Local Gaussian CrossSpectrum Analysis
Abstract
:1. Introduction
2. Definitions
2.1. The Local Gaussian Correlations
2.2. The Local Gaussian CrossSpectrum
 1.
 With ${G}_{k}^{}$ and ${G}_{\ell}^{}$ as the univariate marginal cumulative distributions of, respectively, ${\left\{{Y}_{k,t}^{}\right\}}_{t\in \mathbb{Z}}^{}$ and ${\left\{{Y}_{\ell ,t}^{}\right\}}_{t\in \mathbb{Z}}^{}$, and Φ as the cumulative distribution of the univariate standard normal distribution, define normalized versions ${\left\{{Z}_{k,t}^{}\right\}}_{t\in \mathbb{Z}}^{}$ and ${\left\{{Z}_{\ell ,t}^{}\right\}}_{t\in \mathbb{Z}}^{}$ by$$\begin{array}{c}\hfill {\left\{{Z}_{k,t}^{}:={\Phi}_{}^{1}\phantom{\rule{0.166667em}{0ex}}\left({G}_{k}^{}\phantom{\rule{0.166667em}{0ex}}\left({Y}_{k,t}^{}\right)\right)\right\}}_{t\in \mathbb{Z}}^{},\phantom{\rule{2.em}{0ex}}{\left\{{Z}_{\ell ,t}^{}:={\Phi}_{}^{1}\phantom{\rule{0.166667em}{0ex}}\left({G}_{\ell}^{}\phantom{\rule{0.166667em}{0ex}}\left({Y}_{\ell ,t}^{}\right)\right)\right\}}_{t\in \mathbb{Z}}^{}.\end{array}$$
 2.
 For a given point $\mathit{v}=({v}_{1}^{},{v}_{2}^{})$ and for each bivariate lag h pair ${\mathit{Z}}_{k\ell :h:t}^{}:=({Z}_{k:t+h}^{},{Z}_{\ell :t}^{})$, a local Gaussian crosscorrelation ${\rho}_{k\ell :\mathit{v}}\left(h\right)$ can be computed based on a fiveparameter local Gaussian approximation of the bivariate density of ${\mathit{Z}}_{k\ell :h:t}^{}$ at $({v}_{1}^{},{v}_{2}^{})$.
 3.
 When ${\sum}_{h\in \mathbb{Z}}^{}\left{\rho}_{k\ell :\mathit{v}}\left(h\right)\right<\infty $, the local Gaussian crossspectrum at the point $\mathit{v}$ is defined as$$\begin{array}{c}\hfill {f}_{k\ell :\mathit{v}}\left(\omega \right):=\sum _{h=\infty}^{\infty}{\rho}_{k\ell :\mathit{v}}\left(h\right)\xb7{e}^{2\pi i\omega h}.\end{array}$$
 1.
 ${f}_{k\ell :\mathit{v}}\left(\omega \right)$ coincides with ${f}_{k\ell}\left(\omega \right)$ for all $\mathit{v}\in {\mathbb{R}}^{2}$ when ${\left\{{\mathit{Y}}_{t}^{}\right\}}_{t\in \mathbb{Z}}^{}$ is a multivariate Gaussian time series.
 2.
 The following holds when $\stackrel{\u02d8}{\mathit{v}}:=({v}_{2}^{},{v}_{1}^{})$ is the diagonal reflection of $\mathit{v}=({v}_{1}^{},{v}_{2}^{})$:$$\begin{array}{cc}\hfill {f}_{k\ell :\mathit{v}}\left(\omega \right)& =\overline{{f}_{\ell k:\stackrel{\u02d8}{\mathit{v}}}\left(\omega \right)},\hfill \end{array}$$$$\begin{array}{cc}\hfill {f}_{k\ell :\mathit{v}}\left(\omega \right)& ={\rho}_{k\ell :\mathit{v}}\left(0\right)+\sum _{h=1}^{\infty}{\rho}_{\ell k:\stackrel{\u02d8}{\mathit{v}}}\left(h\right)\xb7{e}^{+2\pi i\omega h}+\sum _{h=1}^{\infty}{\rho}_{k\ell :\mathit{v}}\left(h\right)\xb7{e}^{2\pi i\omega h}.\hfill \end{array}$$
2.3. Related Local Gaussian Entities
Algorithm 1 For a sample ${\left\{{\mathit{y}}_{t}^{}=({y}_{1,t}^{},\cdots ,{y}_{d,t}^{})\right\}}_{t=1}^{n}$ of size n from a multivariate time series, an mtruncated estimate ${\widehat{f}}_{k\ell :\mathit{v}}^{m}\left(\omega \right)$ of ${f}_{k\ell :\mathit{v}}\left(\omega \right)$ is constructed by means of the following procedure. 

