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A Robust Similarity Estimator

Ilya Archakov

Abstract

We construct and analyze an estimator of association between random variables based on their similarity in both direction and magnitude. Under special conditions, the proposed measure becomes a robust and consistent estimator of the linear correlation, for which an exact sampling distribution is available. This distribution is intrinsically insensitive to heavy tails and outliers, thereby facilitating robust inference for correlations. The measure can be naturally extended to higher dimensions, where it admits an interpretation as an indicator of joint similarity among multiple random variables. We investigate the empirical performance of the proposed measure with financial return data at both high and low frequencies. Specifically, we apply the new estimator to construct confidence intervals for correlations based on intraday returns and to develop a new specification for multivariate GARCH models.

A Robust Similarity Estimator

Abstract

We construct and analyze an estimator of association between random variables based on their similarity in both direction and magnitude. Under special conditions, the proposed measure becomes a robust and consistent estimator of the linear correlation, for which an exact sampling distribution is available. This distribution is intrinsically insensitive to heavy tails and outliers, thereby facilitating robust inference for correlations. The measure can be naturally extended to higher dimensions, where it admits an interpretation as an indicator of joint similarity among multiple random variables. We investigate the empirical performance of the proposed measure with financial return data at both high and low frequencies. Specifically, we apply the new estimator to construct confidence intervals for correlations based on intraday returns and to develop a new specification for multivariate GARCH models.
Paper Structure (13 sections, 4 theorems, 54 equations, 8 figures, 1 table)

This paper contains 13 sections, 4 theorems, 54 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. A Measure of Similarity for Random Variables
  3. The Similarity Estimator
  4. Robust Estimation and Inference for Correlations under Variance Homogeneity
  5. Similarity Estimator under Variance Heterogeneity
  6. Measuring Similarity for Multiple Variables
  7. Equicorrelation Scenario
  8. Empirical Applications
  9. Robust Confidence Intervals for Stock Return Correlations
  10. A Robust Multivariate GARCH Model
  11. Conditional Correlations for Bivariate Model
  12. Dynamic Equicorrelation Model
  13. Conclusion

Key Result

Proposition 1

Assume that $x=(x_{1},x_{2})^{\prime}$ is a bivariate random vector which follows some elliptical distribution with zero mean and positive-definite covariance matrix $\Sigma$ with homogeneous variances, and let $\rho$ denotes the Pearson correlation between $x_{1}$ and $x_{2}$. Denote the measure of where $\phi_{\rho}=\frac{1}{2}\log\Bigl(\frac{1+\rho}{1-\rho}\Bigl)$ is the Fisher transformation o

Figures (8)

  • Figure 1: A heatmap plot for $\phi_{r}$ as a function of $x_{1}$ and $x_{2}$. Areas with the red color indicate higher values of $\gamma$, while areas with the blue color indicate lower (negative) values of $\phi_{r}$. The white lines corresponding to $x_{1}=x_{2}$ ($r=1$) and $x_{1}=-x_{2}$ ($r=-1$) represent the loci where the function is undefined.
  • Figure 2: The probability density functions of $z_{\hat{\gamma}}=\frac{\sqrt{T}(\hat{\gamma}-\phi_{\rho})}{\pi/2}$ for $T=1,3,10$ (colored lines) and the standard normal probability density (dashed line) which is the distribution limit of $z_{\hat{\gamma}}$ for $T\rightarrow\infty$.
  • Figure 3: Finite sample distributions of $\phi_{\hat{\rho}}$ (red dashed lines) and $\hat{\gamma}$ (blue solid lines) obtained on 10,000 simulated samples of sizes $T=8$ (top plots) and $T=40$ (bottom plots). Data vectors $x_{t}$ were simulated out of the three selected distributions -- normal, $t$-distribution with 5 degrees of freedom, and Cauchy -- with the true correlation parameter $\rho=0.5$.
  • Figure 4: Probability density functions of $\phi_{r}-\phi_{\rho}$ for different dimensions $n$ of vector $x$.
  • Figure 5: Confidence intervals for daily correlations between Apple (AAPL) and Exxon Mobil (XOM) estimated using intraday returns. The analyzed period is between February and April 2020 (62 trading days). The intervals are obtained for coverage probabilities 90% (boxes) and 95% (whiskers). Red circles correspond to daily Realized (sample) Correlations, while blue squares correspond to daily correlations estimated by the Kendall tau coefficient.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma A.1
  • proof