Clustering Financial Time Series by Dependency

Abstract

In this paper, we propose a procedure for clustering financial time series by dependency on their volatilities. Our procedure is based on the generalized cross correlation between the estimated volatilities of a time series. Monte Carlo experiments are carried out to analyze the improvements obtained by clustering using the squared residuals instead of the levels of the series. Our procedure was able to recover the original clustering structures in all cases in our Monte Carlo study. Finally, the methodology is applied to a set of financial data.

Appendix

Correlation of Two Dependent Chi-square Variables

Let $\eta _{t,x}$ and $\eta _{t,y}$ be normal variables with mean 0, variance 1, and ${\text {cor}}(\eta _{t,x}, \eta _{t,y}) = \rho$. Suppose that $\eta _{t,y} = \rho \eta _{t,x} + a \eta _{t,z}$ with $\eta _{t,z}$ a standard normal variable independent of $\eta _{t,x}$ and $a^{2} = 1- \rho ^{2}$. Thus

$$\begin{aligned} \eta ^{2}_{t,x}\eta ^{2}_{t,y} = \eta _{t,x}^{2}(\rho ^{2} \eta ^{2}_{t,x} + 2\rho a \eta _{t,x}\eta _{t,z}+a^{2} \eta ^{2}_{t,z}) = \eta _{t,x}^{4}\rho ^{2} + 2 a \rho \eta ^{3}_{t,x}\eta _{t,z}+a^{2} \eta ^{2}_{t,z } \eta ^{2}_{t,x}.\end{aligned}$$

Using the moments of standard normal variables and the independence between $\eta _{t,x}$ and $\eta _{t,z}$, we have that $E(\eta ^{2}_{t,x}\eta ^{2}_{t,y}) = 3\rho ^{2}+ a^{2} = 2\rho ^{2}+1$. Thus, we conclude that

$$\begin{aligned} \text {cov}(\eta ^{2}_{t,x}, \eta ^{2}_{t,y}) = E(\eta ^{2}_{t,x}\eta ^{2}_{t,y}) - E(\eta ^{2}_{t,x})E(\eta ^{2}_{t,y}) = E(\eta ^{2}_{t,x}\eta ^{2}_{t,y}) -1 = 2\rho ^{2}. \end{aligned}$$

On the other hand, we know that $var(\eta _{t,x}^{2}) = var(\eta _{t,y}^{2}) = 2$, therefore,

$$\begin{aligned} \text {cor}(\eta ^{2}_{t,x}, \eta ^{2}_{t,y}) = \frac{2\rho ^{2}}{2} = \rho ^{2}. \end{aligned}$$