Possibilistic IV Regression

• Possibilistic Instrumental Variable Regression is a method for causal inference that relaxes exogeneity assumptions by allowing for arbitrary instrument invalidity.
• It leverages possibility theory to perform sensitivity analysis and derive posterior possibility inferences and valid confidence sets for treatment effects.
• Empirical evaluations show that adjusting the violation set A can maintain near-nominal coverage even when instruments are weak or potentially invalid.

Possibilistic instrumental variable regression is a methodology for causal inference in structural models with endogenous treatments when the validity of instrumental variables (IVs) is uncertain. Grounded in possibility theory rather than classical probability, this approach allows principled posterior inference on treatment effects under user-specified relaxations of the exogeneity assumption, thus facilitating sensitivity analysis even in the presence of arbitrary instrument invalidity. The method offers valid confidence sets for the treatment effect that remain informative with a single, potentially invalid, instrument and does not require specification of prior distributions or reliance on Markov chain Monte Carlo (MCMC) (Steiner et al., 20 Nov 2025).

1. Structural Model and Problem Formulation

The observable data comprise independent, identically distributed $(Y_i, X_i, Z_i)$ generated by the triangular structural model: $$Y_i = \beta X_i + Z_i \alpha + \epsilon_i, \qquad X_i = Z_i \gamma_2 + \eta_i,$$ where $Z_i \in \mathbb{R}^p$ is a vector of $p$ instruments, $X_i$ is the treatment, $Y_i$ is the outcome, and $(\epsilon_i, \eta_i)^\top$ is a mean-zero jointly Gaussian error $$(\epsilon_i, \eta_i)^\top \sim N\left(0, \Sigma\right)$$ with

$$\Sigma = \begin{pmatrix} \sigma_{11} & \sigma_{12} \ \sigma_{12} & \sigma_{22} \end{pmatrix}.$$

The classical IV assumptions are: relevance ($\gamma_2 \neq 0$), exogeneity ($\alpha = 0$), and instrument validity ($Z_i \perp (\epsilon_i, \eta_i)$). However, in possibilistic IV regression, exogeneity is not assumed a priori; instead, $\alpha$ is allowed to be nonzero, encoding the potential invalidity of instruments.

Key to the approach is the incorporation of a violation set $A \subset \mathbb{R}^p$, representing plausible values for $\alpha$ and thus for the degree and direction of exogeneity violations. Sensitivity analysis is then performed by conditioning inference on the event $\alpha \in A$.

2. Possibility Theory Foundations

A possibility function $f: \Theta \to [0, 1]$ on parameter space $\Theta$ encodes uncertainty by satisfying $\sup_{\theta \in \Theta} f(\theta) = 1$. The associated outer measure is

$$\overline{\mathbb{P}}_f(A) = \sup_{\theta \in A} f(\theta), \quad A \subset \Theta.$$

For a random variable $\boldsymbol{\theta}$ with uncertainty $f_{\boldsymbol{\theta}}$, joint and conditional outer measures are defined via suprema analogous to the above, for instance

$$f_{\boldsymbol{\theta} \mid \boldsymbol{\psi}}(\theta \mid \psi) = \frac{f_{\boldsymbol{\theta}, \boldsymbol{\psi}}(\theta, \psi)}{\sup_{\theta'} f_{\boldsymbol{\theta}, \boldsymbol{\psi}}(\theta', \psi)}.$$

This framework enables uncertainty quantification and posterior inference without full probabilistic modeling, aligning with the modeling uncertainty inherent in IV settings with ambiguous exogeneity.

3. Posterior Possibility Inference

3.1 Reduced-Form Posterior

The reduced-form likelihood is modeled with $W = [Y, X] \sim MN(Z \Gamma, I_n, \Psi)$, where $\Gamma = [\gamma_1, \gamma_2]$ and $\Psi = R(\beta) \Sigma R(\beta)^\top$ with $$R(\beta) = \begin{bmatrix} 1 & \beta \ 0 & 1 \end{bmatrix}$$.

With a vacuous prior $f(\Gamma, \Psi) = 1$, the reduced-form posterior possibility is

$$f_{\text{RF}}(\Gamma, \Psi \mid W) = \frac{p(W \mid \Gamma, \Psi)}{\sup_{\Gamma', \Psi'} p(W \mid \Gamma', \Psi')}.$$

3.2 Structural Posterior

The parameters $(\Gamma, \Psi)$ are reparameterized as $(\alpha, \beta, \Sigma)$ via $\gamma_1 = \beta \gamma_2 + \alpha$, $\Psi = R(\beta)\Sigma R(\beta)^\top$. The structural posterior possibility is then

$$f_{\text{S}}(\alpha, \beta, \Sigma \mid W) = \sup_{(\Gamma, \Psi): \Gamma [1, -\beta]^\top = \alpha,~ \Psi = R(\beta)\Sigma R(\beta)^\top} f_{\text{RF}}(\Gamma, \Psi \mid W).$$

Under an uninformative prior, a closed-form solution is available by profiling out $\Sigma$ and using

$$\log f_{\mathrm{S}}(\alpha, \beta \mid W) = -\frac{1}{2} (\alpha - t(\beta))^\top \left( \frac{Z^\top Z}{\sigma_{11}} \right) (\alpha - t(\beta)),$$

where $t(\beta) = \hat{\gamma}_1 - \beta \hat{\gamma}_2$ and $$\sigma_{11} = \hat{\Psi}_{11} - 2\beta \hat{\Psi}_{12} + \beta^2\hat{\Psi}_{22}$$. Here, $\hat{\Gamma} = (Z^\top Z)^{-1} Z^\top W$, $$\hat{\Psi} = \frac{1}{n}(W - Z\hat{\Gamma})^\top (W - Z\hat{\Gamma})$$.

