Multiple and Partial Correlations

In addition to Optimizer Inputs (Mean, Standard Deviations, and Correlations), the Estimator calculates estimates of Multiple and Partial Correlations, which can be viewed in the Results Worksheet. These statistics provide more information about the relationships between asset pairs and the ensemble of assets as a whole, and can be used when considering including or excluding assets based on their risk characteristics.

Mutiple Correlations show the joint correlation of each assets with all of the other assets. It is equal to the square root of the R-squared statistic from Multiple Regression, which shows the strength of the relationship between a variable and its best prediction based on all of the other variables, or the proportion of the variance explained by the other variables. Therefore, assets with low multiple correlation scores are less well explained by the other variables in the case, and providing more unique signal to the case. Assets with values of multiple correlation close to 1.0 (or -1.0) will provide less in the way of risk reduction to the case.

Mathematically, the multiple correlation for an asset can be computed from the correlation matrix. It is computed as the square root of the triple matrix product of the row vector of correlations between that asset and the other assets, times the inverse of the correlation matrix of the other assets, times the column vector of the correlations between the asset and the other assets.

The multiple correlations can be found in a column to the right of the Means and Standard Deviations on the Results Worksheet. 

The partial correlation matrix can be viewed by toggling the drop-down at the top of the correlations on the Results Worksheet, all the way to the right. By default this drop-down is set to display the full correlations. 

Partial correlations show the strength of the linear relationship between a pair of assets in the presence of all of the other assets. Equivalently, the statistic measures the correlation between the residuals of two regressions, where each of the two assets is regressed against the other assets (than the pair). Partial correlations show what remains after eliminating the relations due to common relationships with other assets. Note that these calculated statistics are for informational purposes only, and not suitable for use as Optimizer inputs.