Regression coefficients! Everybody loves 'em.[Citation needed]
You know what else everybody loves? Test statistics for regression coefficients![Disputed, quite disputed, but with pushback]
Test statistics for regression coefficients are standard in statistical software output (typically t or z test statistics, depending on the nature of the estimator), and are just dandy for testing null hypotheses of the form $H_{0}\text{: }\beta = 0$ vs $H_{A}\text{: }\beta\ne 0$. [Hot damn!]
However… what if we want to test null hypotheses of the form $H_{0}\text{: }\beta = c$ for $c \ne 0$?
I know that a t test for a Pearson's correlation coefficient is pretty straightforward when the null hypothesis being tested is of the form $H_{0}\text{: }\rho =0$ vs $H_{A}\text{: }\rho \ne 0$, but that the math gets wonkier when the null hypothesis being tested is of the form $H_{0}\text{: }\rho =c$ for $c\ne 0$. So I wonder…
How to calculate the t or z test statistic for the null hypothesis $\boldsymbol{H_{0}\textbf{: }\beta = c}$ for $\boldsymbol{c \ne 0}$? Is it as simple as $(\hat{\beta}-c)/\hat{\sigma}_{\beta}$ or does it get funky?
Note: This is not a question about a test of one regression coefficient equaling another regression coefficient (i.e. not a test of $H_{0}\text{: }\beta_{A} = \beta_{B}$).
Edit: I bring up Pearson's $\rho$ in motivating my question, because (in the OLS case) $\rho$ and $\beta$ share the same numerator ($\rho$ is just the numerator normalized over the sample size), but calculating a t statistic for $\rho$ gets funky as noted above. I assume this is because the standard error no longer has the form under the $c=0$ null, and wonder why the standard error of $\beta$ does not likewise get funky.