Nonlinear spectral analysis: A local Gaussian approach

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 Tj{\o}stheim 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.