Dependence Estimation for High Frequency Sampled Multivariate CARMA Models

TL;DR

This paper analyzes the asymptotic behavior of sample autocovariance and cross-covariance functions in high-frequency sampled multivariate MCARMA models, establishing normal limit distributions and extending classical formulas.

Contribution

It derives the asymptotic distribution of sample autocovariance functions for multivariate MCARMA models and extends Bartlett's formula to the continuous-time setting.

Findings

Asymptotic normality of sample autocovariance functions

Extension of Bartlett's formula to continuous-time models

Limit results for multivariate MA processes

Abstract

The paper considers high frequency sampled multivariate continuous-time ARMA (MCARMA) models, and derives the asymptotic behavior of the sample autocovariance function to a normal random matrix. Moreover, we obtain the asymptotic behavior of the cross-covariances between different components of the model. We will see that the limit distribution of the sample autocovariance function has a similar structure in the continuous-time and in the discrete-time model. As special case we consider a CARMA (one-dimensional MCARMA) process. For a CARMA process we prove Bartlett's formula for the sample autocorrelation function. Bartlett's formula has the same form in both models, only the sums in the discrete-time model are exchanged by integrals in the continuous-time model. Finally, we present limit results for multivariate MA processes as well which are not known in this generality in the multivariate setting yet.