Generates data from a interactive regression (IRM) model. The data generating process is defined as

$$d_i = 1\left\lbrace \frac{\exp(c_d x_i' \beta)}{1+\exp(c_d x_i' \beta)} > v_i \right\rbrace,$$

$$y_i = \theta d_i + c_y x_i' \beta d_i + \zeta_i,$$

with $$v_i \sim \mathcal{U}(0,1)$$, $$\zeta_i \sim \mathcal{N}(0,1)$$ and covariates $$x_i \sim \mathcal{N}(0, \Sigma)$$, where $$\Sigma$$ is a matrix with entries $$\Sigma_{kj} = 0.5^{|j-k|}$$. $$\beta$$ is a dim_x-vector with entries $$\beta_j = \frac{1}{j^2}$$ and the constancts $$c_y$$ and $$c_d$$ are given by

$$c_y = \sqrt{\frac{R_y^2}{(1-R_y^2) \beta' \Sigma \beta}},$$

$$c_d = \sqrt{\frac{(\pi^2 /3) R_d^2}{(1-R_d^2) \beta' \Sigma \beta}}.$$

The data generating process is inspired by a process used in the simulation experiment (see Appendix P) of Belloni et al. (2017).

make_irm_data(
n_obs = 500,
dim_x = 20,
theta = 0,
R2_d = 0.5,
R2_y = 0.5,
return_type = "DoubleMLData"
)

## Arguments

n_obs (integer(1)) The number of observations to simulate. (integer(1)) The number of covariates. (numeric(1)) The value of the causal parameter. (numeric(1)) The value of the parameter $$R_d^2$$. (numeric(1)) The value of the parameter $$R_y^2$$. (character(1)) If "DoubleMLData", returns a DoubleMLData object. If "data.frame" returns a data.frame(). If "data.table" returns a data.table(). If "matrix" a named list() with entries X, y, d and z is returned. Every entry in the list is a matrix() object. Default is "DoubleMLData".

## References

Belloni, A., Chernozhukov, V., Fernández-Val, I. and Hansen, C. (2017). Program Evaluation and Causal Inference With High-Dimensional Data. Econometrica, 85: 233-298.