Direction dependence in a regression line

By Y. Dodge, V. Rousson. 2000

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Analysis of directional dependence using asymmetric copula-based regression models (2014, methodology): “von Eye and DeShon [31] analysed the same data to determine the direction of dependence using the linear regression method proposed by Dodge and Rousson.[10] Assuming that the error term in a linear regression model is symmetric, the cube of the Pearson’s correlation between two variables X and Y (d”
A Note on Directional Dependence in Regression Setting (2005, methodology): “…random variable, which implies that 2r+1 = 0, r = 0 k− 1 /2, we end up with kXY = ∑k r=0 k−r even k−r r Y / kY k X / k X For k = 3, 3XY = 3 0 Y + 2 1 Y + 1 2 Y + 0 3 Y / 3Y 3 X / 3 X = 3 Y / 3 Y 3 X / 3 X which is the same as the ones derived by Dodge and Rousson (2000) and Muddapur (2003).”
Asymmetric properties of the Pearson correlation coefficient: Correlation as the negative association between linear regression residuals (2016, methodology): “Table 1 shows the parameter recovery of Equation (5) and the asymmetric formulas of the Pearson correlation coefficient based on manifest variables X and Y given in Dodge and Rousson (2001) for samples sizes of 50, 250, and 500.”