Make an econometric theorem for dummy

1. Problem framed: what “dummy” usually means in econometrics

Applied econometricians commonly use a dummy variable to represent a binary treatment or group indicator in a linear regression. Textbook and course descriptions that teach distribution theory and asymptotics—law of large numbers, convergence in probability, and the central limit theorem—are the foundations for deriving sampling distributions and inference for coefficients on dummy variables ^{[2] [3]}.

2. Theorem template: what a “theorem for dummy” would assert

A standard theorem for the OLS coefficient on a binary regressor states: under standard assumptions (exogeneity/zero conditional mean or weak exogeneity, finite second moments, and regularity conditions for LLN/CLT), the OLS estimator of the dummy’s coefficient is consistent and asymptotically normal, with variance that can be estimated consistently. The ingredients for proving such a result are central limit and law-of-large-numbers arguments emphasized in econometric theory courses and texts ^{[2] [3]}.

3. Modern perspective: finite-sample efficiency and estimator class

Recent work reframes classical efficiency results. Hansen’s “A Modern Gauss–Markov Theorem” shows finite-sample efficiency bounds for estimating linear regression coefficients without restricting attention to only linear estimators; the least squares (or GLS when appropriate) attains these lower bounds depending on error covariance assumptions ^[1]. That implies any theorem about the dummy coefficient should state both the asymptotic distribution and the finite-sample efficiency context in which OLS is optimal ^[1].

4. Exogeneity caveat and weak exogeneity literature

If the dummy is endogenous, the simple OLS theorem fails. Newer papers on weak exogeneity and linear regression analyze how bias and asymptotics change when exogeneity assumptions are weakened; they derive asymptotic bias as a function of primitives and give conditions under which inference remains valid or must be adjusted ^[4]. Any formal theorem must declare its exogeneity assumptions up front and, if relaxing them, cite appropriate weak-exogeneity results ^[4].

5. What a concise, provable theorem might look like (journalistic style)

Under i.i.d. sampling, E[ε | D, X] = 0, and finite second moments, the OLS estimator β̂d on a binary regressor D satisfies β̂d →p βd and √n(β̂d − βd) →d N(0, σ^2·V), where σ^2·V is the usual asymptotic variance that can be consistently estimated. Proof uses law of large numbers and central limit theorem machinery taught in econometric theory courses ^{[2] [3]}. If the errors exhibit heteroskedasticity or autocorrelation, variance estimation and efficiency claims must be adjusted; Hansen’s work shows the least-squares variance corresponds to an efficiency bound under particular covariance assumptions ^[1].