moderndid.compute_identified_set_sdrm

moderndid.compute_identified_set_sdrm(m_bar, true_beta, l_vec, num_pre_periods, num_post_periods)

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

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

The identified set is the union of identified sets for each component polyhedron, following Equation (7) in [1],

\[\mathcal{S}(\beta, \Delta) = \bigcup_{k=1}^{K} \mathcal{S}(\beta, \Delta_k).\]

For \(\Delta^{SDRM}(\bar{M})\), this union is taken over all possible pre-treatment periods s where the maximum second difference could occur, and for both positive and negative signs of this maximum,

\[\mathcal{S}(\beta, \Delta^{SDRM}(\bar{M})) = \bigcup_{s<0} \left( \mathcal{S}(\beta, \Delta_{s,+}^{SDRM}(\bar{M})) \cup \mathcal{S}(\beta, \Delta_{s,-}^{SDRM}(\bar{M})) \right),\]

where \(\mathcal{S}(\beta, \Delta_{s,\text{sign}}^{SDRM}(\bar{M}))\) is the identified set when the maximum pre-period second difference occurs at period \(s\) with a given sign.

For each component, we solve linear programs to find the bounds of \(l'\tau_{post}\) subject to \(\delta \in \Delta_{s,\text{sign}}^{SDRM}(\bar{M})\) and \(\delta_{pre} = \beta_{pre}\).

Parameters:

m_barfloat

Relative magnitude parameter \(\bar{M}\). 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 \(\beta = (\beta_{pre}', \beta_{post}')'\).

l_vecnumpy.ndarray

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

num_pre_periodsint

Number of pre-treatment periods \(\underline{T}\).

num_post_periodsint

Number of post-treatment periods \(\bar{T}\).

Returns:

DeltaSDRMResult

Lower and upper bounds of the identified set.

References

[1]

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