Dynamic Double Machine Learning explained

External question — Cross Validated Stack Exchange Author: jbuddy_13 Original post: https://stats.stackexchange.com/questions/676185 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I'm familiar with Double Machine Learning. The idea is intuitive and we only need five components: Two ML models, two residual arrays, and one regression model. The two ML ("nuisance") models predict the propensity of treatment and outcome attributable to common covariates, $X$ . The residual arrays then articulate, what variance in treatment exposure and outcome cannot be attributed to $X$ . This is the secret sauce behind the orthogonalization strategy. We finally regress the unexplained variance on the outcome on the unexplained variance in treatment exposure, effectively isolating an RCT hidden in observational data. (The R-Learner follows a very similar design deviating only in that an $\tau(X)$ ML regressor predicts the residual ratio where the regressor's residuals are weighted by the squared treatment residuals. Which effectively trades a slope for an ML modeling and by extension, CATE in lieu of ATE.) Now, Dynamic DML introduces panel data where a given unit can have periods of either treatment exposure or non-exposure, indexed by time. Without knowledge of the effect duration, a deflation strategy is needed such that we don't overestimate the cumulative effects of early exposures. This sounds great! But the specific implementation is not intuitive to me. The EconML library describes the following implementation where $Z_t$ is the concatenation of two feature vectors, $X_t$ and $W_t$ indexed by time, $t$ . $$\begin{split}Z_t =~& A \cdot T_{t-1} + B \cdot Z_{t-1} + \eta_t\\ T_t =~& p(T_{t-1}, Z_t, \zeta_t) \\ Y_t =~& \theta_0(X_0)'T_t + \mu'Z_t + \epsilon_t\end{split}$$ So the current state, $Z_t$ is a function of the previous treatment exposure, previous state, and error. But beyond this, I'm not certain of the authors' intent. In plain English, how does DDML deviate from DML?