moderndid.compute_identified_set_sd

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

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

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

Following Lemma 2.1 in [2], if \(\Delta\) is closed and convex, then \(\mathcal{S}(\beta, \Delta)\) is an interval, \([\theta^{lb}(\beta, \Delta), \theta^{ub}(\beta, \Delta)]\), where

\[ \begin{align}\begin{aligned}\theta^{lb}(\beta, \Delta) := l'\beta_{post} - \max_{\delta} \{l'\delta_{post} : \delta \in \Delta, \delta_{pre} = \beta_{pre}\},\\\theta^{ub}(\beta, \Delta) := l'\beta_{post} - \min_{\delta} \{l'\delta_{post} : \delta \in \Delta, \delta_{pre} = \beta_{pre}\}.\end{aligned}\end{align} \]

The identified set is constructed subject to \(\delta \in \Delta^{SD}(M)\) and \(\delta_{pre} = \beta_{pre}\).

Under the decomposition \(\beta = \tau + \delta\) with \(\tau_{pre} = 0\), the causal parameter \(\theta = l'\tau_{post}\) is partially identified. Since \(\delta_{pre} = \beta_{pre}\) is point identified, the restriction \(\delta \in \Delta^{SD}(M)\) constrains the possible values of \(\delta_{post}\).

Parameters:

m_barfloat

Smoothness parameter M. Bounds the second differences: \(|\delta_{t-1} - 2\delta_t + \delta_{t+1}| \leq M\).

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:

DeltaSDResult

Lower and upper bounds of the identified set.

Notes

The constraint \(\delta_{pre} = \beta_{pre}\) reflects that pre-treatment event study coefficients identify the pre-treatment trend under the no-anticipation assumption.

References

[1]

Andrews, I., Roth, J., & Pakes, A. (2021). 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.