# Why fixed design analysis for observational data [duplicate]

Why do we use a fixed design analysis of regression coefficients, even for observational data, where the design is not fixed?

For instance: $Var[\hat \beta]=(X'X)^{-1}\sigma^2$ is conditional on $X$. Since $X$ is random in observational studies, this is an under estimate of the true $Var[\hat \beta]$.

Edit: As pointed out by @christoph-hanck, $(X'X)^{-1}$ cannot be, by definition, systematically smaller than $\mathbb{E}(X'X)^{-1}$. Question remains: why do we use fixed design standard errors, instead of random design standard errors?

• What is the alternative? You do not have the random design matrix, so you can't estimate unconditional variance Commented Aug 2, 2018 at 13:30
• The issue isn't with how the data collected so much as it concerns how the results will be used. Exactly what use do you have in mind for a regression analysis where you are concerned about the distribution of the regressors?
– whuber
Commented Aug 2, 2018 at 14:07
• @whuber: I want to do inference on the $\beta$'s. Commented Aug 2, 2018 at 14:10
• @Aksakal: we know from Stein's paradox that when estimating many parameters at once (such as $Var[X]$), we can do improve accuracy by regularizing. Commented Aug 2, 2018 at 14:11
• what does regularization have to do with this? it's a wholly different subject Commented Aug 2, 2018 at 14:13

By the law of total variance, we may write $$Var(\widehat{\beta})=E[Var(\widehat{\beta}|X)]+Var[E(\widehat{\beta}|X)]$$ As OLS is (under suitable assumptions) unbiased, $E(\widehat{\beta}|X)=\beta$, so that $Var[E(\widehat{\beta}|X)]=Var[\beta]=0$, implying $$Var(\widehat{\beta})=E[Var(\widehat{\beta}|X)]$$ so that I am not sure in which sense $Var(\widehat{\beta}|X)$ would systematically underestimate $Var(\widehat{\beta})$.
• Are you asking why $(X'X)^{-1}-\mathbb{E}[(X'X)^{-1}]$ is negative definite matrix? I suspect it can be shown using Anderson's Lemma. Commented Aug 2, 2018 at 11:53
• I suppose so. For concreteness, let us say we have iidness with a single standard normal regressor, so that $(X'X)^{-1}$ is inverse chi square with $E[(X'X)^{-1}]=1/(n-2)$ and we may have $(X'X)^{-1}>1/(n-2)$ or $(X'X)^{-1}<1/(n-2)$, no?` Commented Aug 2, 2018 at 12:00