Sampling. \(\big(Y_i(0), Y_i(1), X_i\big) \overset{\text{iid}}{\sim} \mu\), a population measure over (control outcome, treated outcome, covariate) triples.

Design (50:50). A random half of the units is assigned to treatment (complete randomization), so \(\Pr(W_i=1)=\tfrac12\) and \(W_i \perp \big(Y_i(0),Y_i(1),X_i\big)\).

We will focus on \(\tau_c\) (cleaner); estimators today are also valid for \(\tau\) but carry extra noise term, \(\mathrm{Var}(\text{CATE})/n\), from treatment-effect heterogeneity entering through the random draw of covariates.

The estimator is just two group averages (\(n_1=n_0=\tfrac n2\)): \[ \hat\tau_{\text{DM}} = \bar Y_1 - \bar Y_0, \qquad \bar Y_w = \frac{1}{n_w}\sum_{i:\,W_i=w} Y_i = \frac{1}{n_w}\sum_{i:\,W_i=w} Y_i(w). \]

Split each potential outcome into a part predictable from \(X\) and a residual: \[ Y_i(w) = \underbrace{m_{w}(X_i)}_{\mathbb{E}[Y_i(w)\mid X_i]} + \varepsilon_i(w), \qquad \mathbb{E}[\varepsilon_i(w)\mid X_i]=0 . \]

Average within each group and subtract (\(\langle\cdot\rangle_{T},\langle\cdot\rangle_{C}\): treated/control averages): \[ \hat\tau_{\text{DM}} = \big[\,\langle m_1(X_i)\rangle_{T} - \langle m_0(X_i)\rangle_{C}\,\big] \;+\; \big[\,\langle\varepsilon(1)\rangle_{T} - \langle\varepsilon(0)\rangle_{C}\,\big]. \]

\[ \hat\tau_{\text{DM}} = \underbrace{(\bar m_1 - \bar m_0)}_{\textstyle \tau_c} \;+\; \underbrace{(\langle m_1\rangle_{T}-\bar m_1) - (\langle m_0\rangle_{C}-\bar m_0)}_{\textstyle \text{imbalance}} \;+\; \underbrace{\langle\varepsilon(1)\rangle_{T} - \langle\varepsilon(0)\rangle_{C}}_{\textstyle \text{noise}} . \]

• \(\tau_c=\bar m_1-\bar m_0\) is the conditional ATE (\(\mathbb{E}[\tau_c]=\tau\)).

• Imbalance. Say the outcome grows with a covariate \(X\), but in our sample the treated group (\(\langle m_1\rangle_{T}\))is mostly low-\(X\) units. Then the treated look worse than a representative group(\(\bar{m}_1\)) would; that gap is the imbalance.

• Noise is orthogonal to \(X\) (\(\mathbb{E}[\varepsilon\mid X]=0\)): covariates cannot touch it.

Adjustment models \(m_w=\mathbb{E}[Y_i(w)|X_i]\) and subtracts the imbalance, removing that variance. That is the whole payoff of covariates.

Combine an outcome model \(\hat m_w(x)\approx\mathbb{E}[Y\mid X=x,W=w]\) with an estimated propensity \(\hat e(x)\approx\Pr(W=1\mid X=x)\): \[ \hat\tau_{\text{AIPW}} = \frac1n\sum_{i}\!\Big[\hat m_1(X_i)-\hat m_0(X_i) + \frac{W_i\big(Y_i-\hat m_1(X_i)\big)}{\hat e(X_i)} - \frac{(1-W_i)\big(Y_i-\hat m_0(X_i)\big)}{1-\hat e(X_i)}\Big]. \]

Doubly robust: consistent if either \(\hat m_w\) or \(\hat e\) is correct.

Personal Learning: the source of bias comes from imbalance; the imbalance that stems from covariates’ direct effects to the potential outcomes and the imbalance of covariate space between treated and control groups.

