Empirical likelihood intervals for conditional Value‐at‐Risk in ARCH/GARCH models

Value‐at‐Risk (VaR) is a simple, but useful measure in risk management. When some volatility model is employed, conditional VaR is of importance. As autoregressive conditional heteroscedastic (ARCH) and generalized ARCH (GARCH) models are widely used in modelling volatilities, in this article, we propose empirical likelihood methods to obtain an interval estimation for the conditional VaR with the volatility model being an ARCH/GARCH model.     

Effects of outliers on the identification and estimation of GARCH models

Abstract. This paper analyses how outliers affect the identification of conditional heteroscedasticity and the estimation of generalized autoregressive conditionally heteroscedastic (GARCH) models. First, we derive the asymptotic biases of the sample autocorrelations of squared observations generated by stationary processes and show that the properties of some conditional homoscedasticity tests can be distorted. Second, we obtain the asymptotic and finite sample biases of the ordinary least squares (OLS) estimator of ARCH(p) models. The finite sample results are extended to generalized least squares (GLS), maximum likelihood (ML) and quasi‐maximum likelihood (QML) estimators of ARCH(p) and GARCH(1,1) models. Finally, we show that the estimated asymptotic standard deviations are biased estimates of the sample standard deviations.     

Infinitely Divisible Distributions in Integer‐Valued Garch Models

We propose an integer‐valued stochastic process with conditional marginal distribution belonging to the class of infinitely divisible discrete probability laws. With this proposal, we introduce a wide class of models for count time series that includes the Poisson integer‐valued generalized autoregressive conditional heteroscedastic (INGARCH) model (Ferland et al., 2006) and the negative binomial and generalized Poisson INGARCH models (Zhu, 2011, 2012a). The main probabilistic analysis of this process is developed stating, in particular, first‐order and second‐order stationarity conditions. The existence of a strictly stationary and ergodic solution is established in a subclass including the Poisson and generalized Poisson INGARCH models.     

Integer‐Valued GARCH Process

Abstract. An integer‐valued analogue of the classical generalized autoregressive conditional heteroskedastic (GARCH) (p,q) model with Poisson deviates is proposed and a condition for the existence of such a process is given. For the case p = 1, q = 1, it is explicitly shown that an integer‐valued GARCH process is a standard autoregressive moving average (1, 1) process. The problem of maximum likelihood estimation of parameters is treated. An application of the model to a real time series with a numerical example is given.     

Functional Generalized Autoregressive Conditional Heteroskedasticity

Heteroskedasticity is a common feature of financial time series and is commonly addressed in the model building process through the use of autoregressive conditional heteroskedastic and generalized autoregressive conditional heteroskedastic (GARCH) processes. More recently, multivariate variants of these processes have been the focus of research with attention given to methods seeking an efficient and economic estimation of a large number of model parameters. Because of the need for estimation of many parameters, however, these models may not be suitable for modelling now prevalent high‐frequency volatility data. One potentially useful way to bypass these issues is to take a functional approach. In this article, theory is developed for a new functional version of the GARCH process, termed fGARCH. The main results are concerned with the structure of the fGARCH(1,1) process, providing criteria for the existence of strictly stationary solutions both in the space of square‐integrable and continuous functions. An estimation procedure is introduced, and its consistency and asymptotic normality are verified. A small empirical study highlights potential applications to intraday volatility estimation.     

Evaluating Specification Tests for Markov‐Switching Time‐Series Models

Abstract. We evaluate the performance of several specification tests for Markov regime‐switching time‐series models. We consider the Lagrange multiplier (LM) and dynamic specification tests of Hamilton (1996) and Ljung–Box tests based on both the generalized residual and a standard‐normal residual constructed using the Rosenblatt transformation. The size and power of the tests are studied using Monte Carlo experiments. We find that the LM tests have the best size and power properties. The Ljung–Box tests exhibit slight size distortions, though tests based on the Rosenblatt transformation perform better than the generalized residual‐based tests. The tests exhibit impressive power to detect both autocorrelation and autoregressive conditional heteroscedasticity (ARCH). The tests are illustrated with a Markov‐switching generalized ARCH (GARCH) model fitted to the US dollar–British pound exchange rate, with the finding that both autocorrelation and GARCH effects are needed to adequately fit the data.     

Stability of nonlinear AR‐GARCH models

Abstract. This article studies the stability of nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a nonlinear autoregression of order p [AR(p)] with the conditional variance specified as a nonlinear first‐order generalized autoregressive conditional heteroskedasticity [GARCH(1,1)] model. Conditions under which the model is stable in the sense that its Markov chain representation is geometrically ergodic are provided. This implies the existence of an initial distribution such that the process is strictly stationary and β‐mixing. Conditions under which the stationary distribution has finite moments are also given. The results cover several nonlinear specifications recently proposed for both the conditional mean and conditional variance, and only require mild moment conditions.     

