moderndid.compute_identified_set_sdb

moderndid.compute_identified_set_sdb(m_bar, true_beta, l_vec, num_pre_periods, num_post_periods, bias_direction='positive')

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

Computes the identified set for \(l'\tau_{post}\) under the restriction that the underlying trend \(\delta\) lies in \(\Delta^{SDB}(M)\), which combines second differences bounds with a sign restriction.

The identified set is an interval \([\theta^{lb}, \theta^{ub}]\) derived from Lemma 2.1 in [2]. The bounds are given by

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

where \(\Delta\) is \(\Delta^{SDB}(M)\), the intersection of the smoothness and sign restrictions.

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 (pre and post periods).

l_vecnumpy.ndarray

Vector defining parameter of interest.

num_pre_periodsint

Number of pre-treatment periods.

num_post_periodsint

Number of post-treatment periods.

bias_direction{‘positive’, ‘negative’}, default=’positive’

Direction of bias sign restriction.

Returns:

DeltaSDBResult

Lower and upper bounds of the identified set.

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.