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

$$d_i = 1\left\lbrace \alpha_x Z + v_i > 0 \right\rbrace,$$

$$y_i = \theta d_i + x_i' \beta + u_i,$$

$$Z \sim \textstyle{Bernoulli} (0.5)$$ and

$$\left(\begin{array}{c} u_i \\ v_i \end{array} \right) \sim \mathcal{N}\left(0, \left(\begin{array}{cc} 1 & 0.3 \\ 0.3 & 1 \end{array} \right) \right).$$

The covariates :$$x_i \sim \mathcal{N}(0, \Sigma)$$, where $$\Sigma$$ is a matrix with entries $$\Sigma_{kj} = 0.5^{|j-k|}$$ and $$\beta$$ is a dim_x-vector with entries $$\beta_j=\frac{1}{j^2}$$.

The data generating process is inspired by a process used in the simulation experiment of Farbmacher, Gruber and Klaaßen (2020).

make_iivm_data(
n_obs = 500,
dim_x = 20,
theta = 1,
alpha_x = 0.2,
return_type = "DoubleMLData"
)

## Arguments

n_obs

(integer(1))
The number of observations to simulate.

dim_x

(integer(1))
The number of covariates.

theta

(numeric(1))
The value of the causal parameter.

alpha_x

(numeric(1))
The value of the parameter $$\alpha_x$$.

return_type

(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

Farbmacher, H., Guber, R. and Klaaßen, S. (2020). Instrument Validity Tests with Causal Forests. MEA Discussion Paper No. 13-2020. Available at SSRN:doi:10.2139/ssrn.3619201 .