Generate Qualitative Data (Instrumental Variables)

Outcome type

Potential outcomes are generated differently according to outcome_type. If outcome_type == "multinomial", generate_qualitative_data_iv computes linear predictors for each class using the covariates:

$$\eta_{mi} (d) = \beta_{m1}^d X_{i1} + \beta_{m2}^d X_{i2} + \beta_{m3}^d X_{i3}, \quad d = 0, 1,$$

and then transforms $\eta_{mi} (d)$ into valid probability distributions using the softmax function:

$$P(Y_i(d) = m | X_i) = \frac{\exp(\eta_{mi} (d))}{\sum_{m'} \exp(\eta_{m'i}(d))}, \quad d = 0, 1.$$

It then generates potential outcomes $Y_i(1)$ and $Y_i(0)$ by sampling from {1, 2, 3} using $P_i(Y(d) = m | X), \, d = 0, 1$.

If instead outcome_type == "ordered", generate_qualitative_data_iv first generates latent potential outcomes:

$$Y_i^* (d) = \tau d + X_{i1} + X_{i2} + X_{i3} + N (0, 1), \quad d = 0, 1,$$

with $\tau = 2$. It then constructs $Y_i (d)$ by discretizing $Y_i^* (d)$ using threshold parameters $\zeta_1 = 2$ and $\zeta_2 = 4$. Then,

$$P(Y_i(d) = m | X_i) = P(\zeta_{m-1} < Y_i^*(d) \leq \zeta_m | X_i) = \Phi (\zeta_m - \sum_j X_{ij} - \tau d) - \Phi (\zeta_{m-1} - \sum_j X_{ij} - \tau d), \quad d = 0, 1,$$

which allows us to analytically compute the local probabilities of shift.

Treatment assignment and instrument

The instrument is always generated as $Z_i \sim \text{Bernoulli}(0.5)$. Treatment is always modeled as $D_i \sim \text{Bernoulli}(\pi(X_i, Z_i))$, with $\pi(X_i, Z_i) = P ( D_i = 1 | X_i, Z_i)) = (X_{i1} + X_{i3} + Z_i) / 3$. Thus, $Z_i$ can increase the probability of treatment intake but cannot decrease it.

Other details

The function always generates three independent covariates from $U(0,1)$. Observed outcomes $Y_i$ are always constructed using the usual observational rule.

The DGPs outlined above ensure that $Z_i$ is a valid instrument for D_i.