It’s never been very clear to me what distinguishes econometrics from statistics. My preliminary understanding was that statistics is data-focused whereas econometrics always starts from theory. But is that it? How are the two disciplines different?
I think it’s helpful to think of econometrics as an application of statistics that is well suited to deal with problems economists typically encounter in their research. So they are certainly very related in some sense, but the focus is on the connection between economics and statistics. One way to alternatively think about this is that econometrics combines statistics with assumptions that come from economic theory or reasoning, and econometrics is about studying to what extent these economic assumptions buy information in a statistical context. Three ways this manifests itself are: 1. statistical models fall out of economic models, rather than starting with a statistics model, 2. the focus is on issues that are particularly salient for economists, and 3. re-contextualizing statistical assumptions and approaches as economic assumptions (and vice-versa)
To expand on these points, the first point emphasizes that the statistical model are typically motivated from of an economics model. For example, you may be studying markets, and a classic result from economic theory is market clearing, which states that supply of a good equals demand of that good, and so when you have data on firms producing goods and consumers purchasing them, you may want to impose this condition in your statistical model, and this can be stated as a moment condition, and thus is a subset of Generalized Method of Moments (GMM), which was developed in econometrics because so many economic models have some moment conditions that must hold, and we can use that information with our statistical models.
The second point is an obvious one, and you could maybe think of the first point as a case of it, but it really emphasizes that econometrics develops statistical tools in the context of what economists are interested in, and one classic interest is in causality rather than correlation in situations. For example, the development of instrumental variable approaches that allow for heterogeneity in potential outcomes is largely driven by econometricians, since it’s a common problem in that field: economists typically studies individuals (or individual firms), and it’s very reasonable that each individual has a different treatment effect. Additionally, unlike some fields, it may be harder to run RCTs in some contexts, and so classic papers like Imbens and Angrist (1994) analyze what IV methods identify when you have an instrument without full support.
A final point should be made that econometrics also focuses on relating statistical models to economics. This is the reverse direction of the first point: given a statistical model, what assumptions would you have to place on individuals so that the model holds, and are these assumptions sensible from an economics perspective. For example, Vytlacil (2002) showed that the classic IV assumptions and monotonicity are equivalent to a Roy model with an index switching threshold (a variant of a classic economic model), which allows economists to understand statistical assumptions from an economics perspective.