In this article, we extend the earlier work of Freeland and McCabe [Journal of time Series Analysis (2004) Vol. 25, pp. 701-722] and develop a general framework for maximum likelihood (ML) analysis of higher-order integer-valued autoregressive processes. Our exposition includes the case where the innovation sequence has a Poisson distribution and the thinning is binomial. A recursive representation of the transition probability of the model is proposed. Based on this transition probability, we derive expressions for the score function and the Fisher information matrix, which form the basis for ML estimation and inference. Similar to the results in Freeland and McCabe (2004), we show that the score function and the Fisher information matrix can be neatly represented as conditional expectations. Using the INAR(2) specification with binomial thinning and Poisson innovations, we examine both the asymptotic efficiency and finite sample properties of the ML estimator in relation to the widely used conditional least squares (CLS) and Yule-Walker (YW) estimators. We conclude that, if the Poisson assumption can be justified, there are substantial gains to be had from using ML especially when the thinning parameters are large. Copyright 2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd

We propose a new approach for modeling non-linear multivariate interest rate processes based on time-varying copulas and reducible stochastic differential equations (SDEs). In the modeling of the marginal processes, we consider a class of non-linear SDEs that are reducible to Ornstein-Uhlenbeck (OU) process or Cox, Ingersoll, and Ross (1985) (CIR) process. The reducibility is achieved via a non-linear transformation function. The main advantage of this approach is that these SDEs can account for non-linear features, observed in short-term interest rate series, while at the same time leading to exact discretisation and closed form likelihood functions. Although a rich set of specifications may be entertained, our exposition focuses on a couple of nonlinear constant elasticity volatility (CEV) processes, denoted OU-CEV and CIR-CEV, respectively. These two processes encompass a number of existing models that have closed form likelihood functions. The statistical properties of the two processes are investigated. In order to obtain more flexible functional form over time, we allow the transformation function to be time-varying. Results from our study of US and UK short term interest rates suggest that the new models outperform existing parametric models with closed form likelihood functions. We also find the time-varying effects in the transformation functions statistically significant. We study the conditional dependence structure of the two rates using Patton (2006a) time-varying Symmetrised JoeClayton copula. We find evidence of asymmetric dependence between the two rates, and that the level of dependence is positively related to the level of the two rates. (1985) (CIR). La reduction est réalisée via une fonction de transformation non-linéaire. L'avantage principal de cette approche consiste en ce que ces SDES peuvent représenter des caractéristiques non-linéaires, observées dans les séries de taux d'intérêt a court terme, tout en conduisant en même tempsà une discretisation exacte età fonctions de vraisemblance analytique. Bien qu'une riche palette de spécifications puisseêtre considérée, nous nous concentrons sur deux fonctions de volatilitéàélasticité constante nonlinéaire (CEV), dénotées respectivement OU-CEV et CIR-CEV. Ces deux processus enveloppent un nombre important de modèles existants qui possèdent des fonctions de vraisemblance analytiques. Nous examinons les propriétés statistiques de ces deux processus. Afin d'obtenir des formes fonctionnelles flexibles, nous autorisons la fonction de transformationà varier dans le temps. Nos résultats empiriques portant sur les taux courts américains et britaniques suggèrent que nos nouveaux modèles dominent les modèles existants ayant des fonctions de vraisemblance analytiques. Nous trouvons aussi que les effets variables dans le temps de la fonction de transformation sont statistiquement significatifs. Nousétudions la structure de dépendance conditionnelle des deux taux au moyen de la copule de Joe-Clayon symétriséeà paramètres variables proposée par Patt...

Summary  We examine the ability of two recent methods – the smoothed implied volatility smile method (SML) and the density functionals based on confluent hypergeometric functions (DFCH) – for estimating implied risk‐neutral densities (RNDs) from European‐style options. Two complementary Monte Carlo experiments are conducted and the performance of the two RND estimators is evaluated by the root mean integrated squared error (RMISE) criterion. Results from both experiments show that the DFCH method outperforms the SML method for the overall quality of the estimated RNDs concerning both accuracy and stability. An application of the two methods to the OTC currency options market is also presented.

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We examine model specification in regime-switching continuous-time diffusions for modeling S&P 500 Volatility Index (VIX). Our investigation is carried out under two nonlinear diffusion frameworks, the NLDCEV and the CIRCEV frameworks, and our focus is on the nonlinearity in regime-dependent drift and diffusion terms, the switching components, and the endogeneity in regime changes. While we find strong evidence of regime-switching effects, models with a switching diffusion term capture the VIX dynamics considerably better than models with only a switching drift, confirming the presence and importance of volatility regimes. Strong evidence of nonlinear endogeneity in regime changes is also detected. Meanwhile, we find significant nonlinearity in the regime-dependent diffusion specification, suggesting that the nonlinearity in the VIX dynamics cannot be accounted for by regime-switching effects alone. Finally, we find that models based on the CIRCEV specification are significantly closer to the true data generating process of VIX than models based on the NLDCEV specification uniformly across all regime-switching specifications.