Statistical modeling with COGARCH(p,q) processes
Date
2009
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Abstract
In this paper, a family of continuous time GARCH processes, generalizing the COGARCH(1, 1) process of Klüppelberg, et al. (2004), is introduced and studied. The resulting COGARCH(p,q) processes, q ≥ p ≥ 1, exhibit many of the characteristic features of observed financial time series, while their corresponding volatility and squared increment processes display a broader range of autocorrelation structures than those of the COGARCH(1, 1) process. We establish sufficient conditions for the existence of a strictly stationary non-negative solution of the equations for the volatility process and, under conditions which ensure the finiteness of the required moments, determine the autocorrelation functions of both the volatility and squared increment processes. The volatility process is found to have the autocorrelation function of a continuous-time ARMA process while the squared increment process has the autocorrelation function of an ARMA process.
To estimate the parameters of the COGARCH(2, 2) processes, the least-squares method is used. We give conditions under which the volatility and the squared increment processes are strongly mixing, from which it follows that the least-squares estimators are strongly consistent and asymptotically normal. Finally, the model is fitted to a high frequency dataset.
To estimate the parameters of the COGARCH(2, 2) processes, the least-squares method is used. We give conditions under which the volatility and the squared increment processes are strongly mixing, from which it follows that the least-squares estimators are strongly consistent and asymptotically normal. Finally, the model is fitted to a high frequency dataset.
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Subject
GARCH processes
least-squares estimators
volatility process
statistics
finance
applied mathematics