Interpreting the average correlation figure in Pozzi et al. (2013) paper

Has anyone read the paper "Spread of risk across financial markets:better to invest in the peripheries" by F. Pozzi, T. Di Matteo & T. Aste? If so, how do you interpret equation (1) in the dependency measure subsection of the Methods section? It looks like it produces an average correlation figure:

$$$$\bar{R}_{ij}^w(t)=\frac{1}{2(\tau + 1)} \left( \sum_{s=t-\tau}^t R^w_{ij}(s) + \sum_{i=1}^{j-1} \sum_{j=2}^{N} \sum_{s=t-\tau}^t \frac{2 R_{ij}^w(s)}{N(N-1)} \right).\\$$$$

That average figure can then be used (using methods from elsewhere in the paper and the cited papers, particularly citation 9, which I'm comfortable with) to form a shrunk covariance and correlation matrix. Is that right? OR, does the equation produce average values for each of the pairwise correlations, resulting, by itself, in a shrunk correlation matrix?

Not really expecting many answers, but anything is appreciated, thanks.

• Thank you very much for this. I can't see an option to upvote the response, or anything. I'm still making my way through it, especially the triple summation notation later in the response. Commented Dec 16, 2021 at 19:06
• Awesome, that makes sense. While we're here, question on the triple sum in the middle of your last formulation. Does that mean, in words, the sum, over six months, of the average pairwise correlation? Commented Dec 16, 2021 at 19:20
• Yes, this is my understanding.
– Pleb
Commented Dec 16, 2021 at 19:27
• Hi, @user60352, I've taken another look at this and have posted an answer below. Thus I have deleted my comments above, since they can be found in the answer :-).
– Pleb
Commented Dec 18, 2021 at 0:02

Interpretation:

The authors shrink the 6 month average of the exponentially smoothed correlations towards the average sample correlation. Thus your second formulation is correct, in the sense that equation (1) already contains shrinkage.

The basis of my statement comes from the derivations below. Here, I highlight how you can recover equation (1) using the shrinkage equation found in the paper of Ledoit and Wolf (2003).

Basis of my statement:

I will redefine some of the statements from the paper for completeness. They define the exponentially smoothed weighted moments as:

$$\sum_{s=t-\tau}^t w_s f^w\left(r(s)\right),$$

with $$w_s$$ being the exponential weights, $$r(s)$$ being the daily returns and $$f(\cdot)$$ being (in our case) the summand of the empirical correlation function. If we would take the empirical average over the above function we get:

$$\bar{f}^{w}(t)=\frac{1}{\tau + 1}\sum_{s=t-\tau}^t w_s f^w\left(r(s)\right).$$

The above formulation sums over $$\tau + 1$$ elements, since for $$\tau = 0$$ the sum contains one element.

The shrinkage equation:

We can define the shrinkage equation with $$F$$ being the target and $$S$$ being the exponentially smoothed sample correlation. Here, we set $$\delta = \frac{1}{2}$$ and see that:

\begin{align} \bar{R}_{ij}^w(t) &= \delta F + (1-\delta) S\\ &= \frac{1}{2} \left(F + S\right)\\ &= \frac{1}{2} \left( \frac{2}{(N-1)N}\sum_{i=1}^{j-1} \sum_{j=2}^{N} R_{ij}^w(t) + R^w_{ij}(t)\right), \end{align}

where you can find the formula for the average sample correlation in the original paper of Ledoit and Wolf (2003) (appendix A).

Averaging over the past $$\tau$$ months give us equation(1):

\begin{align} \bar{R}_{ij}^w(t)&=\frac{1}{2} \left( \frac{2}{(N-1)N} \frac{1}{\tau + 1}\sum_{i=1}^{j-1} \sum_{j=2}^{N}\sum_{s=t-\tau}^t R_{ij}^w(s) + \frac{1}{\tau + 1}\sum_{s=t-\tau}^t R^w_{ij}(s)\right)\\ &=\frac{1}{2(\tau + 1)} \left( \frac{2}{(N-1)N} \sum_{i=1}^{j-1} \sum_{j=2}^{N}\sum_{s=t-\tau}^t R_{ij}^w(s) + \sum_{s=t-\tau}^t R^w_{ij}(s)\right),\\ \end{align}

where the interpretation of the triple sum can be viewed as the 6 month average of the averaged pairwise sample correlations. The authors set $$\tau = 125$$ days which is 6 months.

As an added bonus, the authors further define the averaged weighted covariance matrix with shrinkage in the supplementary material chapter S.5. They further define $$P^w$$ as a diagonal matrix with averaged weighted variances over the main diagonal defined as $$(\bar{s}_{kh}^w)^2=\frac{1}{\tau + 1}\sum_{k=t-\tau}^t (\hat{s}_{kh}^w)^2$$.