Difference-in-Differences

The DiD family in this package covers three estimators: a closed-form 2x2 case, a heterogeneous neural 2x2 case, and a two-way fixed-effects panel case. All three return an InferenceResult with a valid influence-function standard error. A single entry point did() auto-selects the right estimator from the arguments you pass.

Source Papers

The saturated 2x2 DiD regression and two-way FE panel DiD follow the standard DiD literature. Angrist, J. D. and Pischke, J. S. (2009), Mostly Harmless Econometrics, Chapter 5 covers both specifications. The influence-function methodology that produces valid standard errors follows Farrell, Liang, Misra (2021), “Deep Neural Networks for Estimation and Inference,” Econometrica.

The closed-form 2x2 estimator is proved in src/deep_inference/_did_closed.py to equal the HC0 robust OLS standard error of the saturated cell-mean regression to machine precision.

When to Use

Use DiD when you have a binary group indicator \(G\) (treated vs control) and a binary time indicator \(P\) (pre vs post period), and you are willing to invoke the parallel trends assumption: absent the treatment, the treated group’s outcome would have evolved the same way as the control group’s outcome.

Choose the sub-method based on your data:

• did(Y, group, post), the closed-form exact method, has no covariates and estimates one average treatment effect for all individuals.

• did(Y, group, post, X=X), the neural method, lets the treatment effect \(\tau(X)\) vary with covariates and targets \(E[\tau(X)]\).

• did(Y, group, post, X=X, unit=unit, time=time), the panel_fe method, absorbs unit and time fixed effects and targets \(E[\tau(X)]\) in a panel.

Do not use the closed-form exact method if you believe \(\tau\) is heterogeneous. Do not use neural if you have a genuine panel (the same units observed over time); use panel_fe, which absorbs unit-level confounders.

Model

Saturated 2x2 DiD (methods exact and neural).

The saturated regression is

\[Y_i = \alpha(X_i) + \gamma(X_i)\,G_i + \lambda(X_i)\,P_i + \tau(X_i)\,(G_i \cdot P_i) + \varepsilon_i\]

where \(G_i \in \{0,1\}\) is the group indicator and \(P_i \in \{0,1\}\) is the post-period indicator. In the neural method, all four coefficients \([\alpha, \gamma, \lambda, \tau]\) vary with covariates \(X_i\) via the neural network. In the exact method, they are constants estimated from cell means.

The per-observation treatment variable packed into T is \(t_i = [G_i, P_i, G_i P_i]\) (dimension 3). The design vector with intercept is \(W_i = [1, G_i, P_i, G_i P_i]\) (dimension 4). The structural parameter vector is \(\theta_i = [\alpha, \gamma, \lambda, \tau]\), so theta_dim = 4.

Panel FE DiD (method panel_fe).

The two-way FE estimator removes unit-specific effects \(\mu_i\) and time-specific effects \(\nu_t\) by within-transforming. Define \(\tilde{Y}_{it} = Y_{it} - \bar{Y}_i - \bar{Y}_t + \bar{Y}\) and \(\tilde{D}_{it} = D_{it} - \bar{D}_i - \bar{D}_t + \bar{D}\) as the within-transformed outcome and treatment indicator. After this transformation, the model reduces to an intercept-free regression through the origin:

\[\tilde{Y}_{it} = \tau(X_{it})\,\tilde{D}_{it} + \varepsilon_{it}\]

There is only one structural parameter per observation: \(\tau(X_{it}) = \theta_0(X_{it})\), so theta_dim = 1. The neural network maps covariates \(X_{it}\) to the scalar effect.

Loss / Likelihood

Both the saturated 2x2 and the panel FE model use squared-error loss.

Saturated 2x2:

\[\ell(y, t, \theta) = \frac{1}{2}(y - W^\top\theta)^2, \quad W = [1, G, P, G \cdot P]^\top\]

Panel FE:

\[\ell(\tilde{y}, \tilde{d}, \theta) = \frac{1}{2}(\tilde{y} - \tilde{d}\,\theta_0)^2\]

The factor \(\frac{1}{2}\) is cosmetic and cancels in the gradient.

