Correlation influence on two variable coefficients Consider the linear model $Y = B_{0} + B_{1}X_{1} + B_{2}X_{2} + e$,
where the columns $X_{1}$ and $X_{2}$ of the design matrix have mean $0$ and length $1$. That is $X_{i}'X_{i} = 1$ and $X_{i}'J = 0$ where $J$ is a column entirely of ones. Let $\rho$ be the correlation between $X_{1}$ and $X_{2}$.
Q: Determine what values of $\rho$ will make the variance of $\hat{B}_1$ and $\hat{B}_2$ larger than $5\sigma^2$. A: My thought is that this is a question related to the variance inflation factor.  Therefore, I could plug in $1 / (1 - z^2) = 5$.  Here, $z$ is the correlation I am trying to solve for.  After performing a few algebra steps, I get $z = 0.8$ (Correlation).
Please let me know if I am on the right track with the variance inflation factor or if I am supposed to use another methodology (covariance matrix, etc.)
Thank you.
A: The variance of $\hat{B}_1$ is
$$
{\rm var}(\hat{B}_1) = \sigma^2 ([X^{'}X]^{-1})_{22},
$$
where $\sigma^2 = {\rm var}(e)$, and $X = [\boldsymbol{1}, X_1, X_2]$
denotes the design matrix (see for example here).
From this question, we have 
$$\begin{align}
X'X & = 
\begin{pmatrix}
n & 0 & 0 \\
0 & 1 & \rho \\
0 & \rho & 1
\end{pmatrix},
\end{align} 
$$
where $n$ denotes the number of observations, and hence
$$
[X^{'}X]^{-1} =
\frac{1}{n (1 - \rho^2)}
\begin{pmatrix}
(1 - \rho^2) & 0 & 0 \\
0 & n & -n \rho \\
0 & -n \rho & n
\end{pmatrix} .
$$
Therefore, 
$$
{\rm var}(\hat{B}_1) = \frac{\sigma^2}{ (1 - \rho^2)}.
$$
So, ${\rm var}(\hat{B}_1) > 5 \sigma^2$ if and only if $|\rho| > \sqrt{0.8}$. The result is the same for $\hat{B}_2$.
