I would build a bivariate response model simultaneously for $Y_{i1}$ and $Y_{i2}$, introducing an underlying propensity $Y^*_{i1}$, $Y^*_{i2}$ such that $$Y_{i1}=\min(a,Y^*_{i1}), Y_{i2} = \max(b,Y^*_{i2}).$$ If $X_i$ is a treatment variable, then a version of your model is $$Y^*_{i2} = Y^*_{i1} + \beta_0 + \beta_1 X_i + \epsilon_i.$$
I think you use Stata, so you would want to look at cmp package that fits models of this kind. There might be some caveats, as Roodman talks about "fully observed" models there, in the sense that nothing depends on the latent variables $Y^*_1, Y^*_2$. You might have to change somewhat the model to make sure it is estimable.