Suppose that I have a linear regression: $ y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \epsilon $
I would like to demonstrate that one regression coefficients is significantly greater than the other:
$H_{0} = \beta_{1} > \beta_{2}$ or $H_{0} = \beta_{1} - \beta_{2} > 0$.
Therefore, I would conduct a t-test:
$t = \mu_{d} / \sigma_{d} $
with $\mu_{d} = \beta_{1} - \beta_{2} $
and $\sigma_{d} = \sqrt{\sigma_{\beta_{1}}^2 + \sigma_{\beta_{2}}^2 - Cov(\beta_{1}, \beta_{2})}$
and $Cov (\beta_{1}, \beta_{2}) = \sigma_{\beta_{1}} * \sigma_{\beta_{2}} * \rho_{\beta_{1}, \beta_{2}} $
As $\beta_{1}$, $\beta_{2}$, $\sigma_{\beta_{1}}$ and $\sigma_{\beta_{2}}$ are known from the regression results, the only mising part is the correlation of $\beta_{1}$ and $\beta_{2}$. However, I don't know how this can be computed and would be really thankful for any help!
