Econometric estimation in long-range dependent volatility models: Theory and practice

Abstract

It is commonly accepted that some financial data may exhibit long-range dependence, while other financial data exhibit intermediate-range dependence or short-range dependence. These behaviours may be fitted to a continuous-time fractional stochastic model.The estimation procedure proposed in this paper is based on a continuous-time version of the Gauss-Whittle objective function to find the parameter estimates that minimize the discrepancy between the spectral density and the data periodogram. As a special case, the proposed estimation procedure is applied to a class of fractional stochastic volatility models to estimate the drift, standard deviation and memory parameters of the volatility process under consideration. As an aplication, the volatility of the Dow Jones, S&P 500, CAC 40, DAX 30, FTSE 100 and NIKKEI 225 is estimated.

On the robustness to small trends of parameter estimation for continuous-time stationary models with memory

“…Continuous versions of Whittle estimation procedure have been considered, for example, in Anh et al [3,4], Avram et al [5], Casas and Gao [9], Gao [14], Gao et al [15,16], Leonenko and Sakhno [26].…”

“…It can be shown that the standard Taylor expansion methods based on the smoothed periodogram convergence results of type (3.4) and (3.5) with a general smoothing function g(λ; θ) and with the contaminated periodogram I T,X (λ) instead of I T,Y (λ), lead consistent and asymptotically normally distributed estimators of θ. We will not pursue this matter here (the details will be reported elsewhere), however, notice that in the special case of Whittle procedure, where g(λ; θ) = ∂ ∂ 1 f (λ,θ) · w(λ) the results of Anh et al [3], Avram et al [5], Casas and Gao [9], Gao [14], Gao et al [15,16], Leonenko and Sakhno [26] concerning consistency and asymptotic normality of the Whittle minimum contrast estimators constructed on the basis of the periodogram I T,Y (λ), continue to hold without change for estimators calculated on the basis of the contaminated periodogram I T,X (λ), under appropriate assumptions imposed on the model Y (t) on the smoothing function g(λ, θ) and on the trend M (t).…”

“…Taking into account that β + γ > 1, from (4.8) we get 9) and hence for T > 2, I 4 (T ) can be estimated as follows…”

“…However, data in many fields of science (economics, finance, hydrology, telecommunications, etc.) is well modeled by a stationary process with unbounded or vanishing at the origin spectral density (see, e.g., Beran et al [7], Casas and Gao [9], Gao [14], Tsai and Chan [29], and references therein).…”

Maximum Likelihood Estimation With Dynamic Measurement Errors and Application to Interest Rate Modeling

“…Some considered jumps in the dynamics of asset prices Andersen and Bollerslev (1997)]. Recent works on this problem can be found in [Casas and Gao (2008); Hu (2004)]. Most of these works assumed the model to be generated by the finite parameter model.…”