Score and Hessian

Saturated 2x2:

\[\ell_\theta = (W^\top\theta - y)\, W, \qquad \ell_{\theta\theta} = W W^\top\]

The Hessian \(W W^\top\) is constant: it does not depend on \(\theta\) or \(Y\).

Panel FE:

\[\ell_\theta = (\tilde{d}\,\theta_0 - \tilde{y})\,\tilde{d}, \qquad \ell_{\theta\theta} = \tilde{d}^2\]

The scalar Hessian \(\tilde{d}^2\) is also constant in \(\theta\) and \(Y\).

Both Hessians are theta-independent, so \(\Lambda(X) = E[\ell_{\theta\theta} \mid X]\) can be estimated without any additional data split. This places both models in Regime B.

Target

The target is \(\mu^* = E[\tau(X)]\), the population average treatment effect (equal to the ATT under parallel trends).

For the saturated 2x2, this is AverageParameter(param_index=3), which picks out \(\theta_3 = \tau\).

For the panel FE, this is AverageParameter(param_index=0), which picks out \(\theta_0 = \tau\) (the only parameter).

For the closed-form exact method, the estimand is the cell-mean DiD contrast:

\[\beta = \mu_{11} - \mu_{10} - \mu_{01} + \mu_{00}, \qquad \mu_{gp} = E[Y \mid G=g, P=p]\]

Influence Function

Closed-form 2x2. For observation \(i\) in cell \((g, p)\), the influence function is

\[\psi_i = \hat\beta + s_{g_i, p_i}\,\frac{Y_i - \hat\mu_{g_i, p_i}}{\hat p_{g_i, p_i}}\]

where \(s_{gp} \in \{+1, -1\}\) is \(+1\) for cells \((0,0)\) and \((1,1)\) and \(-1\) for cells \((0,1)\) and \((1,0)\), \(\hat\mu_{gp}\) is the cell mean, and \(\hat p_{gp} = n_{gp}/n\) is the cell proportion. Then \(\hat\beta = \text{mean}(\psi_i)\) and \(\hat\sigma^2 = \text{var}(\psi_i)/n\) (HC0 denominator). The package docstring proves this equals the HC0 robust OLS SE of the saturated regression to machine precision.

Neural 2x2 and Panel FE. The FLM influence function applies with the constant Hessian:

\[\psi_i = H(\hat\theta(X_i)) - e_\tau^\top\, \Lambda(X_i)^{-1}\, \ell_\theta(Y_i, T_i, \hat\theta(X_i))\]

where the plug-in \(H(\hat\theta(X_i)) = \hat\tau(X_i)\) is the per-observation treatment effect, \(e_\tau\) is the unit selector picking the \(\tau\) component of \(\theta\), and \(\ell_\theta\) is the squared-error score. The correction enters with a minus sign. The point estimate is \(\hat\mu = (1/n)\sum_i \psi_i\) and the standard error is \(\mathrm{sd}(\psi)/\sqrt{n}\), which is heteroskedasticity-robust.

Algorithm and Regime

All three DiD methods use Regime B (analytic Lambda, two-way cross-fitting). The Hessian is constant in \(\theta\) and \(Y\), so Lambda requires no additional estimation step beyond averaging:

• Saturated 2x2: \(\hat\Lambda = \frac{1}{n}\sum_i W_i W_i^\top\), the sample average of \(WW^\top\).

• Panel FE: \(\hat\Lambda(X) = E[\tilde{D}^2 \mid X]\), estimated by ridge regression without an intercept (the within-transform absorbed the intercept; the panel_fe model signals this via analytic_intercept=False).

No three-way split is needed. Each fold uses a two-way split: one half fits \(\hat\theta(X)\) and the other half computes the influence function.

The did() function inspects its arguments to auto-select the method:

1. unit and time both provided: selects panel_fe.

2. X provided but no panel ids: selects neural.

3. Neither: selects exact.

Override with method='exact', method='neural', or method='panel_fe'.

