Nonlinear spectral analysis via the local Gaussian correlation

Abstract

The spectral distribution \(f(\omega)\) of a stationary time series \(\{Y_t\}_{t\in\mathbb{Z}}\) can be used to investigate whether or not periodic structures are present in \(\{Y_t\}_{t\in\mathbb{Z}}\), but \(f(\omega)\) has some limitations due to its dependence on the autocovariances \(\gamma(h)\), For example, \(f(\omega)\) can not distinguish white i.i.d. noise from GARCH-type models (whose terms are dependent, but uncorrelated), which implies that \(f(\omega)\) can be an inadequate tool when \(\{Y_t\}_{t\in\mathbb{Z}}\) contains asymmetries and nonlinear dependencies. Asymmetries between the upper and lower tails of a time series can be investigated by means of the local Gaussian autocorrelations \(\gamma_{v}(h)\) introduced in Tjøstheim and Hufthammer (2013), and these local measures of dependence can be used to construct the local Gaussian spectral density \(f_{v}(\omega)\) that is presented in this paper. A key feature of \(f_{v}(\omega)\) is that it coincides with \(f(\omega)\) for Gaussian time series, which implies that \(f_{v}(\omega)\) can be used to detect non-Gaussian traits in the time series under investigation. In particular, if \(f(\omega)\) is flat, then peaks and troughs of \(f_{v}(\omega)\) can indicate nonlinear traits, which potentially might discover local periodic phenomena that goes undetected in an ordinary spectral analysis.