numerical verification of equality of coefficient and SE per FWL

fwl_estimates_and_se.R

library(estimatr)
data(auto)
# %% FWL regression coefficient
auto$ytil = lm(price ~ displacement, auto)$resid
auto$x2til = lm(weight ~ displacement, auto)$resid
(fwlest = lm_robust(ytil ~ x2til, auto, se_type = "HC0")
  %>% summary %>% .$coefficients %>% .[2, 1:2])
# %%
(fullest =
  lm_robust(price ~ weight + displacement, auto, se_type = "HC0") %>%
  summary %>% .$coefficients %>% .[2, 1:2]
)
# %% theorem 1 and theorem 2 - Ding 2021 Statistics and Probability Letters
all.equal(fwlest, fullest)

# %%