Same target \(\tau_c\), two estimators. Compare their limiting variances (50:50):

\[ \sqrt{n}\,(\hat\tau_{\text{DM}} - \tau_c) \to \mathcal{N}(0, V_{\text{DM}}), \qquad \sqrt{n}\,(\hat\tau_{\text{AIPW}} - \tau_c) \to \mathcal{N}(0, V). \] \[ \underbrace{V_{\text{DM}} = 2\,\mathrm{Var}(Y_i(1)) + 2\,\mathrm{Var}(Y_i(0))}_{\textstyle \text{full outcome variance}} \quad \underbrace{V = 2\,\mathbb{E}[\varepsilon_i(1)^2] + 2\,\mathbb{E}[\varepsilon_i(0)^2]}_{\textstyle \text{residual variance}} \]

Since \(\mathrm{Var}(Y_i(w))=\mathrm{Var}(m_w(X_i))+\mathbb{E}[\varepsilon_i(w)^2]\), \[ V_{\text{DM}} - V = 2\,\mathrm{Var}(m_1(X_i)) + 2\,\mathrm{Var}(m_0(X_i)) \;\ge\; 0, \] exactly the outcome variation the covariates explain. Better \(m_w\) ⇒ smaller \(V\).

Caveat: asymptotic.

• Additive: “how many more/fewer?” Effect in levels, \(Y_i(1)=Y_i(0)+\tau+\varepsilon_i\). → diff-in-means / OLS / AIPW.
• Multiplicative: “what % change vs. baseline?” \(\tau_i = (1+\tau) Y_i(0)\eta_i\)
  • baseline level itself is the effect modifier: knowing \(Y_i(0)\) becomes essential.

• Binary outcomes, extensive margin (whether):
  • logit \(\;\mathrm{logit}\,\Pr(Y_i(t)=1\mid X)=\alpha(X)+\beta t\;\) → constant in log-odds, but the effect on the probability is largest at intermediate baseline. (FAFSA; Athey, Keleher, and Spiess(2025))
  • LPM \(\;\Pr(Y_i(t)=1\mid X)=\alpha(X)+\beta t\;\) → additive, flat in baseline.
• Intensive margin (how much, given \(Y>0\)): additive, multiplicative, …

In Part (a), the variance \(V\) shrinks with how well we predict the baseline outcome \(Y_i(0)\). So rather than adjust for the whole vector \(X_i\), build the single best predictor of \(Y_i(0)\) and use that as the covariate.

That predictor is \(m_0(x)=\mathbb{E}[Y_i(0)\mid X_i=x]\): it minimizes the residual variance, and collapsing \(X_i\) into this one number loses nothing for the baseline mean: \(\mathbb{E}[Y_i(0)\mid X_i]=\mathbb{E}[Y_i(0)\mid m_0(X_i)]\).

Hansen (2008) calls it the prognostic score.

Each unit has a stable type (e.g. a habitual sharer) shaping both its past and its \(Y_i(0)\), so its past is a noisy signal of \(Y_i(0)\).

Now consider ATE estimation with two different sets of covariates: the full history or an out-of-fold \(\hat Y_i(0)\) (predicted from the history).

History is a superset of information to \(\hat Y_i(0)\). Would it be better?

Consider two cases of \(\tau_i\)

• Case 1: \(\tau_i \propto Y_i(0)\) (the level).
• Case 2: \(\tau_i \propto \mathrm{sd}\big(\{Y_{i,t}\}\big)\) (volatility).

Heterogeneous effect, two channels:

• Case 1 (level): \(\tau_i\propto Y_{i,T+1}(0)\).
• Case 2 (volatility): \(\tau_i\propto \mathrm{sd}(\{Y_{i,t}\})\).

How we score a covariate set \(S\). Because it is a simulation, each unit’s true effect \(\tau_i=Y_i(1)-Y_i(0)\) is known.

