AverageDerivative

Average derivative estimator for continuous treatments

from _api_doc_utils import *

1 Where it fits

Group: Causal inference

AverageDerivative targets an average marginal effect of a scalar continuous treatment. The class exposes three related estimating equations through method='ob', 'ipw', or 'dr'.

The doubly robust option combines outcome-bridge and weighting components, while the other options expose the individual pieces.

2 Linear average-derivative estimators

Let \(\mu=E_n[W]\), \(W_i^c=W_i-\mu\), and

\[ R_i=(1,\ {W_i^c}',\ D_i{W_i^c}',\ D_i)'. \]

Centering makes the final coefficient in the linear interaction regression equal the sample-average derivative with respect to \(D\).

The outcome-based method fits

\[ \hat\lambda_{\mathrm{OB}} = \arg\min_\lambda\sum_i(Y_i-R_i'\lambda)^2 \]

and reports the last component. Its joint moments estimate \(\mu\) from \(W_i-\mu\) and use \(R_i(Y_i-R_i'\lambda)\) for the regression.

The IPW method first projects \(D\) on \(\tilde W=[\mathbf1,W]\), sets

\[ v_i=D_i-\tilde W_i'\hat\pi, \qquad \hat\sigma^2=E_n[v_i^2], \qquad g_i=\frac{v_i}{\hat\sigma^2}, \]

and reports

\[ \hat\beta_{\mathrm{IPW}} = \frac{\sum_i g_iY_i}{\sum_i g_iD_i}. \]

Its stacked moments include \(\tilde W_iv_i\), \(v_i^2-\sigma^2\), and \(g_i(Y_i-D_i\beta)\).

The DR method combines both structures. It uses regressors \(R_i\) and instruments

\[ Z_i=(1,\ {W_i^c}',\ D_i{W_i^c}',\ g_i)' \]

in a just-identified linear IV fit and reports the coefficient on \(D\). The full moment system jointly accounts for treatment-projection coefficients, \(\log\sigma^2\), control means, and outcome coefficients. These are linear-basis OB, IPW, and DR estimators; the class does not fit generic nonparametric derivatives.

3 Inference

All three methods use the same exactly identified sandwich machinery. The Jacobian of the full stacked mean-moment vector is computed by central differences, and the empirical moment outer product supplies the meat. Vanilla is the uncorrected iid sandwich; HC1, Bartlett Newey-West, and cluster covariance add the same finite-sample corrections described on the EPLM page. Only the scalar average-derivative covariance block is returned.

The method accepts continuous treatment values and does not enforce a binary treatment. No cross-fitting, bootstrap, or built-in Wald method is provided. Validity depends on the selected linear outcome and treatment structures and, for IPW/DR, nondegenerate residual treatment variance.

4 Performance and numerical behavior

OB is a dense least-squares fit; IPW adds a treatment projection; DR adds a just-identified IV system. Summary-time differentiation is the dominant risk: the DR nuisance vector has \(4p+4\) elements for \(p\) raw controls, requiring two full moment evaluations per element followed by a dense cubic inversion. Interactions also double the outcome-design width. Scaling and collinearity matter, and there is no regularization.

5 Python API

Constructor: cm.AverageDerivative

Call fit(y, d, w). The summary reports method, coef, se, and vcov. There is no predict() method because the object is a semiparametric target rather than a full conditional mean model.

print(inspect.signature(cm.AverageDerivative))

(method='dr', fd_eps=1e-06)

cls = cm.AverageDerivative
display(HTML(html_table(["Public method"], public_methods(cls))))

Public method
fit(self, /, y, d, w)
summary(self, /, vcov=None, lags=None, clusters=None)

6 Minimal example

rng = np.random.default_rng(12)
w = rng.normal(size=(320, 2))
d = 0.2 + w @ np.array([0.5, -0.3]) + rng.normal(scale=0.7, size=320)
y = 0.8 * d + w @ np.array([0.2, -0.1]) + rng.normal(scale=0.5, size=320)
model = cm.AverageDerivative(method='dr')
model.fit(y, d, w)
print(model.summary())

{'method': 'dr', 'coef': 0.775623613574262, 'se': 0.04377879620939335, 'vcov': array([[0.00191658]])}

7 summary() contract

The table below is generated by fitting the live class in this repository and then inspecting summary(). Shapes are shown because most values are plain NumPy arrays or scalars.

rng = np.random.default_rng(112)
w = rng.normal(size=(120, 2))
d = 0.2 + w @ np.array([0.5, -0.3]) + rng.normal(size=120) * 0.7
y = 0.8 * d + w @ np.array([0.2, -0.1]) + rng.normal(size=120) * 0.5
model = cm.AverageDerivative(method='dr')
model.fit(y, d, w)
summary = model.summary()
display(HTML(html_table(["summary() key", "shape"], summary_shape_rows(summary))))

summary() key	shape
method	()
coef	()
se	()
vcov	(1, 1)