Confidence Intervals for β1 and β2

Question

I'm having a bit of trouble with the question below.

Using this Data -> Data

Question: Establish $95\%$ confidence intervals for $β_2$ and $β_3$.

I know I use the formula: $β_2 ±$ (t-value)(standard error)

How do I find the standard error using the data given in the pic?

Answer 1

I assume that you want to construct two different CIs (for each parameter) and not simultaneously for the vector $(\beta_2, \beta_3$). As such, you can use the result $$ \text{var}(\hat{\beta}) = \sigma^2(X'X)^{-1} . $$ When instead of $\sigma^2$ you can plug-in its estimator, namely $\hat{\sigma}^2=\frac{1}{10-3}\sum_{i=1}^{10} e_i^2$. And you should invert the following matrix $$ X'X= \begin{bmatrix} N & \sum X_{2t} & \sum X_{3t} \\ \sum X_{2t} & \sum X^2_{2t} & \sum X_{2t}X_{3t} \\ \sum X_{3t} & \sum X_{3t}X_{3t} & \sum X^2_{3t} \end{bmatrix}, $$ then, $\text{var}(\hat{\beta}_i) = \hat{\sigma}^2(X'X)^{-1}_{ii}$.