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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.

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    $\begingroup$ 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 $V(\delta) = V(\beta_2) + V(\beta_1) - 2\operatorname{Cov}(\beta_1,\beta_2).$ $\endgroup$
    – whuber
    Commented Jun 1, 2022 at 21:42
  • $\begingroup$ 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? $\endgroup$
    – pep
    Commented Jun 2, 2022 at 1:20

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