Usage

Closed-form 2x2 (method exact):

import numpy as np
from deep_inference import did

np.random.seed(42)
n = 2000
G = (np.random.random(n) < 0.5).astype(float)   # treated group
P = (np.random.random(n) < 0.5).astype(float)   # post period

# True cell means: beta* = 3.1 - 2.0 - 1.5 + 1.0 = 0.6
cell_means  = {(0,0): 1.0, (0,1): 1.5, (1,0): 2.0, (1,1): 3.1}
cell_sigmas = {(0,0): 1.2, (0,1): 1.0, (1,0): 1.4, (1,1): 1.8}
Y = np.array([
    cell_means[(int(G[i]), int(P[i]))]
    + cell_sigmas[(int(G[i]), int(P[i]))] * np.random.normal()
    for i in range(n)
])

result = did(Y, group=G, post=P)
print(result.summary())
print(f"DiD estimate: {result.mu_hat:.4f} +/- {result.se:.4f}")

Heterogeneous neural 2x2 (method neural):

from deep_inference import did
import numpy as np

np.random.seed(42)
n = 5000
X = np.random.normal(0, 1, (n, 3))
G = (np.random.random(n) < 0.5).astype(float)
P = (np.random.random(n) < 0.5).astype(float)

# True tau(X) = 0.5 + 0.3*X[:,0], so E[tau] = 0.5
tau_true   = 0.5 + 0.3 * X[:, 0]
alpha_true = 1.0 + 0.2 * X[:, 0]
gamma_true = 0.3 + 0.1 * X[:, 0]
lam_true   = 0.2 - 0.1 * X[:, 0]
Y = (alpha_true + gamma_true*G + lam_true*P + tau_true*(G*P)
     + np.random.normal(0, 1, n))

result = did(Y, group=G, post=P, X=X, n_folds=50, epochs=200)
print(f"E[tau(X)]: {result.mu_hat:.4f} +/- {result.se:.4f}")
print(f"95% CI: [{result.ci_lower:.4f}, {result.ci_upper:.4f}]")

# theta_hat has four columns: [alpha, gamma, lambda, tau]
tau_hat = result.theta_hat[:, 3]
print(f"tau(X) mean: {tau_hat.mean():.4f}  (true {tau_true.mean():.4f})")

Panel FE DiD (method panel_fe):

from deep_inference import did
import numpy as np

np.random.seed(42)
N_units, N_periods = 400, 6
n = N_units * N_periods

unit = np.repeat(np.arange(N_units), N_periods)
time = np.tile(np.arange(N_periods), N_units)
X    = np.random.normal(0, 1, (n, 2))

G_unit = (np.arange(N_units) % 2 == 1).astype(float)
G    = G_unit[unit]               # half the units treated
Post = (time >= 3).astype(float)  # last three periods post
D    = G * Post                   # treatment indicator

tau_true = 0.5 + 0.3 * X[:, 0]
unit_fe  = 0.5 * unit / N_units
time_fe  = 0.3 * time / N_periods
Y = unit_fe + time_fe + tau_true * D + np.random.normal(0, 1, n)

result = did(Y, group=G, post=Post, X=X, unit=unit, time=time,
             n_folds=50, epochs=200)
print(f"E[tau(X)]: {result.mu_hat:.4f} +/- {result.se:.4f}")

# theta_hat is (n, 1); column 0 is tau(X)
print(f"tau(X) sample mean: {result.theta_hat[:, 0].mean():.4f}")

Via inference() directly:

from deep_inference import inference
import numpy as np

# Saturated 2x2 neural: pack [G, P, G*P] into T
T_did = np.column_stack([G, P, G * P])
result = inference(Y, T=T_did, X=X, model='did', target='tau', t_tilde=0.0)

# Panel FE: residualize first
from deep_inference.utils import residualize_fixed_effects
Y_tilde = residualize_fixed_effects(Y, unit, time)
D_tilde = residualize_fixed_effects(D, unit, time)
result  = inference(Y_tilde, T=D_tilde, X=X, model='did_fe', target='fe_effect',
                    t_tilde=0.0)

Evidence

All numbers are read directly from eval reports under evals/reports/.

