How do I model a paired experiment with pre-post data? I have designed an experiment with 20K treated units matched to 20K control units. Every treated unit is matched to a control unit that is as similar as possible on a number of metrics.
The treated units are subjected to a treatment for 4 weeks, while the control units are untouched. I measure $Y_\text{post}$, the dependent variable, at the end of the experiment. I also measure $Y_\text{pre}$, the same dependent variable, measured over the 4 weeks preceding the experiment.
How do I take into account the simultaneous pre-post and paired (or matched) experiment design?
From some searching I've come up with the following model in lme4 syntax:
lmer(Y_post ~ arm + offset(Y_pre) + (1 | pair_id), df)

where arm indicates if the unit was in the control or treated group, and pair_id is a unique identifier for each pair of units.
Is this approach correct? Will the fixed coefficient for arm correctly represent the treatment effect?

The model proposed is not incorrect but it can be improved.
I would suggest including the matching variables in the regression model too. The main reasons are two: 1. we might have a poor balance on the other covariates, even if the exact matching variables are perfectly balanced and 2.  some of the covariates might have substantial effects on the outcome too. The inclusion of these covariates, if anything, will allow us to reduce the standard error of the final estimate $\beta$ estimate. Yes, we will lose some degrees of freedom but with ~20K pairs that's should not be a huge concern. If you haven't seen it already I would suggest a quick read of Stuart (2010) "Matching Methods for Causal Inference: A Review and a Look Forward", Section 5. "ANALYSIS OF THE OUTCOME". The exceptional vignette of the MatchIt R package "Estimating Effects After Matching" written by Greifer should be you some greater pointer too.
That said, be aware of the "Table 2 Fallacy" as interpreting the $\beta$ coefficients of these matching variables is problematic because these additional variables are still subject to confounding; "The Table 2 Fallacy: Presenting and Interpreting Confounder and Modifier Coefficients" (2013) by Westreich & Greenland cover this in detail.
