Nonlinear Spectral Analysis: A Local Gaussian Approach

Abstract

The spectral distribution $f(\omega )$ of a stationary time series ${\left\{Y_t\right\}}_{t\in \mathbb{Z}}^{}$ can be used to investigate whether or not periodic structures are present in ${\left\{Y_t\right\}}_{t\in \mathbb{Z}}^{}$, but $f(\omega )$ has some limitations due to its dependence on the autocovariances $\gamma (h)$. For example, $f(\omega )$ can not distinguish white iid noise from GARCH-type models (whose terms are dependent, but uncorrelated), which implies that $f(\omega )$ can be an inadequate tool when ${\left\{Y_t\right\}}_{t\in \mathbb{Z}}^{}$ 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, and these local measures of dependence can be used to construct the local Gaussian spectral density presented in this paper. A key feature of the new local spectral density is that it coincides with $f(\omega )$ for Gaussian time series, which implies that it 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 the new local spectral density can indicate nonlinear traits, which potentially might discover local periodic phenomena that remain undetected in an ordinary spectral analysis.

KEYWORDS:

• GARCH models
• Graphical tools
• Local periodicities

Supplementary Materials

The online supplementary materials contain the appendices. The scripts needed for the reproduction of the examples in this article is contained in the R-package localgaussSpec, cf. Appendix G for further details.

Acknowledgments

The authors are most grateful for the valuable comments and suggestions from the referees and the associate editor.

Notes

1 Use remotes::install_github(”LAJordanger/localgaussSpec”) to install the package. See the documentation of the function LG_extract_scripts for further details. See also Appendix G.

2 Strict stationarity is necessary in order for the machinery of the local Gaussian approximations to be feasible, since Gaussian pdfs will be used to locally approximate the pdfs corresponding to the bivariate pairs $(Y_{t+h},Y_t)$.

3 The vector $\mathit{\theta }$ is a function of the point v , but this will henceforth be suppressed in the notation.

4 The solution ${\mathit{\theta }}_0$ will always satisfy Equation (5a), but it will in general not satisfy Equation (5b).

5 Confer Appendix B.1.2 in the supplementary materials for a detailed exposition.

6 For a proof of this statement; see, for example, Nelsen (2006, Theorem 2.4.3).

7 See Berentsen, Tjøstheim, and Nordbø (2014b) for an approach where this is used to construct a canonical local Gaussian correlation for the copula C.

8 The matrices then becomes 1 × 1, so the singularity problems does not occur.

9 Note that it is not the local Gaussian correlation that is the target of interest when this simplified approach is used for density estimation.

10 The theory for the normalized one-free-parameter version of LGC is available in the first authors PhD-thesis, https://bora.uib.no/handle/1956/16950. This also contains a discussion with regard to why an approach based on the normalized one-free-parameter approach fails to produce decent results.

11 This is the Deutschemark/British pound Exchange Rate (dmbp) data from Bollerslev and Ghysels (1996), which is a common benchmark dataset for GARCH-type models, and as such models are among the motivating factors for the study of the local Gaussian spectral density, it seems natural to test the method on dmbp. The data plotted here was found in the R-package rugarch, see Ghalanos (2020), where the following description was given: “The daily percentage nominal returns computed as $100\left[\text{ln}\hspace{0.17em}(P_t)-\hspace{0.17em}\text{ln}\hspace{0.17em}(P_t-1)\right]$, where P_t is the bilateral Deutschemark/British pound rate constructed from the corresponding U.S. dollar rates.”

12 Consider the function to be a constant with respect to all the new variables that are introduced.

13 Notational convention: “$\vee$” denotes the maximum of two numbers, whereas “$\wedge$” denotes the minimum.

14 The corresponding coordinates are $(-1.28,-1.28)$, (0, 0) and $(1.28,1.28)$.

15 See Chang et al. (2017) for details about shiny.

16 A further investigation of this is easy when the shiny-application in the R-package localgaussSpec is used, since it then is possible to immediately switch to an investigation of the corresponding spectra.

17 If you have a black and white copy of this article, then read “red” as “dark” and “blue” as “light.”

18 Solid lines are always used by the R-package localgaussSpec when ${\widehat{f}}_v^m(\omega )$ is based on real data.

19 The R-package rugarch, Ghalanos (2020) was used to find the parameters of a multitude of GARCH-type models, and the asymmetric power ARCH model with the best fit was then selected.