# Standard error of regression coefficients? [duplicate]

I'm trying to test the hypothesis that $$\beta_{1} = \beta_{2}$$ for the SLR $$y_{i} = \beta_{1} + \beta_{2}x_{i} + \epsilon_{i}.$$ I start by letting $$\delta = \beta_{2} - \beta_{1}$$, so my null hypothesis is that $$\delta = 0$$. When I use the Wald test to complete the test, however, I'm stuck trying to find $$\hat{SE}[\delta]$$ for the denominator. I tried

$$\hat{V}[\delta] = \hat{V}[\beta_{2} - \beta_{1}] = \hat{V}[\beta_{2}] - \hat{V}[\beta_{1}],$$

but I'm realizing these coefficients likely aren't independent. Anyone know how to approach this? I'm getting lost in the expressions.

• One way: rewrite the model (equivalently) as $y_i = \alpha_1 z_i + \alpha_2 x_i + \epsilon_i$ with $z_i = 1+x_i$ and test $\alpha_2=0.$ Alternatively, use the correct formula $V(\delta) = V(\beta_2) + V(\beta_1) - 2\operatorname{Cov}(\beta_1,\beta_2).$
– whuber
Commented Jun 1, 2022 at 21:42
• Hi, more detail here would help. What is $B_1$ supposed to represent? Usually the first term of a regression equation is the intercept ($B_0$ or $a$). Are you trying to compare a regression coefficient to the intercept of the same model? If so, what is the reasoning behind the comparison?
– pep
Commented Jun 2, 2022 at 1:20