2.4. Estimation
The Input Parameters and Some Other Technical Details
2.5. Asymptotic Theory for ${\widehat{f}}_{k\ell :\mathit{v}}^{m}\left(\omega \right)$
2.5.1. A Brief Sketch of the Requirements for ${\mathit{Y}}_{t}^{}=({Y}_{1,t}^{},\cdots ,{Y}_{d,t}^{})$
2.5.2. Convergence Theorems for ${\widehat{f}}_{k\ell :\mathit{v}}^{m}\left(\omega \right)$, ${\widehat{\alpha}}_{k\ell :\mathit{v}}^{m}\left(\omega \right)$, and ${\widehat{\varphi}}_{k\ell :\mathit{v}}^{m}\left(\omega \right)$
3. Visualizations and Interpretations
3.1. Sanity Testing the Implemented Estimation Algorithm
3.1.1. Bivariate Gaussian White Noise
3.1.2. Bivariate Local Trigonometric Examples
 Select $r\ge 2$ bivariate time series ${\left\{({C}_{1,i}^{}\left(t\right),{C}_{2,i}^{}\left(t\right))\right\}}_{i=1}^{r}$.
 Select a random variable I with values in the set $\{1,\cdots ,r\}$, and use this to sample a collection of indices ${\left\{{I}_{t}^{}\right\}}_{t\in \mathbb{Z}}^{}$ (that is, for each t, an independent realization of I is taken). Let ${p}_{i}^{}:=\mathrm{P}({I}_{i}^{}=i)$ denote the probabilities for the different outcomes.
 Define ${Y}_{t}^{}$ by means of the equation$${Y}_{1,t}^{}:=\sum _{i=1}^{r}{C}_{1,i}^{}\left(t\right)\xb7\U0001d7d9\{{I}_{t}^{}=i\},$$$${Y}_{2,t}^{}:=\sum _{i=1}^{r}{C}_{2,i}^{}\left(t\right)\xb7\U0001d7d9\{{I}_{t}^{}=i\}.$$The indicator function 𝟙{·} ensures that only one of the bivariate $({C}_{1,i}^{}\left(t\right),{C}_{2,i}^{}\left(t\right))$components contributes for a given value t, that is, it is also possible to write $({Y}_{1,t}^{},{Y}_{2,t}^{})=({C}_{1,{I}_{t}^{}}^{}\left(t\right),{C}_{2,{I}_{t}^{}}^{}\left(t\right))$.
3.2. A Real Multivariate Time Series and a Poorly Fitted GARCHType Model
3.2.1. The DAX–CAC Subset of the EuStockMarketsLogReturns
3.2.2. A Simple CopulaGARCHModel Fitted to the EuStockMarkets LogReturns
 Fit the selected model to the data.
 Perform a local Gaussian spectrum investigation based on simulated samples from the fitted model. The parameters should match those used in the investigation of the original data.
 Compare the plots based on the original data with corresponding plots based on the simulated data from the model. It can be of interest to not only compare the Co, Quad, and Phaseplots, but also include plots that show the traces and the estimated local Gaussian auto and crossspectra.
4. Conclusions
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Notes
1  This multivariate approach was initiated in the first author’s Ph.D. thesis, available at https://bora.uib.no/handle/1956/16950. The version in this paper has been extended with new methods and visualizations that were developed due to review comments related to the univariate theory published in JT22. 
2  This is due to the way the local Gaussian correlation is defined; see Tjøstheim and Hufthammer (2013) for details. 
3  The corresponding coordinates are $(1.28,1.28)$, $(0,0)$, and $(1.28,1.28)$. 
4  The Amplitudeplots are not included here since the interesting details (in most cases) would already have been detected by the other plots. 
5  If you have a black and white copy of this paper, then read “blue” as “light” and “red” as “dark”. 
6  The dotted lines represent the means of the estimated values, whereas the 90% pointwise confidence intervals are based on the 5% and 95% quantiles of these samples. 
7  In this respect, the situation is similar to the detection of a pure sinusoidal for the global spectrum. 
8  The corresponding script in the Rpackage localgaussSpec enables an investigation of all the combinations between DAX, SMI, CAC, and FTSE, but only the DAX–CAC subset will be discussed here. 
9  
10  Use remotes::install_github("LAJordanger/localgaussSpec") to install the package. See Section S6.1 in the online Supplementary Material for further details. 
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Jordanger, L.A.; Tjøstheim, D. Local Gaussian CrossSpectrum Analysis. Econometrics 2023, 11, 12. https://doi.org/10.3390/econometrics11020012
Jordanger LA, Tjøstheim D. Local Gaussian CrossSpectrum Analysis. Econometrics. 2023; 11(2):12. https://doi.org/10.3390/econometrics11020012
Chicago/Turabian StyleJordanger, Lars Arne, and Dag Tjøstheim. 2023. "Local Gaussian CrossSpectrum Analysis" Econometrics 11, no. 2: 12. https://doi.org/10.3390/econometrics11020012