2. Target Functionals
We are usually not interested in \(\theta^*(X_i)\) for every individual, but rather in some summary: an average of a function of \(\theta^*\) across the population,
where \(H\) is the target function and \(\tilde{t}\) is an evaluation point. This \(\mu^*\) is the scalar we want a valid confidence interval for.
Common targets
Target |
\(H(x, \theta, \tilde{t})\) |
Interpretation |
|---|---|---|
Average parameter |
\(\theta_k\) |
\(\mathbb{E}[\beta_{\text{price}}]\): mean price sensitivity |
Average marginal effect |
\(g'(\alpha + \beta \tilde{t}) \cdot \beta\) |
Mean marginal effect at \(\tilde{t}\) |
Elasticity |
\(\beta \cdot \tilde{t} \cdot (1-p)\) |
Price elasticity of demand |
Dose-response |
\(g(\alpha + \beta \tilde{t})\) |
Predicted outcome at \(\tilde{t}\) |
Profit |
\(\tilde{t} \cdot g(\alpha + \beta \tilde{t})\) |
Expected revenue |
DID / ATE |
\(g(\theta'\tilde{t}_1) - g(\theta'\tilde{t}_0)\) |
Treatment effect contrast |
Custom |
Any \(H(x, \theta, \tilde{t})\) |
User-specified |
The Jacobian requirement
The Jacobian \(H_\theta = \partial H / \partial \theta\) is required for the influence function correction introduced in The Influence Function Correction. For built-in targets, closed-form Jacobians are provided. For custom targets, the package computes them via automatic differentiation — the user supplies only the target function \(H\).
Tip
Choosing a target is an economic modeling decision, not a statistical one. The same fitted neural network \(\hat\theta(X)\) can answer many questions — average price sensitivity, elasticity at a chosen price, expected profit — simply by swapping the target \(H\).