Estimating correlations among elliptically distributed random variables under any form of heteroskedasticity

Abstract

The paper introduces a semiparametric estimator of the correlations among elliptically distributed random variables invariant to any form of heteroscedasticity, robust to outliers, and asymptotically normal. Our estimator is particularly fit for financial applications as vectors of stock returns are generally well approximated by heteroskedastic processes with elliptical (conditional) distributions and heavy tails. The superiority of our estimator with respect to Pearson's sample correlation in financial applications is illustrated using simulated data and real high-frequency stock returns. Using simple exponentially weighted moving averages, we extend our estimator to the case of time-varying correlations and compare it to the popular GARCH-DCC model. We show that the two approaches have comparable performances through simulations and a simple application. However, our estimator is extremely fast to compute, computationally robust, and straightforward to implement.

Keywords:

• Correlation
• Heteroscedasticity
• Stochastic volatility
• Elliptical distributions
• Heavy tails
• Quadrant correlation
• Dynamic conditional correlation

JEL Classifications:

• C13
• C14
• C32
• C46
• G1

Acknowledgments

The author gratefully acknowledges the Bicocca Data Science Lab for providing computational resources.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The i.i.d. hypothesis can be relaxed provided that the marginal distribution of the processes $Z_{it}$ is continuous and elliptical with correlation matrix $\mathit{R}$ and that a law of large number (LLN) and a central limit theorem (CLT) apply. For example, $Z_{it}$ could be a martingale difference sequence, and we could exploit the LLN and CLT for these kinds of processes (see Hall and Heyde 1980, for a complete treatment). Alternatively, $Z_{it}$ could be a weakly dependent process, such as a stationary ergodic sequence, a mixing, or a near-epoch dependent process, as LLN and CLT also exist under these hypotheses (see Davidson 1994, in particular Chapters 20 and 24). Under weak dependence, the asymptotic variances of Theorem 2 would have to be adjusted to take into account the possible autocorrelation of the sequence $\left(X_{it}{X}_{jt}\right)$ but all the rest would remain valid.

2 Notice that in our notation the 5% trimmed correlation is more trimmed than the 10% correlation as percentages refer to returns and not to the share of omitted observations.

3 The DCC has been estimated using the rmgarch package (Ghalanos 2019) for R (R Core Team 2020).

4 The symbols of the selected stocks are RTX, LMT, WBA, DHR, SO, MSFT, UNH, JPM, KO, SBUX.

5 The CBOE Volatility Index (VIX) peaks on day 2020-03-16.