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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!

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You're missing the point of the question, I think. The issue isn't whether the estimators converge in probability to the things they estimate, it's whether they estimate the right things. The second estimator is consistent but for something other than $\delta$.

Suppose, so as not to prejudice things, we write $\xi=\lim_{n\to\infty} \hat\delta$ in the first question and $\zeta=\lim_{n\to\infty} \hat\delta$ in the second question (limits in probability). The question wants you to show that $\xi=\delta$ and that $\zeta\neq\delta$.

The first one is fairly easy; it follows from OLS consistency. For the second one it's probably easiest to write down a model including $W$ and work out what $\zeta$ is in terms of the coefficients in that model and show it isn't $\delta$.

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