Consider the model $$y_i = x_{i}'\beta + e_i.$$
In the discussion of instrumental variables estimator it can be shown that OLS estimator $b$ is biased and inconsistent estimator of $\beta$. Nonetheless, $b$ does estimate something: $plim_{n\to\infty}b = \beta + Q^{-1}\gamma = \theta$ with $Q = \mathbb{E}(x_i'x_i)$ and $\mathbb{E}(x_ie_i) = \gamma \neq 0.$
Assume data is i.i.d. and that $\mathbb{E}(x_i'x_i)$ has full rank.
- Show that $b$ is asymptotically normally distributed.
- Derive the asymptotic covariance matrix of $b$.
My attempt. Write $b - \beta = \left(\frac{1}{n}\sum_ix_i'x_i\right)^{-1}\left(\frac{1}{n}\sum_ix_ie_i\right)$. Since $\mathbb{E}(x_ie_i) \neq 0$, we can't use CLT directly on the RHS. But observe that $$\sqrt{n}\left(\frac{1}{n}\sum_ix_ie_i - \mathbb{E}(x_ie_i)\right) \xrightarrow{d} N(0, V),$$ where $V = Var(x_ie_i)$. Now transform expression for $b - \beta$ :
\begin{equation} \begin{aligned} b - \beta - \left(\frac{1}{n}\sum_ix_i'x_i\right)^{-1}\gamma &= \left(\frac{1}{n}\sum_ix_i'x_i\right)^{-1}\left(\frac{1}{n}\sum_ix_ie_i\right) - \left(\frac{1}{n}\sum_ix_i'x_i\right)^{-1}\gamma \\ & = \left(\frac{1}{n}\sum_ix_i'x_i\right)^{-1}\left(\frac{1}{n}\sum_ix_ie_i - \gamma\right) \end{aligned} \end{equation}
Multiply both sides by $\sqrt{n}$ to get $$\sqrt{n}\left[(b - \beta - \left(\frac{1}{n}\sum_ix_i'x_i\right)^{-1}\gamma)\right] = \left(\frac{1}{n}\sum_ix_i'x_i\right)^{-1}\sqrt{n}\left(\frac{1}{n}\sum_ix_ie_i - \gamma\right)$$
$\left(\frac{1}{n}\sum_ix_i'x_i\right)^{-1} \xrightarrow{p} Q^{-1} = \mathbb{E}(x_i'x_i)$ and $\sqrt{n}\left(\frac{1}{n}\sum_ix_ie_i - \gamma\right) \xrightarrow{d} N(0, V)$. Then, by Slutsky theorem $$\sqrt{n}\left[(b - \beta - \left(\frac{1}{n}\sum_ix_i'x_i\right)^{-1}\gamma)\right] \xrightarrow{d} N(0, Q^{-1}VQ^{-1})$$
I am not quite sure that the result is correct and makes sense. Any hints or suggestions are welcome.