So we agree that, provided y is the variable of interest -- not censored -- a linear model estimated by OLS is a good starting point. But other functional forms can be better, such as logistic if y is binary or fractional, exponential if y is nonnegative.
In many cases one should include the covariates flexibly -- such as squares and interactions. This is especially true in treatment effect contexts. If w is the treatment, interact it with the controls when estimating the average treatment effect.
As @TymonSloczynski showed in his elegant 2020 REStat paper, if d is the treatment, just adding x as in the regression y on d, x can produce a badly biased estimate of the ATE. Interacting d and elements of x is generally better. Same is true for nonlinear regression adjustment.
The best RA choices in Stata:
teffects ra (y x1 ... xk) (d) (any y)
teffects ra (y x1 ... xk, logit) (d) (binary y)
teffects ra (y x1 ... xk, flogit) (d) (fractional y)
teffects ra (y x1 ... xk, poisson) (d) (y >= 0)
teffects ra (y x1 ... xk) (d) (any y)
teffects ra (y x1 ... xk, logit) (d) (binary y)
teffects ra (y x1 ... xk, flogit) (d) (fractional y)
teffects ra (y x1 ... xk, poisson) (d) (y >= 0)
These choices are best because (1) they are still consistent for an RCT. (2) Each can be combined with inverse propensity score weighting to produce "doubly robust" estimators. Replacing logit with probit or Poisson with NegBin fails on both counts.
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