Consistency of covariance matrix estimate in linear regression 
Show that $\hat{\theta} =\frac{1}{N} \sum\limits_{i=1}^{N}\hat{u_i}^{2}x_i'x_i $ is a consistent estimator for $E(u^2x_i'x_i)$, by showing:
$$\frac{1}{N} \sum\limits_{i=1}^{N}\hat{u_i}^{2}x_i'x_i = \frac{1}{N} \sum\limits_{i=1}^{N}{u_i^{2}}x_i'x_i + o_p(1),$$
where $\hat{u}_i=y_i-x_i\hat\beta$ are the OLS residuals of the following regression model
$$y_i = x_i\beta + u_i,$$
where $x_i$ is a $1\times K$ vector, $\beta$ is a $K\times 1$ vector and $y_i$ with $u_i$ are scalars. We assume that $(y_i,x_i,u_i)$, $i=1,...,N$ forms an iid sample.

We should use the following hints:

*

*$\hat{u}_i^2$ = $u_i^2 - 2u_ix_i(\hat{\beta}-\beta) + [x_i(\hat{\beta}-\beta)]^2$


*$\hat{\beta} - \beta$ = $o_p(1)$


*Sample averages are $O_p(1)$.


*We assume all necessary expectations exist and are finite.
I'm getting stuck with this. I know I must be missing some simple sort of substitution or there is some gap in my knowledge or understanding preventing me from making the necessary manipulations.
I would love if someone could walk me through this and explain the intuition a bit here.
 A: I am sorry I don't have any reputation to write it as a comment so I have to write it as an answer. First of all, you need to be careful with not forgetting to write $x_i'$ rather than $x_i$ in some instances, especially in 1.) I think  by $\hat{\beta}$ Wooldridge means the OLS estimator of the parameter of interest in your regression equation, not what you denoted as $\hat{\beta}$ in the first line.  
To solve this problem it is sufficient  to assume several things: 
$$E[x_iu_i]=0$$
$$E[|x_{il}x_{im}|^2]<\infty \text{ for any } l,m$$
$$E[u_i^4]<\infty$$
under the condition that $\{(x_i,u_i)\}_{i=1}^n$ are i.i.d. across $i$. 
Try using inequalities such as Cauchy-Schwarz or inequalities for matrix norms and then using laws of large numbers and Slutsky theorem. 
A: Read the Wooldridge passage on proving that feasible GLS estimate is consistent. The proof relies on application on law of large numbers and Slutsky lema. It is actually not necessary to use the fact $Eu_ix_i=0$, as long as you have that $\hat\beta-\beta = o_p(1)$. 
Using the hint number one insert the formula for $\hat{u_i}$ into left hand side of your main formula.
You will get three terms in a form $\frac{1}{N}\sum f(x_i,u_i,\hat\beta)$. Since $(x_i,u_i)$ form an iid sample, the law of large numbers gives you that $\frac{1}{N}\sum_{i=1}g(x_i,u_i) \to Eg(x_i,u_i)$. 
Your main problem is that you need separate $\hat\beta - \beta$ from $x_i$ and $u_i$, only then you can use law of large numbers. For that use Slutsky lema. When you have separated $\hat\beta-\beta$, exploit the fact that it is $o_p(1)$, the fact that converging sequence (for which you have applied the law of large number) is $O_p(1)$ (the hint 3) and the fact that $o_p(1)O_p(1)=o_p(1)$.
