Solved – Compare coefficients of two independent variable from two regressions models with the same dependent variable

Question

I have the two following regression:
$$y=X_1\beta_1+\varepsilon_1$$
and
$$y=X_2\beta_2+\varepsilon_2$$

$x_1$ and $x_2$ are indicator variables and very similar. One expert claim that the two coefficients are the same. I want to know what kind of test I can perform to see if $β_1=β_2$.

Answer 1

If the two predictors are orthogonal, an optimal test is to compare the model

$$ y_i \sim \beta_0 + \beta_1 X_1 + \beta_2 X_2 $$

with the submodel

$$ y_i \sim \beta_0 + \beta_1 (X_1 + X_2) $$

with the likelihood ratio test. The null hypothesis is that $\beta_1 = \beta_2$.