Influence diagnostics for multivariate GARCH processes

This article presents diagnostics for identifying influential observations when estimating multivariate generalized autoregressive conditional heteroscedasticity (GARCH) models. We derive influence diagnostics by introducing minor perturbations to the conditional variances and covariances. The derived diagnostics are applied to a bivariate GARCH model of daily returns of the S&P500 and IBM. We find that univariate diagnostic procedures may be unable to identify the influential observations in a multivariate model. Importantly, the proposed curvature‐based diagnostic identified influential observations where the correlation between the two series had a major change. These observations were not identified as influential using the univariate diagnostics for each asset separately. When estimating the bivariate GARCH model allowing for weights at influential observations, we found that the time‐varying correlations behaved differently from that implied by the model ignoring influential observations. The application therefore highlights the importance of extending univariate diagnostic procedures to multivariate settings.     

Time‐series models with an EGB2 conditional distribution

A time‐series model in which the signal is buried in noise that is non‐Gaussian may throw up observations that, when judged by the Gaussian yardstick, are outliers. We describe an observation‐driven model, based on an exponential generalized beta distribution of the second kind (EGB2), in which the signal is a linear function of past values of the score of the conditional distribution. This specification produces a model that is not only easy to implement but which also facilitates the development of a comprehensive and relatively straightforward theory for the asymptotic distribution of the maximum‐likelihood (ML) estimator. Score‐driven models of this kind can also be based on conditional t distributions, but whereas these models carry out what, in the robustness literature, is called a soft form of trimming, the EGB2 distribution leads to a soft form of Winsorizing. An exponential general autoregressive conditional heteroscedastic (EGARCH) model based on the EGB2 distribution is also developed. This model complements the score‐driven EGARCH model with a conditional t distribution. Finally, dynamic location and scale models are combined and applied to data on the UK rate of inflation.     

A light‐tailed conditionally heteroscedastic model with applications to river flows

Abstract. A conditionally heteroscedastic model, different from the more commonly used autoregressive moving average–generalized autoregressive conditionally heteroscedastic (ARMA‐GARCH) processes, is established and analysed here. The time‐dependent variance of innovations passing through an ARMA filter is conditioned on the lagged values of the generated process, rather than on the lagged innovations, and is defined to be asymptotically proportional to those past values. Designed this way, the model incorporates certain feedback from the modelled process, the innovation is no longer of GARCH type, and all moments of the modelled process are finite provided the same is true for the generating noise. The article gives the condition of stationarity, and proves consistency and asymptotic normality of the Gaussian quasi‐maximum likelihood estimator of the variance parameters, even though the estimated parameters of the linear filter contain an error. An analysis of six diurnal water discharge series observed along Rivers Danube and Tisza in Hungary demonstrates the usefulness of such a model. The effect of lagged river discharge turns out to be highly significant on the variance of innovations, and nonparametric estimation approves its approximate linearity. Simulations from the new model preserve well the probability distribution, the high quantiles, the tail behaviour and the high‐level clustering of the original series, further justifying model choice.     

Local Likelihood for non‐parametric ARCH(1) models

Abstract. We propose a non‐parametric local likelihood estimator for the log‐transformed autoregressive conditional heteroscedastic (ARCH) (1) model. Our non‐parametric estimator is constructed within the likelihood framework for non‐Gaussian observations: it is different from standard kernel regression smoothing, where the innovations are assumed to be normally distributed. We derive consistency and asymptotic normality for our estimators and show, by a simulation experiment and some real‐data examples, that the local likelihood estimator has better predictive potential than classical local regression. A possible extension of the estimation procedure to more general multiplicative ARCH(p) models with p > 1 predictor variables is also described.     

Poisson QMLE of Count Time Series Models

Regularity conditions are given for the consistency of the Poisson quasi‐maximum likelihood estimator of the conditional mean parameter of a count time series model. The asymptotic distribution of the estimator is studied when the parameter belongs to the interior of the parameter space and when it lies at the boundary. Tests for the significance of the parameters and for constant conditional mean are deduced. Applications to specific integer‐valued autoregressive (INAR) and integer‐valued generalized autoregressive conditional heteroscedasticity (INGARCH) models are considered. Numerical illustrations, Monte Carlo simulations and real data series are provided.     

ON MIXTURE MEMORY GARCH MODELS

We propose a new volatility model, which is called the mixture memory generalized autoregressive conditional heteroskedasticity (MM‐GARCH) model. The MM‐GARCH model has two mixture components, of which one is a short‐memory GARCH and the other is the long‐memory fractionally integrated GARCH. The new model, a special ARCH( ∞ ) process with random coefficients, possesses both the properties of long‐memory volatility and covariance stationarity. The existence of its stationary solution is discussed. A dynamic mixture of the proposed model is also introduced. Other issues, such as the expectation–maximization algorithm as a parameter estimation procedure, the observed information matrix, which is relevant in calculating the theoretical standard errors, and a model selection criterion, are also investigated. Monte Carlo experiments demonstrate our theoretical findings. Empirical application of the MM‐GARCH model to the daily S&P 500 index illustrates its capabilities.     

