## DDML Algorithm #

DDML estimators proceed in two stages:

**Cross-fitting**to estimate conditional expectation functions.**Second stage estimation**based on Neyman orthogonal scores.

Chernozhukov et al. (2018) show that cross-fitting ensures that we can leverage a large class of machine learners for causal inference – including popular machine learners such as random forests or gradient boosting. Cross-fitting ensures independence between the estimation error from the first step and the regression residual in the second stage.

To illustrate the estimation methodology, let us consider the **Partially Linear Model**:
\[ Y = a.D + g(X) + U \\
D = m(X) + V \quad\quad~~\]
Under an conditional orthogonality, we can write
\[a = \frac{E\left[\big(Y - \ell(\bm{X})\big)\big(D - m(\bm{X})\big)\right]}{E\left[(D - m(\bm{X}))^2\right]}.\]
where
\(m(\bm{X})\equiv E[D\vert X] \)
and
\(\ell(\bm{X})\equiv E[Y\vert X]\)
.

DDML uses cross-fitting to estimate the conditional expectation functions, which are then used to obtain the DDML estimate of \(a\) .

To implement **cross-fitting**, we randomly split the sample into
\(K\)
evenly-sized folds, denoted as
\(I_1,\ldots, I_K\)
. For each fold
\(k\)
, the conditional expectations
\(\ell_0\)
and
\(m_0\)
are estimated using only observations *not* in the
\(k\)
th fold – i.e., in
\(I^c_k\equiv I \setminus I_k\)
– resulting in
\(\hat{\ell}_{I^c_{k}}\)
and
\(\hat{m}_{I^c_{k}}\)
, respectively, where the subscript
\({I^c_{k}}\)
indicates the subsample used for estimation. The out-of-sample predictions for an observation
\(i\)
in the
\(k\)
th fold are then computed via
\(\hat{\ell}_{I^c_{k}}(\bm{X}_i)\)
and
\(\hat{m}_{I^c_{k}}(\bm{X}_i)\)
. Repeating this procedure for all
\(K\)
folds then allows for computation of the DDML estimator for
\(a\)
:
\[ \hat{a}_n = \frac{\frac{1}{n}\sum_{i=1}^n \big(Y_i-\hat{\ell}_{I^c_{k_i}}(\bm{X}_i)\big)\big(D_i-\hat{m}_{I^c_{k_i}}(\bm{X}_i)\big)}{\frac{1}{n}\sum_{i=i}^n \big(D_i-\hat{m}_{I^c_{k_i}}(\bm{X}_i)\big)^2},\]
where
\(k_i\)
denotes the fold of the
\(i\)
th observation.