Solved – Testing the general linear hypothesis: $H_0: \beta_1 = \beta_2 = \beta_3 = \beta_4 = \beta$

hypothesis testingmultiple regressionself-studyt-test

Again, we are testing the linear hypothesis;

$H_0: \beta_1 = \beta_2 = \beta_3 = \beta_4 = \beta$

for the model,

$$y = \beta_0 + \beta_1x_1+\beta_2x_2+\beta_3x_3+\beta_4x_4+\epsilon$$

I know how to solve this for testing if two betas are equal but I don't quite understand the equality of all four. I imagine this is simpler than I'm envisioning but I can't seem to piece it together.

I know I can develop a matrix $T$ of 1's and 0's and multiply it by my vector of prediction coefficients $\mathbf{\beta}$

The problem is that I'm not entirely sure how to correctly construct the T matrix to test this hypothesis. More specifically, I'm not confident in the output vector.

Here is what I did:

$$T =
\begin{pmatrix}
0 & 1 & 0 & 0 & 0 & -1 \\
0 & 0 & 1 & 0 & 0 & -1 \\
0 & 0 & 0 & 1 & 0 & -1 \\
0 & 0 & 0 & 0 & 1 & -1 \\
\end{pmatrix}
$$

$$ \mathbf{\beta} =
\begin{pmatrix}
\beta_0\\
\beta_1\\
\beta_2\\
\beta_3\\
\beta_4\\
\beta
\end{pmatrix}
$$
NOTE: The $\beta$ to the left of the equality should be bold. I'm not implying an equality between the left side of the equality with the sixth element in the beta vector.

When I multiply these I get a $4\times1$ vector of 0's. Is this the proper way to set up the test? I don't need to actually test this hypothesis. I just need to properly setup T and beta. Thanks in advance.

Best Answer

Based on your comment, you do not have idea what four $\beta$s should be, and just want to test if they are the same.

It equals to $\beta_1 = \beta_2 = \beta_3 =\beta_4$, and can be write in different ways. One of them is:

$\beta_1 = \beta_2$

$\beta_1 = \beta_3$

$\beta_1 = \beta_4$

Based on these 3 equations, the $T$ matrix is $$T = \begin{pmatrix} 0 & 1 & -1 & 0 & 0 \\ 0 & 1 & 0 & -1 & 0\\ 0 & 1 & 0 & 0 & -1 \\ \end{pmatrix} $$

$$ \mathbf{\beta} = \begin{pmatrix} \beta_0\\ \beta_1\\ \beta_2\\ \beta_3\\ \beta_4 \end{pmatrix} $$

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