Least tail‐trimmed squares for infinite variance autoregressions

We develop a robust least squares estimator for autoregressions with possibly heavy tailed errors. Robustness to heavy tails is ensured by negligibly trimming the squared error according to extreme values of the error and regressors. Tail‐trimming ensures asymptotic normality and super‐‐convergence with a rate comparable to the highest achieved amongst M‐estimators for stationary data. Moreover, tail‐trimming ensures robustness to heavy tails in both small and large samples. By comparison, existing robust estimators are not as robust in small samples, have a slower rate of convergence when the variance is infinite, or are not asymptotically normal. We present a consistent estimator of the covariance matrix and treat classic inference without knowledge of the rate of convergence. A simulation study demonstrates the sharpness and approximate normality of the estimator, and we apply the estimator to financial returns data. Finally, tail‐trimming can be easily extended beyond least squares estimation for a linear stationary AR model. We discuss extensions to quasi‐maximum likelihood for GARCH, weighted least squares for a possibly non‐stationary random coefficient autoregression, and empirical likelihood for robust confidence region estimation, in each case for models with possibly heavy tailed errors.     

Postmodel selection estimators of variance function for nonlinear autoregression

We consider a problem of estimating a conditional variance function of an autoregressive process. A finite collection of parametric models for conditional density is studied when both regression and variance are modelled by parametric functions. The proposed estimators are defined as the maximum likelihood estimators in the models chosen by penalized selection criteria. Consistency properties of the resulting estimator of the variance when the conditional density belongs to one of the parametric models are studied as well as its behaviour under mis‐specification. The autoregressive process does not need to be stationary but only existence of a stationary distribution and ergodicity is required. Analogous results for the pseudolikelihood method are also discussed. A simulation study shows promising behaviour of the proposed estimator in the case of heavy‐tailed errors in comparison with local linear smoothers.     

Tests for conditional heteroscedasticity of functional data

Functional data objects derived from high-frequency financial data often exhibit volatility clustering. Versions of functional generalized autoregressive conditionally heteroscedastic (FGARCH) models have recently been proposed to describe such data, however so far basic diagnostic tests for these models are not available. We propose two portmanteau type tests to measure conditional heteroscedasticity in the squares of asset return curves. A complete asymptotic theory is provided for each test. We also show how such tests can be adapted and applied to model residuals to evaluate adequacy, and inform order selection, of FGARCH models. Simulation results show that both tests have good size and power to detect conditional heteroscedasticity and model mis-specification in finite samples. In an application, the tests show that intra-day asset return curves exhibit conditional heteroscedasticity. This conditional heteroscedasticity cannot be explained by the magnitude of inter-daily returns alone, but it can be adequately modeled by an FGARCH(1,1) model.     

A Stationary Spatio-Temporal GARCH Model

We introduce a lagged nearest-neighbour, stationary spatio-temporal generalized autoregressive conditional heteroskedasticity (GARCH) model on an infinite spatial grid that opens for GARCH innovations in a space-time ARMA model. This is illustrated by a real data application to a classical dataset of sea surface temperature anomalies in the Pacific Ocean. The model and its translation invariant neighbourhood system are wrapped around a torus forming a model with finite spatial domain, which we call circular spatio-temporal GARCH. Such a model could be seen as an approximation of the infinite one and simulation experiments show that the circular estimator with a straightforward bias correction performs well on such non-circular data. Since the spatial boundaries are tied together, the well-known boundary issue in spatial statistical modelling is effectively avoided. We derive stationarity conditions for these circular processes and study the spatio-temporal correlation structure through an ARMA representation. We also show that the matrices defined by a vectorized version of the model are block circulants. The maximum quasi-likelihood estimator is presented and we prove its strong consistency and asymptotic normality by generalizing results from univariate GARCH theory.     

Inference for pth‐order random coefficient integer‐valued autoregressive processes

Abstract. A pth‐order random coefficient integer‐valued autoregressive [RCINAR(p)] model is proposed for count data. Stationarity and ergodicity properties are established. Maximum likelihood, conditional least squares, modified quasi‐likelihood and generalized method of moments are used to estimate the model parameters. Asymptotic properties of the estimators are derived. Simulation results on the comparison of the estimators are reported. The models are applied to two real data sets.     

On the Distributional Properties of GARCH Processes

In this paper we study the distributional properties of the generalized autoregressive conditional heteroskedasticity (GARCH) model often being applied in economics. For a large class of non-normal distributions of the noise process various inequalities on the distribution of the GARCH process are established. Moreover, these results are used to derive useful conclusions about the behavior of the average run length of a Shewhart control chart for time series.