4. Conditional Inference and Computation

To assess $\beta$ given possible instrument invalidity, the posterior possibility conditional on $\alpha \in A$ is

$$f(\beta \mid \alpha \in A, W) = \frac{\sup_{\alpha \in A,\, \Sigma} f_{\text{S}}(\alpha, \beta, \Sigma \mid W)}{\sup_{\beta',\, \alpha \in A,\, \Sigma} f_{\text{S}}(\alpha, \beta', \Sigma \mid W)}.$$

The supremum in $\Sigma$ is solved via the MLE, as $$R(\beta)\hat{\Sigma}(\beta) R(\beta)^\top = \hat{\Psi}$$. The optimization in $\alpha$ simplifies to projecting $t(\beta)$ onto $A$ under the $Z^\top Z$ metric:

• If $t(\beta) \in A$, the maximizer $\hat{\alpha}(\beta) = t(\beta)$;
• Otherwise, $\hat{\alpha}(\beta)$ is the projection of $t(\beta)$ onto $A$ in the $Z^\top Z$ norm.

The practical computation reduces to:

1. Estimating reduced form parameters;
2. For each candidate $\beta$, projecting $t(\beta)$ onto $A$;
3. Normalizing to form $f(\beta \mid \alpha \in A, w)$.

The overall complexity is dominated by a $p$-dimensional quadratic program and a $2 \times 2$ covariance update for each $\beta$.

5. Validified Confidence Sets and Sensitivity Analysis

Following Martin–Liu (2013), the validified posterior possibility is defined as

$$\pi_w(\beta \mid A) = \mathbb{P}_\beta\{ f(\beta \mid \alpha \in A, W) \leq f(\beta \mid \alpha \in A, w) \},$$

where $\mathbb{P}_\beta$ denotes the sampling distribution under $\beta$. This yields a valid $(1-\delta)$ confidence set $\{ \beta : \pi_w(\beta \mid A) \geq \delta \}$, satisfying

$$\sup_{\beta} \mathbb{P}_\beta\{ \pi_W(\beta \mid A) \leq \delta \} \leq \delta, \quad \forall\, \delta \in [0, 1].$$

Empirically, these intervals attain near-nominal frequentist coverage when $A$ contains the true $\alpha$.

Sensitivity analysis is facilitated by varying the violation set $A$. Setting $A_\tau = \{\alpha : \|\alpha\| \leq \tau\}$ transitions inference from point-identified as $\tau \to 0$ to uninformative as $\tau \to \infty$. Graphically displaying $f(\beta \mid \alpha \in A_\tau, w)$ as a function of $\tau$ produces a "sensitivity curve" indexing the stability of causal conclusions to exogeneity violations.

6. Empirical Evaluation

Simulation experiments address single- and multiple-instrument settings:

• For a single instrument ($p=1$) with possible violations $\alpha \in \{0, 0.25, 0.5\}$ and $n=100$, nominal $95\%$ coverage is maintained if $A$ is correctly specified. Allowing for plausible $\alpha$ values, i.e., $A = [-0.5, 0.5]$, restores coverage when true $\alpha \neq 0$, though the confidence interval widens.
• For multiple instruments ($p=5$) with up to all instruments invalid ($\alpha_i = 0.1$), possibilistic IV regression using $A = [-0.1, 0.1]^p$ or $A = [0, 0.2]^p$ achieves near-nominal coverage, unlike competing methods, which fail when instrument invalidity is widespread.

A real-data example using the Acemoglu–Johnson–Robinson (AJR) dataset ($n=64$ countries; $Y = \log$ GDP/capita; $X =$ institutional quality; $Z = \log$ settler mortality) demonstrates that with $A = \{0\}$, inference on $\beta$ is tight, but relaxing to $A = [-0.4, 0.4]$ leads the confidence set to include zero. Posterior probabilities for $\beta > 0$ remain robust under moderate exogeneity violations.

7. Advantages, Limitations, and Methodological Comparison

Possibilistic IV regression offers several advantages:

Feature	Possibilistic IV Regression	Existing Alternatives
Handles arbitrary invalidity	Yes (even single instrument)	Often fails
Interval estimation	Yes (possibility/confidence sets)	Rare or conservative
Sensitivity analysis	Natural, via violation set $A$	Typically ad hoc
Optimization complexity	Finite-dimensional (no MCMC)	Often requires MCMC
Frequentist calibration	Validified intervals for all $\beta$	May not hold

• Limitations: Requires explicit specification of a plausible violation set $A$; computational burden grows with instrument dimension $p$, especially for complex $A$; as $A$ expands, inference becomes uninformative.
• Comparison with Existing Methods:
  • Two-stage least squares (TSLS): Fails with invalid or weak instruments.
  • Plausible GMM (PGMM): Dependent on Gaussian priors.
  • BudgetIV: Relies on a "budget" hyperparameter, often overly conservative.
  • CIIV: Restricted to certain linear settings.
  • gIVBMA: Sensitive to prior choices.
  • Partial IV: Requires strong instrument strength.

A plausible implication is that possibilistic IV regression generalizes many existing point- and interval-estimation approaches, providing a robust, practical framework for sensitivity analysis under instrument invalidity (Steiner et al., 20 Nov 2025). Empirically, its calibration and interval width are reasonable when $A$ is well-guided, and the method directly connects uncertainty about exogeneity violations with interval estimations for causal effects.