Closed-form 2x2 (evals/reports/eval_13_did_20260608_140151.txt, M=500, n=2000, \(\beta^* = 0.6\)):

Metric	Value	Target	Status
Coverage	96.4% (482/500)	90-99%	PASS
SE Ratio	0.969	0.7-1.5	PASS
Bias	-0.0004	less than 0.05	PASS
z-score mean	-0.003	(-0.3, 0.3)	PASS
z-score std	0.969	0.7-1.5	PASS

EVAL 13: PASS

Neural 2x2 (evals/reports/eval_14_did_nn_20260608_142239.txt, M=25, n=5000, \(E[\tau^*] = 0.5\)):

Metric	Value	Target	Status
Coverage	96.0% (24/25)	90-99%	PASS
SE Ratio	0.847	0.7-1.5	PASS
Bias	+0.026	less than 0.05	PASS
z-score mean	+0.51	(-0.3, 0.3)	FAIL
z-score std	0.850	0.7-1.5	PASS

EVAL 14: FAIL (z-score mean 0.51 exceeds the 0.3 threshold). Parameter recovery, autodiff, and Lambda estimation all pass individually. M=25 is a small number of replications, so the z-score mean estimate is noisy. Coverage is 96.0% but the off-center z-score distribution needs further investigation at larger M.

Panel FE (evals/reports/eval_15_panel_fe_20260609_060522.txt, M=50, 400 units, 6 periods):

Outcome	Coverage	SE Ratio	Bias	Status
Continuous	98.0% (49/50)	1.019	-0.002	PASS
Binary (LPM)	94.0% (47/50)	1.040	-0.003	PASS

EVAL 15: PASS (both continuous and binary outcomes).

Diagnostics and Pitfalls

Parallel trends is untestable. All three DiD methods invoke parallel trends: absent the treatment, the treated group would have evolved like the control group. This is an identifying assumption you must argue for from your research design. The package enforces no pre-trend tests; add them yourself if needed.

exact vs neural. The exact estimator assumes \(\tau\) is the same for all individuals. If \(\tau\) varies with \(X\), the exact method returns an OLS-weighted average that depends on cell sizes and may not equal \(E[\tau(X)]\). The neural method directly targets \(E[\tau(X)]\) with a heteroskedasticity-robust SE.

Panel FE and serial correlation. The panel FE standard error assumes independence across observations. If there is within-unit serial correlation (the common case in panels), cluster-robust SEs are needed. The package does not implement cluster-robust SEs; if serial correlation is present, the reported SE underestimates uncertainty.

Neural 2x2 z-score centering. The z-score mean of 0.51 at M=25, n=5000 (Eval 14) is a soft warning. Coverage is 96.0% and bias is small (+0.026), but the off-center z-score distribution suggests finite-sample bias in the SE estimator at this configuration. Use larger n or M for stronger guarantees.

theta_hat layout for neural 2x2. result.theta_hat has four columns ordered as \([\alpha, \gamma, \lambda, \tau]\) (0-indexed). Access the heterogeneous treatment effect with result.theta_hat[:, 3]. For panel FE, result.theta_hat has one column, accessed as result.theta_hat[:, 0].

Panel FE binary outcomes (LPM). The panel FE method handles binary \(Y\) as a linear probability model after within-transformation. The IF standard error is heteroskedasticity-robust, which correctly handles the \(p(1-p)\) variance structure. Eval 15 confirmed 94.0% coverage on binary outcomes.

References and API

Papers:

• Angrist, J. D. and Pischke, J. S. (2009). Mostly Harmless Econometrics. Princeton University Press. Chapter 5 covers the saturated DiD regression and two-way FE panel DiD.

• Farrell, M., Liang, T., and Misra, S. (2021). “Deep Neural Networks for Estimation and Inference.” Econometrica.

API:

• did(Y, group, post): closed-form 2x2 (src/deep_inference/_did_closed.py)

• did(Y, group, post, X=X): heterogeneous neural 2x2, model class deep_inference.DiDModel (src/deep_inference/models/did.py)

• did(Y, group, post, X=X, unit=unit, time=time): two-way panel FE, model class deep_inference.FEPanelDiDModel (src/deep_inference/models/panel_fe.py)

• inference(Y, T, X, model='did', target='tau'): direct new API for neural 2x2

• inference(Y, T, X, model='did_fe', target='fe_effect'): direct new API for panel FE

• Eval 13: evals/eval_13_did.py (closed-form)

• Eval 14: evals/eval_14_did_nn.py (neural 2x2)

• Eval 15: evals/eval_15_panel_fe.py (panel FE)