Smoothed estimates for nonlinear time series models with irregular data

Time series data observed at unequal time intervals (irregular data) occur quite often and this usually poses problems in its analysis. A recursive form of the exponentially smoothed estimated is here proposed for a nonlinear model with irregularly observed data and its asymptotic properties are discussed An alternative smoother to that of Wright (1985) is also derived. Numerical comparison is made between the resulting estimates and other smoothed estimates.     

Non-parametric estimation of the conditional mode

The problem of estimating the mode of a conditional probability density function is considered. It is shown that under some regularity conditions the estimate of the conditional mode obtained by maximizing a kernel estimate of the conditional probability density function is strongly consistent and asymptotically normally distributed.     

Generalized value at risk forecasting

Abstract

In this paper, using estimating function approach, a new optimal volatility estimator is introduced and based on the recursive form of the estimator a data-driven generalized EWMA model for value at risk (VaR) forecast is proposed. An appropriate data-driven model for volatility is identified by the relationship between absolute deviation and standard deviation for symmetric distributions with finite variance. It is shown that the asymptotic variance of the proposed volatility estimator is smaller than that of conventional estimators and is more appropriate for financial data with larger kurtosis. For IBM, Microsoft, Apple stocks and SP 500 index the proposed method is used to identify the model, estimate the volatility, and obtain minimum mean square error(MMSE) forecasts of VaR.     

Nonlinear recursive estimation of volatility via estimating functions

For certain volatility models, the conditional moments that depend on the parameter are of interest. Following Godambe and Heyde (1987), the combined estimating function method has been used to study inference when the conditional mean and conditional variance are functions of the parameter of interest (See Ghahramani and Thavaneswaran [Combining Estimating Functions for Volatility. Journal of Statistical Planning and Inference, 2009, 139, 1449-1461] for details). However, for application purposes, the resulting estimates are nonlinear functions of the observations and no closed form expressions of the estimates are available. As an alternative, in this paper, a recursive estimation approach based on the combined estimating function is proposed and applied to various classes of time series models, including certain volatility models.     

Inference for linear and nonlinear stable error processes via estimating functions

This paper describes an estimating function approach for parameter estimation in linear and nonlinear times series models with infinite variance stable errors. Joint estimates of location and scale parameters are derived for classes of autoregressive (AR) models and random coefficient autoregressive (RCA) models with stable errors, as well as for AR models with stable autoregressive conditionally heteroscedastic (ARCH) errors. Fast, on-line, recursive parametric estimation for the location parameter based on estimating functions is discussed using simulation studies. A real financial time series is also discussed in some detail.     

Derivation of Kurtosis and Option Pricing Formulas for Popular Volatility Models with Applications in Finance

This article discusses some topics relevant to financial modeling. The kurtosis of a distribution plays an important role in controlling tail-behavior and is used in edgeworth expansion of the call prices. We present derivations of the kurtosis for a number of popular volatility models useful in financial applications, including the class of random coefficient GARCH models. Option pricing formulas for various classes of volatility models are also derived and a simple proof of the option pricing formula under the Black–Scholes model is given.     

A note on recursive smoothers for semimartingales

The kernel function method developed by Yamato (1971) to estimate a probability density function essentially is a way of smoothing the empirical distribution function. This paper shows how one can generalize this method to estimate signals for a semimartingale model. A recursive convolution smoothed estimate is used to obtain an absolutely continuous estimate for an absolutely continuous signal of a semimartingale model. It is also shown that the estimator obtained has a smaller asymptotic variance than the one obtained in Thavaneswaran (1988).     

Combining estimating functions for volatility

Accurate estimates of volatility are needed in risk management. Generalized autoregressive conditional heteroscedastic (GARCH) models and random coefficient autoregressive (RCA) models have been used for volatility modelling. Following Heyde [1997. Quasi-likelihood and its Applications. Springer, New York], volatility estimates are obtained by combining two different estimating functions. It turns out that the combined estimating function for the parameter in autoregressive processes with GARCH errors and RCA models contains maximum information. The combination of the least squares (LS) estimating function and the least absolute deviation (LAD) estimating function with application to GARCH model error identification is discussed as an application.     

Optimal estimation of polynomial hazard functions

The counting process formulation (Aalen, 1978) for the analysis of life time data is briefly reviewed. This formulation is used to arrive at a regression type model and a smooth estimate of the hazard function. In the regression model, the error terms are martingales and Nelson's estimator is the dependent variable. An optimal approach for estimating the parameters of the polynomial is considered, Asymptotic normality of the optimal estimate is proved and an illustrative example is given.