Least squares problem

Question

You are given a "blackbox":

$RSS = (Y - X\beta)'(Y-X\beta)$

for any linear model $Y=X\beta+\epsilon$

You have $n=60$ observations, and two predictor variables. You want to test

$H_0: \beta_1 = \beta_2$ in the model:

$Y = \beta_0+\beta_1X_1+\beta_2X_2+\epsilon$

Describe how you would accomplish this using the "black box"?

My professor has outlined changing

$H_0: \beta_1 = \beta_2$ to be $\theta = \beta_1 - \beta_2 = 0$, so $H_0: \theta = 0$.

Then rewrite the linear regression (scalar) form as:

$Y = \beta_0 + (\beta_2+\theta) X_1 + \beta_2 X_2 + \epsilon$

I don't understand where we take the algebra from here...

Answer 1

In the interests of making sure this question has an answer (and given the comments I made satisfied the OP):

$Y = \beta_0 + (\beta_2+\theta) X_1 + \beta_2 X_2 + \epsilon$

Split the $X_1$ term into two terms.

$Y = \beta_0 +\theta X_1+\beta_2 X_1 + \beta_2 X_2 + \epsilon$

Then collect the two terms in $β_2$ together.

$Y = \beta_0 +\theta X_1+\beta_2 (X_1 + X_2) + \epsilon$

Let $X_3=X_1+X_2$.

$Y = \beta_0 +\theta X_1+\beta_2 X_3 + \epsilon$

You have an ordinary regression equation.

You can do a t-test on the term which has $θ$ by itself. However, the same test can also be done as an F-test, by comparing nested models, one with $θ$ in it, the other without.