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Generalized Measures of Correlation for Asymmetry, Nonlinearity, and Beyond

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  • Shurong Zheng
  • Ning-Zhong Shi
  • Zhengjun Zhang

Abstract

Applicability of Pearson's correlation as a measure of explained variance is by now well understood. One of its limitations is that it does not account for asymmetry in explained variance. Aiming to develop broad applicable correlation measures, we study a pair of generalized measures of correlation (GMC) that deals with asymmetries in explained variances, and linear or nonlinear relations between random variables. We present examples under which the paired measures are identical, and they become a symmetric correlation measure that is the same as the squared Pearson's correlation coefficient. As a result, Pearson's correlation is a special case of GMC. Theoretical properties of GMC show that GMC can be applicable in numerous applications and can lead to more meaningful conclusions and improved decision making. In statistical inference, the joint asymptotics of the kernel-based estimators for GMC are derived and are used to test whether or not two random variables are symmetric in explaining variances. The testing results give important guidance in practical model selection problems. The efficiency of the test statistics is illustrated in simulation examples. In real-data analysis, we present an important application of GMC in explained variances and market movements among three important economic and financial monetary indicators. This article has online supplementary materials.

Suggested Citation

  • Shurong Zheng & Ning-Zhong Shi & Zhengjun Zhang, 2012. "Generalized Measures of Correlation for Asymmetry, Nonlinearity, and Beyond," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(499), pages 1239-1252, September.
    • Handle: RePEc:taf:jnlasa:v:107:y:2012:i:499:p:1239-1252
    DOI: 10.1080/01621459.2012.710509
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    2. Hrishikesh D. Vinod & P. M. Rao, 2019. "Externalities from Intra-Firm Trade by U.S. Multinationals," International Advances in Economic Research, Springer;International Atlantic Economic Society, vol. 25(4), pages 389-397, November.
    3. Jiang, Zhenzhen & Guo, Hongping & Wang, Jinjuan, 2023. "Feature screening for multiple responses," Journal of Multivariate Analysis, Elsevier, vol. 198(C).
    4. Maneejuk, Paravee & Yamaka, Woraphon, 2020. "An analysis of the impacts of telecommunications technology and innovation on economic growth," Telecommunications Policy, Elsevier, vol. 44(10).
    5. H. D. Vinod, 2022. "Generalized, Partial and Canonical Correlation Coefficients," Computational Economics, Springer;Society for Computational Economics, vol. 60(4), pages 1479-1506, December.
    6. Wang, Christina Dan & Chen, Zhao & Lian, Yimin & Chen, Min, 2022. "Asset selection based on high frequency Sharpe ratio," Journal of Econometrics, Elsevier, vol. 227(1), pages 168-188.
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    8. David E Allen & Vince Hooper, 2018. "Generalized Correlation Measures of Causality and Forecasts of the VIX Using Non-Linear Models," Sustainability, MDPI, vol. 10(8), pages 1-15, August.
    9. Manuel de Mier & Fernando Delbianco & Fernando Tohmé & Luisina Patrizio & Facundo Rodriguez & Mauro Romero Stéfani, 2023. "Causality by Vote: Aggregating Evidence on Causal Relations in Economic Growth Processes," Working Papers 260, Red Nacional de Investigadores en Economía (RedNIE).
    10. Polanski, Arnold & Stoja, Evarist, 2017. "Forecasting multidimensional tail risk at short and long horizons," International Journal of Forecasting, Elsevier, vol. 33(4), pages 958-969.
    11. Ćmiel, Bogdan & Ledwina, Teresa, 2020. "Validation of association," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 55-67.
    12. Polanski, Arnold & Stoja, Evarist, 2017. "Forecasting multidimensional tail risk at short and long horizons," Bank of England working papers 660, Bank of England.
    13. Lyu, Yongjian & Yi, Heling & Wei, Yu & Yang, Mo, 2021. "Revisiting the role of economic uncertainty in oil price fluctuations: Evidence from a new time-varying oil market model," Economic Modelling, Elsevier, vol. 103(C).
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    15. Han Lin Shang & Kaiying Ji & Ufuk Beyaztas, 2021. "Granger causality of bivariate stationary curve time series," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 40(4), pages 626-635, July.

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