# Hypothesis testing using only RSS

I have been presented with an interesting regression question:

Suppose I have a "black box" that will calculate the residual sum of squares:

$RSS=(Y-X\hat{\beta})'(Y-X\hat{\beta})$ for any standard linear model of the form $Y=X\beta+e$ that I want to put in it. Further assume that there are n=60 observations, two predictor variables, $\alpha_1,\alpha_2$ and that I am testing the hypothesis:

$H_0: \gamma_1=\gamma_2$ using the model:

$Y_j=\gamma_0+\alpha_{j,1}\gamma_1+\alpha_{j,2}\gamma_2+\epsilon_j, j=1,...,60$.

The trick is not to try and just find the appropriate H-matrix and test accordingly, but that the only tool we have at our disposal is this black box that will give me $RSS$ and nothing else.

Any thoughts on how to solve the puzzle?

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Hint Upon rewriting the model as $Y_j=\gamma_0+(\alpha_{j,1}+\alpha_{j-2})\gamma_1+\alpha_{j,2}(\gamma_2-\gamma_1‌​)+\epsilon_j$ the problem simplifies to testing whether a given coefficient vanishes. Now the Gauss-Markov theorem relates the change in RSS to the change in log likelihood... – whuber Jun 26 '12 at 23:04
interesting...I have not specifically covered Gauss-Markov, but I will think on this. – Justin Jun 26 '12 at 23:14