_did_sim.Rmd

Latent factor model (interactive fixed-effects model) [@xu2017generalized, @athey2022design]

$$
Y_{it} = \gamma_i \upsilon_t^T + \tau W_{it} + \epsilon_{it}
$$

where

$\gamma_i$ = vector of latent unit factors

$\upsilon_t$ = vector of latent time factors

In matrix notation,

$$
\mathbf{Y} = \mathbf{L} + \tau \mathbf{W} + \mathbf{E}
$$

where

$\mathbf{L} = \mathbf{\Gamma\Upsilon}^T$

$\mathbf{E}$ = idiosyncratic component (i.e., error matrix)

$E(\mathbf{E} |\mathbf{W, L}) = 0$

No need for $\mathbf{L} \perp \mathbf{W}$ (treatment assignment may depend on the systematic component)

$\mathbf{E}_i \perp \mathbf{E}_{i'}$

$\mathbf{L}$ = systematic component

When $L_{it} = \alpha_i + \beta_t$ (additive form), DID can consistently estimate $\tau$ [@moon2017dynamic, @moon2015linear]