The doubly robust estimator is a popular method for measuring the average treatment effect with observational data (assuming no unmeasured confounders):
$$ \hat{\Delta}_{DR} = n^{-1}\sum_{i=1}^n \biggl[\frac{Z_iY_i}{e(\mathbf{X_i},\hat{\beta_i})}-\frac{(Z_i-e(\mathbf{X_i},\hat{\beta_i}))}{e(\mathbf{X_i},\hat{\beta_i})}m_1(\mathbf{X_i},\hat{\alpha_1})\biggr] \\ -n^{-1}\sum_{i=1}^n \biggl[\frac{(1-Z_i)Y_i}{1-e(\mathbf{X_i},\hat{\beta_i})}-\frac{(Z_i-e(\mathbf{X_i},\hat{\beta_i}))}{1-e(\mathbf{X_i},\hat{\beta_i})}m_0(\mathbf{X_i},\hat{\alpha_0})\biggr]\\ =\hat{\mu}_{1,DR}-\hat{\mu}_{0,DR}.$$
Consider just the $\hat{\mu}_{1,DR}$ estimate for brevity (the $\hat{\mu}_{0,DR}$ calculations are similar), which can be reduced to the following expression: $$\hat{\mu}_{1,DR}=E(Y_1)+E\biggl[ \frac{(Z-e(\mathbf{X},\mathbf{\beta}))}{e(\mathbf{X},\mathbf{\beta})}(Y_1-m_1(\mathbf{X},\mathbf{\alpha_1}))\biggr].$$
If either the propensity score model $e(\mathbf{X},\bf{\beta})$ or the regression model $m_1(\mathbf{X},\mathbf{\alpha_1})$ is unbiased, the second "augmentation" term reduces to zero and we have $\hat{\mu}_{1,DR}=E(Y_1)$.
However, if both the propensity score and regression models are incorrect, the augmentation term will equal the product of the two model's expected biases, in which case the doubly robust estimator invariably ends up being worse than either the propensity score or regression model alone, is that correct?
Adding in the biased augmentation term for $\hat{\mu}_{0,DR}$ may mitigate the bias in $\hat{\mu}_{1,DR}$, although using the doubly robust estimator still presents a gamble.
