ECON8080 M2 Key Formulas

Variance (Linear Transformation)

Var = R^T V_β R

Variance (Nonlinear Transformation)

∇r(β)^T V_β ∇r(β)

Nonlinear Function MVT

r(β^{^}) = r(β) + ∇r(β^~)(β^{^} - β)

Bootstrap Mean, Var, SE

Mean: θ-hat* = 1/B Σ(b=1 to B) θ^{^}*_b
Var: V^{^}_θ^boot = 1/B Σ(b=1 to B) n(θ^{^}*_b - θ-hat*)(θ^{^}*_b - θ-hat*)^T
SE: s.e.^{^boot} = √V^{^}_θ^boot / √n

Single Index Model CEF (General)

P(Y = 1 | X) = G(X^Tβ)

Single-Index Model + MLE (General)

P_i = G(X^Tβ)^Y_i [1 - G(X^Tβ)]^1-Y_i

L(β) = Σ(i=1 to n) [Y_ilog(G(X_i^Tβ)) + (1-Y_i)log(1-G(X_i^Tβ))]

Individual Marginal Effects (Single-Index)

g(X^Tβ)β

Average Marginal Contrast

AMC^{^} = 1/nΣ(i=1 to n)g(X_i^Tβ^{^})β^{^}

Average Treatment Effect

E[Y(1) - Y(0)]

Average Treatment Effect on the Treated

E[Y(1) - Y(0) | D=1]

Unconfoundedness

(Y(1), Y(0)) ⊥⊥ D | X

Once you group people by their observable characteristics, their potential outcomes are completely independent of whether they actually got treated (D=1). As good as random assignment after conditioning.

Overlap

0 < P(D=1 | X=x) < 1

For all X, there must be some probability of getting the treatment, and some probability of not getting it.

ATT (Regression Adjustment)

ATT = E[Y|D=1] - E[X^T | D=1]β₀

ATT^{^} = Y-bar_D=1 - X-bar_D=1β^{^}_{0
β0: from estimating a regression of Y on X using only the untreated.}

ATE (Regression Adjustment)

ATE = E[E[Y | X, D=1] - E[Y|X, D=0]]

ATE^{^} = 1/n Σ (i=1 to n) (X_i^Tβ₁^{^} - X_i^Tβ₀^{^})

Regression Adjustment (def)

Drawing a regression line through the people who didn't get treated, and using that line to guess what would have happened to the treated people if they never took the treatment. Mathematically, subtract the regression predictions from actual treated outcomes.

IPW (def)

Giving bigger weights to the untreated people who have all the characteristics of someone who should have been treated (vice versa for ATE), so the untreated group acts as a perfect clone of your treated group. Mathematically, weigh people by taking the inverse of the propensity score.

AIPW (def)

A "doubly robust" method that combines both regression and propensity score models, remaining accurate as long as at least one of those models is correctly specified.

ATT (IPW)

ATT = E[(D(¹/_π) - (1-D)(^p(X)/_π(1-p(X)))Y]

ATT^{^} = 1/n Σ (i=1 to n) (D_i(¹/_π) - (1-D_i)(^p(X_i)/_π(1-p(Xi)))Y_i

ATE (IPW)

ATE = E[(D(¹/_p(X)) - (1-D)(¹/_1-p(X))Y]

ATE^{^} = 1/n Σ (i=1 to n) (D_i(¹/_p(Xi)) - (1-D_i)(¹/_1-p(Xi))Y_i

ATT (AIPW)

ATT = E[(D(¹/_π) - (1-D)(^p(x)/_π(1-p(x)))) (Y - E[Y|X, D=0])

ATT^{^} = 1/n Σ (i=1 to n) Di(¹/_π^) - (1-Di)(^{p^(x)}/_π^(1-p^(x)))) (Y - E[Y|X, D=0])

ATE (AIPW)

ATE = E[(E[Y|X,D=1] - E[Y|X,D=0]) + ^D/_p(x)(Y - E[Y|X,D=1]) - ^1-D/1-_p(x)(Y - E[Y|X,D=0])