modeling nonstationarity and heteroscedasticity

This paper presents a double AR model without intercept (DARWIN model) and provides us a new way to study the nonstationary heteroscedastic time series. It is shown that the DARWIN model is always nonstationary and heteroscedastic, and its sample properties depend on the Lyapunov exponent. An easy-to-implement estimator is proposed for the Lyapunov exponent, and it is unbiased, strongly consistent, and asymptotically normal. Based on this estimator, a powerful test is constructed for testing the ordinary oscillation of the model. Moreover, this paper proposes the quasi-maximum likelihood estimator (QMLE) for the DARWIN model, which has an explicit form. The strong consistency and asymptotic normality of the QMLE are established regardless of the sign of the Lyapunov exponent. Simulation studies are conducted to assess the performance of the estimation and testing, and an empirical example is given for illustrating the usefulness of the DARWIN model.     

Inference for the tail index of a GARCH(1,1) model and an AR(1) model with ARCH(1) errors

For a GARCH(1,1) sequence or an AR(1) model with ARCH(1) errors, one can estimate the tail index by solving an estimating equation with unknown parameters replaced by the quasi maximum likelihood estimation, and a profile empirical likelihood method can be employed to effectively construct a confidence interval for the tail index. However, this requires that the errors of such a model have at least a finite fourth moment. In this article, we show that the finite fourth moment can be relaxed by employing a least absolute deviations estimate for the unknown parameters by noting that the estimating equation for determining the tail index is invariant to a scale transformation of the underlying model.     

Relative Risk Regression for Binary Outcomes: Methods and Recommendations

Relative risks are often considered preferable to odds ratios for quantifying the association between a predictor and a binary outcome. Relative risk regression is an alternative to logistic regression where the parameters are relative risks rather than odds ratios. It uses a log link binomial generalised linear model, or log‐binomial model, which requires parameter constraints to prevent probabilities from exceeding 1. This leads to numerical problems with standard approaches for finding the maximum likelihood estimate (MLE), such as Fisher scoring, and has motivated various non‐MLE approaches. In this paper we discuss the roles of the MLE and its main competitors for relative risk regression. It is argued that reliable alternatives to Fisher scoring mean that numerical issues are no longer a motivation for non‐MLE methods. Nonetheless, non‐MLE methods may be worthwhile for other reasons and we evaluate this possibility for alternatives within a class of quasi‐likelihood methods. The MLE obtained using a reliable computational method is recommended, but this approach requires bootstrapping when estimates are on the parameter space boundary. If convenience is paramount, then quasi‐likelihood estimation can be a good alternative, although parameter constraints may be violated. Sensitivity to model misspecification and outliers is also discussed along with recommendations and priorities for future research.     

Asymptotic properties of LS and QML estimators for a class of nonlinear GARCH processes

We consider estimation of a class of power-transformed threshold GARCH models. When the power of the transformation is known, the asymptotic properties of the quasi-maximum likelihood estimator (QMLE) are established under mild conditions. Two sequences of least-squares estimators are also considered in the pure ARCH case, and it is shown that they can be asymptotically more accurate than the QMLE for certain power transformations. In the case where the power of the transformation has to be estimated, the asymptotic properties of the QMLE are proven under the assumption that the noise has a density. The finite-sample properties of the proposed estimators are studied by simulation.