moderndid.compute_identified_set_sdrmm

moderndid.compute_identified_set_sdrmm(m_bar, true_beta, l_vec, num_pre_periods, num_post_periods, monotonicity_direction='increasing')

Compute identified set for \(\Delta^{SDRMM}(\bar{M})\).

Computes the identified set for \(l'\tau_{post}\) under the restriction that the underlying trend delta lies in \(\Delta^{SDRMM}(\bar{M})\).

This set combines the second-differences-with-relative-magnitudes constraint with a monotonicity constraint, as discussed in Section 2.4.4 of [1]. The identified set is the union of identified sets for each component polyhedron,

\[\mathcal{S}(\beta, \Delta^{SDRMM}(\bar{M})) = \bigcup_{s<0, sign \in \{+,-\}} \mathcal{S}(\beta, \Delta^{SDRM}_{s,sign}(\bar{M}) \cap \Delta^{Mon})\]

where each component set is computed by solving linear programs to find the range of \(l'\tau_{post}\) consistent with the constraints.

Parameters:

m_barfloat

Relative magnitude parameter. Second differences in post-treatment periods can be at most \(\bar{M}\) times the maximum absolute second difference in pre-treatment periods.

true_betanumpy.ndarray

True coefficient values (pre and post periods).

l_vecnumpy.ndarray

Vector defining parameter of interest \(\theta = l'\tau_{post}\).

num_pre_periodsint

Number of pre-treatment periods.

num_post_periodsint

Number of post-treatment periods.

monotonicity_direction{‘increasing’, ‘decreasing’}, default=’increasing’

Direction of monotonicity restriction.

Returns:

DeltaSDRMMResult

Lower and upper bounds of the identified set.

Notes

The identified set is computed by solving linear programs for each choice of period \(s \in \{-(T_{pre}-2), ..., 0\}\) and sign (positive/negative maximum), then taking the union of all resulting intervals. The monotonicity constraint is enforced in each linear program, ensuring that treatment effects are either non-decreasing or non-increasing over time.

The linear programs solve for the maximum and minimum of \(l'\delta_{post}\) subject to constraints including \(\delta_{pre} = \beta_{pre}\) and \(\delta \in \Delta^{SDRM}_{s,sign}(\bar{M}) \cap \Delta^{Mon}\).

References

[1]

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