1. We predict the outcomes with 5-fold cross-fitted random forest first for \(\hat{Y_i(0)}\).
2. Interact the covariates with the treatment (\(Y_i \sim W_i + X_i + W_iX_i\))
3. Report the out-of-fold \(R^2\) for \(\tau_i\) and estimated \(\hat{\tau(x)}\) — the share of the effect’s variation \(S\) captures.

• (Full history \(=\) OLS on all lags in Case 1, a random forest in Case 2; prognostic \(=\) a cross-fit \(\hat Y_i(0)\))

\(\sigma_\alpha=1\), \(\sigma_\varepsilon=0.5\); \(n\), \(T\), \(\sigma_\nu\) set per scenario (noted on each plot); 50:50 assignment. Details in appendix.

1. For estimating the ATE, Randomization gives unbiasedness; covariates only buy precision, by shrinking residual variance.

2. The natural primary covariate is the out-of-fold predicted counterfactual outcome \(\hat Y_i(0)\).

3. The assumed functional form also hints us on whether \(\hat Y_i(0)\) is the right summary; the GATE-by-baseline plot could be used as the diagnostic for it.

• Conditional on the realized assignment \(\{W_i\}\): yes. Once you fix which units landed in each arm, the treated and control groups have different covariate profiles, so their baseline means differ. \(\hat\tau_{\text{DM}}\) then blends the true effect with that composition gap, off by the realized imbalance \((\langle m_1\rangle_T-\bar m_1)-(\langle m_0\rangle_C-\bar m_0)\).

• Conditional on the treated fraction (\(\tfrac12\)) and \(\{X_i\}\): no. The imbalance term has expectation zero over randomization.

• Unconditional (overall ATE \(\tau\)): no. \(\mathbb{E}[\hat\tau_{\text{DM}}]=\tau\). The imbalance is variance, not bias; adjustment removes it.

Counterintuitively, plugging in an estimated \(\hat e(x)\) is (weakly) more efficient than using the known \(e\equiv\tfrac12\) (Hirano–Imbens–Ridder 2003).

Intuition: regressing \(W\) on \(X\) lets \(\hat e\) absorb the realized correlation between assignment and covariates: it auto-corrects the chance imbalance. The oracle \(\tfrac12\) ignores that your sample over-treated some \(X\) values.

Finite samples? The HIR theorem is asymptotic. In practice estimated \(\hat e\) usually lowers variance too (it absorbs the realized imbalance), but it is not a universal finite-sample guarantee, and an over-flexible propensity model can backfire.

For AIPW with a consistent outcome model, both already hit the efficiency bound \(V\), so it washes out asymptotically. The gain from estimating \(\hat e\) shows up for plain IPW, or when \(\hat m_w\) is poor. (Distinct from the realized fraction \(n_1/n\) vs design \(\tfrac12\): the Hájek ratio normalization is likewise more stable than Horvitz–Thompson.)

Metric per covariate set \(S\): out-of-fold \[ R^2_S = 1 - \frac{\sum_i \big(\tau_i - \hat g_S(\cdot_i)\big)^2}{\sum_i (\tau_i - \bar\tau)^2}, \] where \(\hat g_S\) is a 5-fold cross-fitted prediction of \(\tau_i\) from \(S\).

Why we can compute it: in the simulation each unit’s true effect \(\tau_i=Y_i(1)-Y_i(0)\) is known, so we regress it directly on \(S\) and read off the heterogeneity \(S\) explains. (In real data \(\tau_i\) is unseen: you’d regress an AIPW pseudo-outcome whose conditional mean is \(\tau_i\) instead.)

Learners. Full history \(\to\) OLS on all lags (Case 1, so it can overfit) or a random forest (Case 2, for the nonlinear volatility). Prognostic \(\hat Y(0)\) \(\to\) a cross-fit prediction of \(Y(0)\), used as one covariate.

Read it: higher \(R^2_S \Rightarrow\) \(S\) carries more of the effect’s variation \(\Rightarrow\) a better basis for CATE estimation / targeting.