I'm stuck on two questions, both of which relate to proving the (in)consistency of the standard OLS estimator.
I'm given the following regression model, where $Y_i$ denotes the outcome measure for the $i^{th}$ individual, $D_i$ is a binary treatment indicator (taking on value $0$ conditional on treatment; $1$ conditional on treatment).
$$ Y_i = \alpha + \delta D_i + \pmb{X^{T}_{i}} \theta_i + \epsilon_i $$
I'm asked to show the following two things: first, that the OLS estimator of $\delta$ is consistent, so long as $Y_i$ and $D_i$ are independent, conditional on $X_i$. Next, I'm supposed to show that OLS is inconsistent, if $Y_i$ and $D_i$ are only independent conditional on $X_i$ and an additional covariate $W_i$.
I know the standard proof for the consistency of the OLS estimator, where we end up with the following expression:
$$ \beta +(plim_{n \rightarrow \infty} (1/n) X^{T}X')^{-1}(plim_{n \rightarrow \infty} (1/n) X^Tu) = 0$$
However, in the proof I'm aware of the intercept is included in the $ \beta X$ matrix product. If I follow similar steps for this question, I end up with:
$$ \beta +(plim_{n \rightarrow \infty} (1/n) \delta^{T}\delta')^{-1}(plim_{n \rightarrow \infty} (1/n) \delta^T\epsilon) + (plim_{n \rightarrow \infty} (1/n) \delta^T X\theta)$$
However, I'm not sure how to show that the final expression is zero. If $Y$ and $D$ are independent conditional on X, does that make: $$(plim_{n \rightarrow \infty} (1/n) \delta^T X\theta) = 0$$ If so, why? Moreover, why does this change when $Y_i$ and $D_i$ are independent only an additional covariate? I'm quite confused, so help understanding would be really amazing!