A Least Squares Estimator for Lévy-driven Moving Averages Based on Discrete Time Observations

This article is concerned with a least squares estimator (LSE) of the kernel function parameter θ for a Lévy-driven moving average of the form X(t) = ∫t− ∞K(θ(t − s)) dL(s), where L={L(t),t∈R}$L=\lbrace L(t),t\in \mathbb {R}\rbrace$ is a Lévy process without the Brownian motion part, K is a kernel function and θ > 0 is a parameter. Let h be the time span between two consecutive observations and let n be the size of sample. As h → 0 and nh → ∞, consistency and asymptotic normality of the LSE are studied. The small-sample performance of the LSE is evaluated by means of a simulation experiment. Finally, two real-data applications show that the Lévy-driven moving average gives a good approximation to the autocorrelation of the process.