Weighting Sector Data in Turbulence and Absorption Ratio Measures

Summary

The document asks whether sector weights should enter covariance-based calculations of two market indicators. Turbulence is described through the Mahalanobis distance between an observation and its mean, using the inverse covariance matrix. The absorption ratio is identified as the share of variance explained by the leading principal components. The author is considering sector data for the S&P 500 and wonders whether equal treatment of sectors misrepresents the index compared with weighting sectors by their starting-period portfolio weights.

No answer, calculation, or empirical comparison is included. The question highlights a modeling choice, but weighting observations or variables changes the covariance structure and the interpretation of the resulting indicators; a weighted measure would need a clearly defined objective and methodology. The document does not establish that index weights should be applied, nor does it specify the estimation window, rebalancing policy, or evidence on how weighting affects either indicator.

Key ideas

• The post compares covariance-based turbulence and absorption ratio indicators.
• Turbulence uses Mahalanobis distance and an estimated inverse covariance matrix.
• The absorption ratio summarizes variance captured by leading principal components.
• The author asks whether sector data should reflect S&P 500 sector weights.
• No analysis establishes which weighting approach is preferable.

Tags

# PCA on covariance matrix with weights on the columns?


# PCA on covariance matrix with weights on the columns?












I'm reading two papers by Mark Kritzman on two indicators (turbulence proxied by the Mahalanobis distance and absorption ratio which is basically the ratio of the variance captured by the top 20% PCA components), with formulas below:

Turbulence: $$ d_t = (y_t - \mu) * \Sigma^{-1} * (y_t - \mu)' $$

Absorption Ratio:

Notice that both formulas involve estimating a covariance matrix. While reading the original papers of the author, I have the impression that he does not consider any weight when estimating the covariance matrix. However, would it make more sense to include a column weight for each column according to the weight at the start of the estimation period ? For example in the Turbulence measure, considering that he used the data for 10 (now 11) sectors in the S&P 500, wouldn't it make more sense if we include the weights of each sector in the calculations because we are measuring the Turbulence measure for the S&P 500 after all, if we treat every component equally then would it not skew the final measure ?

The question is the same for the absorption ratio.

Thanks for your help.

(edited for math, but edits must be more than 6 characters, thus this notice.)

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