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Time-Varying Inputs and Assumptions in the GRS Test

Article Quant Q&A · Author: Richard Hardy
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Summary

The document asks which quantities must be stable over time for a Gibbons–Ross–Shanken joint test of CAPM pricing errors. The setup uses time-series regressions of each asset’s excess return on a factor, then tests whether all estimated intercepts are zero. The question specifically concerns whether factor loadings, average excess returns, and factor means may vary through the sample.

The response offers a tentative view: ordinary least squares estimates the regression beta using sample-wide means and covariances, so the standard implementation does not explicitly account for time-varying means or betas. This is an initial interpretation rather than a complete treatment; the respondent says the reasoning is not thorough enough to be conclusive. The text supplies no formal derivation of the GRS test’s full assumptions or guidance for handling instability, so it is best read as flagging a stationarity concern for further investigation rather than settling validity conditions.

Key ideas

  • The GRS test evaluates whether asset pricing regression intercepts are jointly zero.
  • The question raises possible time variation in betas, asset excess-return means, and factor means.
  • The response observes that pooled OLS uses full-sample averages and does not model changing means explicitly.
  • The response is preliminary and does not provide a complete statement of GRS validity conditions.

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StatisticsFactor investingEquities
Full text
# What quantities (means, betas) must be constant over time for the GRS test to be valid?


# What quantities (means, betas) must be constant over time for the GRS test to be valid?












I am interested in testing the CAPM using the GRS test. Consider $N$ assets observed for $T$ time periods. Using the notation of Cochrane "Asset Pricing" (2005), the GRS test amounts to running $N$ time series regressions of the form $$ R^{ei}_t=\alpha_i+\beta_i f_t+\varepsilon^i_t \tag{12.1} $$ and testing the joint hypothesis $H_0\colon \alpha_1=\dots=\alpha_N=0$. The $\alpha$s are treated as pricing errors, so they better be zero if the CAPM is an adequate model.

What quantities must be constant over time for the GRS test to be valid? I suppose $\beta_i$s should be constant, but what about $E(R_t^{ei})$ and $E(f_t)$; can these be time-varying?

## Answer by Richard Hardy (score 0)

https://quant.stackexchange.com/a/76986

Here is my attempt. I do not find it thorough enough to convince myself, but at least it is a start.

I think the OLS estimator of $\beta_i$ used in the GRS test implies that all three of $\beta_i$, $E(R_t^{ei})$ and $E(f_t)$ are (implicitly) assumed to be constant over time. After all, $$ \hat\beta_i^{OLS}=\frac{ \frac{1}{T-1}\sum_{t=1}^T (R^{ei}_t-\bar{R}^{ei})(f_t-\bar{f}) }{ \frac{1}{T-1}\sum_{t=1}^T (f_t-\bar{f})^2 } $$ where we use $\bar{R}^{ei}$ as the empirical counterpart to $E(R_t^{ei})$ for all $t$ and $\bar{f}$ as the empirical counterpart to $E(f_t)$ for all $t$. There is no accounting (empirically) for the possibility that $E(R_t^{ei})$ or $E(f_t)$ are time varying.

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