Model overview

Supported models #

Throughout we use $$Y$$ to denote the outcome variable, $$X$$ to denote confounders, $$Z$$ to denote instrumental variable(s), and $$D$$ to denote the treatment variable(s) of interest.

For a full discussion, please check our working paper.

Partial linear model [partial] #

$Y = a.D + g(X) + U \\ D = m(X) + V \quad\quad~~$

where the aim is to estimate $$a$$ while flexibly controlling for $$X$$ . ddml allows for multiple treatment variables, which may be binary or continuous.

Interactive model [interactive] #

$Y = g(X,D) + U\\ D = m(X) + V\quad$

which relaxes the assumption that $$X$$ and $$D$$ are separable. $$D$$ is a binary treatment variable. We are interested in the Average Treatment Effect or Average Treatment Effect on the Treated.

Partial linear IV model [iv] #

$Y = a.D + g(X) + U\\ Z = m(X) + V\quad\quad~~$

The parameter of interest is $$a$$ . We leverage instrumental variables $$Z$$ to identify $$a$$ , while flexibly controlling for $$X$$ .

Flexible IV model [fiv] #

$Y = a.D + g(X) + U ~~~\\ D= m(Z) + g(X) + V$

As in the Partial Linear Model, we are interested in $$a$$ . The Flexible Partially Linear IV Model allows for approximation of optimal instruments, but relies on a stronger independence assumption than the Partially Linear IV Model.

Interactive IV model [interactiveiv] #

$Y = g(Z,X) + U\\ D = h(Z,X) + V\\ Z = m(X) + E~~~$

where the aim is to estimate the local average treatment effect, while flexibly controlling for $$X$$ . Both $$Z$$ and $$D$$ are binary.