# Testing $H_0: \beta_1 = \beta_2$ in a model $Y = \alpha + \beta_1x_1+ \beta_2x_2 + \varepsilon$

Consider a linear regression model $$Y = \alpha + \beta_1x_1+ \beta_2x_2 + \varepsilon$$, and suppose we are interested in testing $$H_0: \beta_1 = \beta_2$$. These slopes are two population slopes.

This means that under $$H_0$$, we have $$Y = \alpha + \beta(x_1 + x_2) + \varepsilon$$ where $$\beta$$ is the common value of $$\beta_1$$ and $$\beta_2$$.

My question is, can we just use the incremental F-test, where you compare the incremental sum of squares?

Also, is this test meaningful? I feel like it doesn't make sense to perform this test if $$x_1$$ and $$x_2$$ have different units.

EDIT: I just found in the post

t test of individual coefficient and wald test of euqality of two coefficients

that I should use the Wald test. But how do you compute the test statistic? The one in Wikipedia is for comparing a parameter obtained by MLE with a constant.

• If you are worried about incommensurable units, then you hypothesis makes no sense. However, such hypotheses can make sense in many applications. For instance, you might postulate that the rates at which the growth of a plant changes with respect to standardized amounts of fertilizer (mass) and sunlight (energy) are the same. As far as the computing goes, you could simply reformulate your model as $Y=\alpha + \gamma_1(x_1+x_2)+\gamma_2(x_1-x_2)+\varepsilon,$ where the gammas equal $(\beta_1\pm\beta_2)/2,$ and let the software automatically test $\gamma_2=0.$ – whuber Apr 29 at 13:45