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.