Generalized Residuals For Nonlinear First Stages

Why the First-Stage Residual Needs Care

In a linear first stage, the ordinary residual is a natural measure of how far an observed treatment differs from its fitted value. With binary or count treatments, however, a residual’s meaning depends on the first-stage probability model. A binary treatment residual \(D-\hat p\) is bounded, while an appropriate probit control function can take much larger positive or negative values when the fitted probability is near zero or one. For a count model, the score residual may coincide with the raw residual under one model and differ from it under another.

These distinctions matter when a residual is included in a nonlinear outcome model. A nonlinear link does not generally make different residuals interchangeable: holding the outcome-model coefficients fixed, changing the residual changes the fitted index and therefore the fitted mean or probability. The residual must be justified by the first-stage model and the control-function assumptions, not selected just because it is easy to calculate.

Binary Treatment: The Probit Generalized Residual

Let \(D_i\) be a binary treatment, \(X_i\) observed covariates, and \(Z_i\) an excluded instrument. A probit first stage specifies \(D_i=1[\eta_i+u_i>0]\), where \(\eta_i=X_i^\top\gamma_X+Z_i^\top\gamma_Z\) and \(u_i\mid X_i,Z_i\sim N(0,1)\). Thus \(p_i=P(D_i=1\mid X_i,Z_i)=\Phi(\eta_i)\). This specification fixes the latent-error scale at one.

Under that model, the conditional expectation of the latent error is a natural control-function quantity. For a treated observation, \(u_i>-\eta_i\); for an untreated observation, \(u_i\leq-\eta_i\). The truncated-normal expectations give:

Writing \(p_i=\Phi(\eta_i)\), this can also be written as \(r_{gi}=\phi(\eta_i)(D_i-p_i)/\{p_i(1-p_i)\}\). This is also the Bernoulli probit log-likelihood score with respect to \(\eta_i\). It is not simply \(D_i-p_i\): the denominator rescales the raw discrepancy according to the fitted probability.

Score Residuals for Binary and Count First Stages

The score definition gives a useful way to derive residuals systematically. For a Bernoulli first stage with fitted probability \(p_i\), the contribution to the log likelihood is \(D_i\log(p_i)+(1-D_i)\log(1-p_i)\). Differentiating with respect to the linear predictor \(\eta_i\) gives:

For probit, \(\partial p_i/\partial\eta_i=\phi(\eta_i)\), producing the generalized residual above. For logit, \(p_i=1/\{1+\exp(-\eta_i)\}\) and \(\partial p_i/\partial\eta_i=p_i(1-p_i)\); cancellation leaves \(D_i-p_i\). Thus the logit likelihood-score residual equals the raw binary residual, while the probit score generally does not. This equality does not establish that the logit residual is a conditional expectation of a latent error; that interpretation requires a specified latent-error model.

For a Poisson count first stage with a log link, \(Y_i\mid X_i,Z_i\sim\operatorname{Poisson}(\mu_i)\), where \(\log(\mu_i)=\eta_i\). Differentiating the log likelihood \(Y_i\eta_i-\exp(\eta_i)-\log(Y_i!)\) with respect to \(\eta_i\) gives \(Y_i-\mu_i\). Here the score residual is the raw response residual.

The equality need not hold for another count distribution. Under the negative-binomial NB2 parameterization, \(E(Y_i\mid X_i,Z_i)=\mu_i\) and \(\operatorname{Var}(Y_i\mid X_i,Z_i)=\mu_i+\alpha\mu_i^2\), with \(\alpha>0\). For a log link, the score with respect to \(\eta_i=\log\mu_i\) is:

The denominator reflects the overdispersed likelihood. This score residual is not generally the raw residual \(Y_i-\mu_i\), nor is it automatically the conditional expectation of an unobserved treatment disturbance. Its usefulness as a control function depends on the triangular model and the assumptions connecting first-stage unobservables to outcome unobservables.

When Residual Inclusion Has a Control-Function Interpretation

A generalized residual is not a stand-alone correction for endogeneity. A control-function analysis needs a defensible first-stage model and a structural outcome specification. In broad terms, the instrument must be relevant for treatment, excluded from the outcome equation conditional on covariates, and conditionally exogenous. The first-stage distribution and link must adequately describe treatment given the covariates and instruments. The outcome model must also allow the relevant dependence between treatment and outcome unobservables to be represented by the included control function.

For a binary probit treatment with the latent-normal specification above, the conditional expectation of \(u_i\) has a direct derivation and interpretation. A nonlinear second stage may include \(D_i\), \(X_i\), and \(r_{gi}\) in its index, with a link appropriate to the outcome. For count treatments, a likelihood score can be a useful residual-like control, but it does not, by itself, prove that the score captures all dependence between the treatment and outcome disturbances. That claim requires additional structure. A correct first-stage likelihood alone does not guarantee a correct causal model.

The second-stage coefficient on treatment is conditional on the specified control-function model and link. In a nonlinear outcome model it is generally not a marginal risk difference or another population-average effect. Marginal interpretation requires an additional calculation under the fitted model; the residual’s inclusion does not make the index coefficient a marginal effect.

Diagnostics, Alternatives, and Limitations

Check the first stage before interpreting any second-stage result. Inspect whether fitted probabilities or means are plausible, whether the model represents the observed treatment distribution, and whether a few extreme fitted values dominate the residuals. For binary probit models, probabilities very near zero or one can produce large inverse-Mills-type terms; investigate their source rather than automatically truncating them. For counts, assess overdispersion and excess zeros where relevant, and compare plausible count distributions. A residual fitted from a misspecified likelihood can transmit that misspecification directly into the outcome model.

Also check whether the second-stage specification is sensitive to reasonable control-function choices, such as alternative first-stage links or distributions when substantively defensible. Such sensitivity is a warning about model dependence, not a way to select whichever result is preferred. Instrument relevance and exclusion require substantive and statistical assessment; residual plots cannot establish instrument validity. Because the first-stage residual is estimated, standard errors should account for that estimation, for example through a suitable joint sandwich calculation or a bootstrap that refits both stages.

A frequent mistake is to assume that any first-stage residual will remove endogeneity because it measures “unexplained treatment.” That reasoning is incomplete: residuals from different links and distributions encode different information. Use the residual derived for the model and control-function argument being made, state what it represents, and acknowledge when it is only a score-based device rather than a derived latent-error conditional expectation.