Suppose I have an estimator $\hat{\beta}$ for some parameter $\beta_0$ and I have two hypothesis tests $$ H_0^{(1)}:\hat{\beta}=\beta_0\quad\text{versus}\quad H_1^{(1)}:\hat{\beta}\neq\beta_0 $$ and $$ H_0^{(2)}:\hat{\beta}\neq\beta_0\quad\text{versus}\quad H_1^{(2)}:\hat{\beta}=\beta_0. $$ Here the two tests might be different in their approach, but they still 'test' whether $\hat{\beta}$ equals $\beta_0$ with opposing null hypothesis. An example is the KPSS test and the Dickey-Fuller test for the autoregressive process $x_t=\rho x_{t-1}+\varepsilon_t$, in which the KPSS test has alternative hypothesis of a unit root ($\rho=1$) while the Dickey-Fuller test has null hypothesis of a unit root.
Question: Is it always possible to compare the the two tests with opposing null hypothesis? And if so, how would one compare their size and power?
