Local Gaussian Cross-Spectrum Analysis
Ta-Hsin Li
Round 1
Reviewer 1 Report
The ordinary spectral density function is the Fourier transform of the autocorrelation function. In JT22, the authors proposed a new spectrum by replacing the ordinary autocorrelations with the so-called local Gaussian autocorrelations. The current manuscript generalizes the technique from single time series to multiple time series by introducing a cross-spectrum based on the local Gaussian cross-correlations. The manuscript contains a description of the method, an asymptotic estimation theory, some simulated examples, and a real-data example of financial time series. While the generalization of the methodology is clearly and expertly described, additional work on the presentation would make the manuscript acceptable for publication.
{\bf Major comments:}
\begin{enumerate}
\item Sections 2.1-2.4 are well presented, but Section 2.5 is hard to follow and should be restructures/simplified.
\begin{enumerate}
\item A more concise definition of $\theta_{v|k\ell:h}$ should be given right after Eq (12) as the limiting value of
the minimizer of $q_{v|k\ell:h,b}$ as $b \rightarrow 0$ (if this is the case). Eq (13) adds little value here.
It is better to move the discussion of this equation to the appendix along with definitions 2.5-2.6 and assumptions 2.1-2.3. At this point, it suffices to say that under the technical conditions in the appendix, the quantity $\theta_{v|k\ell:h}$ exists and does not depend on the choice of the kernel function.
\item After the definition of the true parameter and spectrum, Theorems 2.1-2.3 can be stated immediately with the necessary definitions of the variables involved while pointing to the appendix for technical assumptions. A Layman's description of the technical requirements (e.g., the $b$'s have to shrink to zero at a certain rate relative to $n$) could be helpful here.
\end{enumerate}
\item The discussion on the input parameters does not belong to ``visualizations and interpretations''. A better place for this discussion is in ``estimation'' or right after it.
\item The presentation of the bivariate local trigonometric example could be improved.
\begin{enumerate}
\item A better explanation is needed to help understand why a spectral peak is expected at a particular frequency and a particular location $v$. It is inadequate just to say ``it is (by construction) known in advance that some points v will be of interest to inspect.'' A time series plot could be helpful here.
\item The added value of the second experiment with different phases is limited, unless the objective to estimate the phase parameters from the spectra.
\end{enumerate}
\item The financial data example makes one wonder if there are previous studies or theories that point to the possibility of DAX leading CAC
or vise versa in market rallies.
\item Inconsistent and confusing notations.
\begin{enumerate}
\item Except the symmetric property in Lemma 2.1, is it really necessary to introduce $\check{v}$ in (8)-(10) ? Why don't simply use $\rho_{k\ell:v}(-h)$ instead of $\rho_{\ell k:\check{v}}(h)$?
\item What's the difference between subscripts ${}_{k\ell:v}$ and ${}_{k\ell | v}$?
\item Why don't use $\theta_{k\ell:v}(h)$ rather than $\theta_{v|k\ell:h}$ to make it consistent with the notation $\rho_{k\ell:v}(h)$?
\end{enumerate}
\item Some remarks in the beginning of each section are repetitive and could be reduced in order
to shorten the manuscript which seems too long for its content.
\end{enumerate}
{\bf Minor comments:}
\begin{enumerate}
\item Eq (1) is redundant. Eq (2) is sufficient to make the connection between the classical cross spectrum and the proposed cross spectrum.
\item The distance plot in Figure 2 is not defined.
\item If only DAX and CAC are used in the experiment, why don't label them as $(Y_1,Y_2)$ instead of $(Y_1,Y_3)$?
\end{enumerate}
Author Response
Please see the attachment.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
see the report attached
Comments for author File: Comments.pdf
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The authors have addressed my concerns.
