Diagrams of linear regression
I made a big diagram describing some assumptions (MLR1-6) that are used in linear regression. In my diagram, there are five categories (in rectangles with dotted lines) of mathematical facts that follow from different subsets of MLR1-6.
In the diagram, I stick to the brute mathematics, which is entirely independent of its (causal) interpretation. But of course what really matters is the causal interpretation.
As Pearl (2009) writes, “behind every causal claim there must lie some causal assumption that is not discernible from the joint distribution and, hence, not testable in observational studies”. If we wish to interpret (and hence ) causally, we must interpret MLR4 causally; it becomes a (strong) causal assumption.
As far as I can tell, when econometricians give a causal interpretation it is typically done thus (they are rarely explicit about it):
- MLR1 holds in every possible world (alternatively: it specifies not just actual, but all potential outcomes), hence is unobservable even in principle.
- yet we make assumption MLR4 about
This talk of the distribution of a fundamentally unobservable “variable” is a confusing device. Pearl’s method is more explicit: replace MLR with the causal graph below, where is used to make it extra clear that the causation only runs one way. MLR1 corresponds to the expression for (and, redundantly, the two arrows towards ), MLR4 corresponds to the absence of arrows connecting and . We thus avoid “hiding causal assumptions under the guise of latent variables” (Pearl). (Because of the confusing device, econometricians, to put it kindly, don’t always sharply distinguish the mathematics of the diagram from its (causal) interpretation. To see me rant about this, see here.)
The second diagram gives the asymptotic distribution of the IV estimator.