moderndid.compute_vlo_vup_dual

moderndid.compute_vlo_vup_dual(eta, s_t, gamma_tilde, sigma, w_t)

Compute the truncation bounds for the test statistic using dual approach with bisection.

Computes the truncation bounds for the test statistic \(\tilde{\gamma}'Y\) under the conditional distribution using the dual formulation. The bounds \([v_{lo}, v_{up}]\) are determined by the requirement that \(\tilde{\gamma}\) remains the optimal dual solution.

The method uses a hybrid approach that first attempts a shortcut based on the first-order optimality conditions, then falls back to bisection if needed. The shortcut exploits the fact that at the boundary \(v = v_{lo}\) or \(v = v_{up}\), a new constraint becomes binding, allowing direct computation in many cases.

Parameters:

etafloat

Optimal value \(\eta^*\) from the linear program.

s_tnumpy.ndarray

Residual vector \(s = (I - b\gamma')Y\) where \(b = \Sigma\gamma/(\gamma'\Sigma\gamma)\).

gamma_tildenumpy.ndarray

Normalized dual solution \(\tilde{\gamma}\) with \(\sum_i \tilde{\gamma}_i = 1\).

sigmanumpy.ndarray

Covariance matrix \(\Sigma_Y\).

w_tnumpy.ndarray

Constraint matrix \(W = [\text{diag}(\sigma)^{1/2}, X_T]\).

Returns:

dict

Dictionary with:

• vlo: float, lower bound of conditional support

• vup: float, upper bound of conditional support

Notes

The bounds are found by solving \(\max_{\lambda \geq 0} \lambda's\) subject to \(W'\lambda = e_1\) and \(\lambda'\mathbf{1} = 1\), where \(e_1\) is the first standard basis vector. The value \(v\) where this maximum equals \(v\) itself determines the boundary of the support.

References

[1]

Andrews, I., Roth, J., & Pakes, A. (2023). Inference for Linear Conditional Moment Inequalities. Review of Economic Studies.

[2]

Rambachan, A., & Roth, J. (2023). A more credible approach to parallel trends. Review of Economic Studies, 90(5), 2555-2591.