Deriving confidence interval for fitted values from asymptotic distribution Consider linear regression model 
$$y = X\beta + e$$
I want to construct confidence interval for fitted values, i.e. $\hat{y} = X\hat{\beta}$. Define linear fucntion of $\beta$ as 
$$f(\beta) = X\beta$$
and 
$$f'(\beta) = X$$
Now by Delta Method 
$$\sqrt{n}(f(\hat{\beta}) - f(\beta)) \xrightarrow{d} f'(\beta)z$$
where $z \sim N(0, V_{\beta})$
$$Var(\sqrt{n}(f(\hat{\beta}) - f(\beta))) = XV_{\beta}X^T$$ 
Hence 
$$\sqrt{n}(f(\hat{\beta}) - f(\beta)) \xrightarrow{d} N(0, XV_{\beta}X^T) $$
How do I proceed from now? 
I can construct 95% confidence interval for the $\sqrt{n}(f(\hat{\beta}) - f(\beta))$ but it does not give much information since I don't know $f(\beta)$. It is stated in the book by Hansen (freely available online, section 6.14) that this interval should be 
$$CI = x_i^T\hat{\beta} \pm 1.96\sqrt{x_i^T\hat{V}_{\hat{\beta}}x_i}$$
where $\hat{V}_{\hat{\beta}}$ is estimated var-covar matrix for $\hat{\beta}$. Probably $\sqrt{X\hat{V}_{\hat{\beta}}X^T}$ denotes standard error term. 
So far I have two questions: (1) how do you come up with Hansen's formula and (2) why dimensions don't match in this formula. $X\hat{V}X^T$  should have dimension $n \times n$ since $X$ is $n\times k$ and $V$ is $k\times k$. However, $X\beta$ is $n\times 1$ matrix. 
EDIT
When I read my question I realized that if $f(\hat{\beta_i}) - f(\beta_i) \sim N(0, \sigma_i^2)$, then CI for $f(\beta_i)$ is given as
$$CI = f(\hat{\beta}_i) \pm 1.96\sqrt{\sigma_i^2}$$
So, assuming that $n$ is big enough one could claim that 
$$\sqrt{n}(f(\hat{\beta}) - f(\beta)) \sim N(0, XV_{\beta}X^T)$$
But then 
$$(f(\hat{\beta}) - f(\beta)) \sim N(0, \frac{1}{n}XV_{\beta}X^T)$$
So, I would reformulate my first question: why there is no division by $n$ in Hansen's formula?  
 A: So far the questions in this post arised from the confusion of notation from Hansen's book. 


*

*Why there is $\hat{V}_{\hat{\beta}}$ in the final formula when it should be $\frac{1}{n}\hat{V}_{\beta}$? You should be careful with notation of variances since there are many of them (4 to be precise). $V_{\beta}$ denotes asymptotic variance of $\sqrt{n}(\hat{\beta} - \beta)$. While $V_{\hat{\beta}}$ corresponds to exact variance of $\hat{\beta}$. Finally, $\hat{V}_{\hat{\beta}}$ and $\hat{V}_{\beta}$ denote estimators of "usual" and asymptotic variances of $\hat{\beta}$ respectively. Once you observed that $\hat{V}_{\beta} = n\hat{V}_{\hat{\beta}}$ confusion why there is no division by $n$ is gone. 

*The second question again arised because of confusion on the notation. Formula for confidence interval is given as 
$$CI = x_i'\hat{\beta} \pm 1.96\sqrt{x_i'\hat{V}_{\hat{\beta}}x_i}$$
where $x_i$ is not the same as $X_i$. The latter one is usual $n\times k$ matrix where each column corresponds to the regressor, while $x_i'$ denotes $i^{th}$ row of this matrix. After recognizing this fact it is obvious that dimensions match. 

