model design - fixed effects model for paired differences
The constant term in your final FE model has no specific meaning without further restrictions. For Stata, it is only chosen such that the (sample) mean of the estimated individual effects add up to 0. So your testing $\alpha&gt;0$ is in fact based on
$$\frac{1}{n} \sum_{i=1}^n \hat\alpha_i,\quad \hat\alpha_i = \frac{1}{T_i} \sum_{t=1}^{T_i} (x_{it} - y_{it}),$$
where I am using the $T_i$ notation in order to allow for the panel data to be unbalanced.

For balanced panel data, the test is (almost) identical to one based on OLS if you use a cluster standard error for OLS because then
$$\frac{1}{n} \sum_{i=1}^n \hat\alpha_i = \frac{1}{n} \sum_{i=1}^n \frac{1}{T} \sum_{t=1}^T (x_{it}-y_{it}) = \frac{1}{nT} \sum_{i=1}^n \sum_{t=1}^T (x_{it} - y_{it}),$$
where the LHS is a statistic from the FE regression and the RHS is the corresponding one from the OLS regression. OLS, FE and RE (and BE) will all give an identical intercept estimate, I think. Only standard errors will be different. If the panel data set is unbalanced, the method using the FE regression and that using the OLS regression differ in how the average differences ($\hat\alpha_i$) are weighted across different $i$.

The test you mention is about whether the mean of $x_{it}-y_{it}$ is bigger than 0 or not. This looks OK. What else? Which technique to use (OLS or FE or RE or BE) doesn't seem to be critical here. I would just use OLS + cluster se, as it is simple to both the researcher and the reader. Try:

xtset id year
gen d = x - y
reg d, vce(cl id)
xtreg d, fe
xtreg d, re
xtreg d, be