# Clustered vs. GMM-based standard errors: which ones to use in asset pricing?

Consider estimating an asset pricing model such as the CAPM or a multifactor model using monthly data. Petersen (2009) section "Asset pricing application" suggests use of standard errors clustered by time, as this addresses the problem of cross-sectional correlation. (Meanwhile, serial correlation is not much of an issue.) Petersen (2009) discusses how time-clustered standard errors fare against alternative approaches such as Fama-MacBeth (works fine) and several other. However, he does not seem to consider GMM estimation that is discussed extensively in Cochrane "Asset Pricing" (2005) Part II, e.g. Chapter 11.

On the other hand, Cochrane does mention clustered standard errors and Petersen's paper in his video lecture on the topic; see here and again here. However, he does not compare between clustered standard errors and GMM-based ones either.

So how do clustered standard errors and GMM-based standard errors compare? (If needed, we may consider (1) clustering by time and (2) assuming zero autocorrelation in GMM for concreteness.) Is one a special case of the other? Which ones makes more sense (given that 13 years have passed since Petersen's paper, I guess there should be some consensus on the matter)?

Due to lack of answers, this has been reposted on Cross Validated Stack Exchange.

Update (now posted as a separate question)
After some thinking, I am not sure if I can even formulate a statistically adequate panel regression model. Take the case of the CAPM written in terms of excess returns (thus the asterisks): $$R_{i,t}^*=\beta_i\mathbb{E}(R_{m,t}^*)+\varepsilon_{i,t}.$$ We can generalize it to allow for nonzero Jensen's $$\alpha$$ $$R_{i,t}^*=\alpha_i+\beta_i\mathbb{E}(R_{m,t}^*)+\varepsilon_{i,t}$$ (and later test $$H_0\colon \alpha_1=\dots=\alpha_N=0$$), but we have the problem of $$\mathbb{E}(R_{m,t}^*)$$ not being observable. Just replacing $$\mathbb{E}(R_{m,t}^*)$$ with $$R_{m,t}^*$$ would introduce measurement error and bias the estimate of $$\beta_i$$. Adding time fixed effects would not help with the problem. To see that, express $$R_{m,t}^*$$ as $$R_{m,t}^*=\mathbb{E}(R_{m,t}^*)+\varepsilon_{m,t}$$, yielding $$R_{i,t}^*=\alpha_i+\beta_iR_{m,t}^*+\beta_i\varepsilon_{m,t}^*+\varepsilon_{i,t}$$ where we see that the time fixed effect is convoluted with $$\beta_i$$. So how does one even start working out standard errors clustered by time if the model itself is inadequate? Or did I miss something?

References

• I don’t have access to the book now but the OLS type methods described in the article should be special cases of GMM methods.
– fes
Commented Feb 1, 2023 at 11:28
• Optimal clustering depends on application. For cross-sectional regressions one would expect cross-sectional correlation of errors to be a bigger issue though you can also correct for both.
– fes
Commented Feb 1, 2023 at 11:30