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This paper studies the quasi-maximum-likelihood estimator (QMLE) in a general conditionally heteroscedastic time series model of multiplicative form$X_{t}=\\sigma _{t}Z_{t}$, where the unobservable volatility$\\sigma _{t}$is a parametric function of$(X_{t-1},\\ldots ,X_{t-p},\\sigma _{t-1},\\ldots,\\sigma _{t-q})$for some p, q ≥ 0, and$(Z_{t})$is standardized i.i.d. noise. We assume that these models are solutions to stochastic recurrence equations which satisfy a contraction (random Lipschitz coefficient) property. These assumptions are satisfied for the popular GARCH, asymmetric GARCH and exponential GARCH processes. Exploiting the contraction property, we give conditions for the existence and uniqueness of a strictly stationary solution$(X_{t})$to the stochastic recurrence equation and establish consistency and asymptotic normality of the QMLE. We also discuss the problem of invertibility of such time series models.

I introduce the notion of continuous invertibility on a compact set for volatility models driven by a stochastic recurrence equation. I prove strong consistency of the quasi-maximum likelihood estimator (QMLE) when the quasi-likelihood criterion is maximized on a continuously invertible domain. This approach yields, for the first time, the asymptotic normality of the QMLE for the exponential general autoregressive conditional heteroskedastic (EGARCH(1,1)) model under explicit but non-verifiable conditions. In practice, I propose to stabilize the QMLE by constraining the optimization procedure to an empirical continuously invertible domain. The new method, called stable QMLE, is asymptotically normal when the observations follow an invertible EGARCH(1,1) model.

For the quarterly data of 10 Asian economies, ranging from the first quarter of 1991 to last quarter of 2012, we model inflation volatility as a time varying process through different symmetric and asymmetric GARCH specifications. We also propose to model inflation volatility on the basis of cyclic component of inflation obtained from an Hodrick Prescott (HP) filter instead of actual inflation when the latter does not fulfil the criterion of stationarity. Through news impact curves (NICs) we tried to highlight the behaviour of inflation volatility in response to lagged inflation shocks under different GARCH specifications. In our results the leverage parameter shows the expected sign and is significant for almost all countries suggesting strong asymmetry in inflation volatility. The hyperbolic sign integral shape of NICs based on Glosten-Jagannathan-Runkle GARCH (GJR-GARCH) highlights the importance of inflation stabilisation programmes particularly because of the subsequent evidence obtained in favour of bidirectional causality running between inflation and inflation volatility. There is also evidence in favour of the argument that a cyclic component of inflation obtained through an HP filter could be used as a suitable proxy of inflation for volatility estimation.

This paper investigates the role of CDS volatility in providing information concerning the credit quality of a company. In Castellano and D’Ecclesia (J. Financ. Decis. Mak. 2:27, 2011 ) a first analysis of how CDS quotes respond to rating announcements is provided and it showed that market participants do not rely much on Rating Agencies, especially during periods characterized by very high volatility, i.e. during a financial crisis. Here, a more accurate analysis of the CDS’s ability to provide timely information on the creditworthiness of reference entities is performed, estimating the volatility of CDS quotes by using Exponential GARCH(1,1) models. The event study methodology is applied to a sample of CDS quotes for US and European markets, over the period 2004–2009. Results provide an accurate understanding of market behavior in the presence of news released by Rating Agencies. Overall, market participants seem to provide timely reactions around the event date and we show that the key element of signaling is represented by the changing volatility in CDS quotes, before and after the rating event.

We derive the analytical expressions of bias approximations for maximum likelihood (ML) and quasi-maximum likelihood (QML) estimators of the EGARCH (1,1) parameters that enable us to correct after the bias of all estimators. The bias-correction mechanism is constructed under the specification of two methods that are analytically described. We also evaluate the residual bootstrapped estimator as a measure of performance. Monte Carlo simulations indicate that, for given sets of parameters values, the bias corrections work satisfactory for all parameters. The proposed full-step estimator performs better than the classical one and is also faster than the bootstrap. The results can be also used to formulate the approximate Edgeworth distribution of the estimators.

In this paper, a generalized GARCH-based stochastic mortality model is developed, which incorporates conditional heteroskedasticity and conditional non-normality. First, a detailed empirical analysis of the UK mortality rates from 1922 to 2009 is provided, where it was found that both the conditional heteroskedasticity and conditional non-normality are important empirical long-term structures of mortality. To describe conditional non-normality, a double-exponential distribution that allows conditional skewness and the heavy-tailed features found in the datasets was selected. For the practical implementation of the proposed model, a two-stage scheme was introduced to estimate the unknown parameters. First, the Quasi-Maximum Likelihood Estimation (QMLE) method was employed to estimate the GARCH structure. Next, the MLE was adopted to estimate the unknown parameters of the double-exponential distribution using residuals as input data. The model was then back-tested against the previous 10 years of mortality data to assess its forecasting ability, before Monte Carlo simulation was carried out to simulate and produce a table of forecast mortality rates from the optimal distribution.

This paper introduces and discusses some of the statistical properties of a time-varying parameter stochastic volatility (SV) in mean model. We derive the autocovariance function of an observed series, under the assumption that the conditional variance follows a flexible parameterization, which nests the autoregressive SV and the exponential GARCH specifications. Furthermore, the mean parameter can be time varying. We also present the autocovariance functions of higher orders and discuss identification issues. Our result can be applied so that the properties of the observed data may be compared with the theoretical properties of the models, thus facilitating model identification. Furthermore, they can be employed in the estimation and derivation of misspecification tests.

This paper considers the moment structure of the general ARMA—EGARCH model. In particular, we derive the autocorrelation function of any positive integer power of the squared errors. In addition, we obtain the autocorrelations of the squares of the observed process and cross correlations between the levels and the squares of the observed process. Finally, the practical implications of the results are illustrated empirically using daily data on four East Asia Stock Indices.

This article models the conditional mean and variance of real gross national product (GNP) and its components using asymmetric exponential generalized autoregressive conditional hetero-scedasticity, a model previously applied only to financial variables. The results imply that the variance of real GNP is higher following negative innovations than positive innovations and that this asymmetry arises in the cyclically sensitive sectors. Further evidence links this asymmetry to the phase of the business cycle: The conditional variance appears to be largest around business-cycle troughs. In addition, shocks to the conditional variance of GNP and its components typically persist for long periods. The evidence of asymmetry in conditional variance is robust to a variety of alternative specifications.