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GMM, OLS, and HAC Inference in CAPM Tests with Overlapping Returns

Article Quant Q&A · Author: anw
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Summary

The document raises a methodological question about testing whether variance swap returns are explained by a single market factor. The described study regresses overlapping 30-day variance swap returns on excess market returns and tests whether the intercept differs from zero. The question is how generalized method of moments relates to ordinary least squares when the factor is a traded return and the regression uses Newey–West heteroskedasticity and autocorrelation consistent inference.

It points out that the usual two regression moment conditions can identify the intercept and beta, which may make the coefficient estimates coincide with OLS. It asks what role a GMM weighting matrix then plays and whether GMM offers a distinct advantage over OLS with HAC standard errors. The document contains no answer or empirical resolution; its main value is framing the distinction between parameter estimation, moment weighting, and robust inference when observations overlap.

Key ideas

  • A single-factor CAPM test can be framed as a regression of asset excess returns on the market excess return.
  • Overlapping return windows can induce serial dependence in regression errors.
  • With just-identified regression moments, GMM coefficient estimates may coincide with OLS.
  • The question distinguishes the GMM weighting matrix from HAC adjustments used for inference.

Tags

StatisticsVolatilityOptionsBacktesting
Full text
# Generalized method of moments concept in CAPM testing


# Generalized method of moments concept in CAPM testing












In the course of my master thesis I’ve come across a paper by Carr and Wu (2009) where the authors evaluate whether returns on variance swaps can be explained by the simple CAPM. (really only market factor). They do this by regressing overlapping 30-day returns on the variance swap on the excess market return and checking whether the alpha is significant. So here is the point that I do not understand. They use the generalized method of moments (GMM) and say that the weighting matrix is computed according to Newey-West (1987). I have read in several books and works (Cochrane (2001)), (also here https://www.kevinsheppard.com/images/5/55/Chapter6.pdf) that to evaluate a beta-factor model where the factor itself is a traded asset with GMM, the two moment conditions are I: (ri-alpha-betaft)=0 II: (ri-alpha-betaft)*ft=0 , where ri denotes the excess return on asset i and ft the market excess return. If I understood the concept of GMM correctly, these two moment conditions would just identify the system and for this case, GMM would collapse to OLS. Then, for my understanding, there would be no weighting matrix. I am really no expert on GMM and pretty sure that I got something wrong or missed something, so I have two questions and would be grateful for an answer:

1) Is there a special reason to prefer GMM over OLS when you use Newey-West HAC errors in combination with OLS to control for the correlation in the residuals in the context outlined above? (on every single day, I have overlapping 30-day returns to regress on the market)

2) What could the described weighting matrix refer to in this context or did I understand the concept of just defined wrongly?

Thank you very much in advance and best regards, Sven

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.