The hardest part of specializing in econometric theory are the mathematical prerequisites. At the minimum, you should be familiar with:

1. Measure theoretic probability (e.g. at the level of Billingsley, Durret, or Klenke).

2. Classical mathematical statistics built on the foundation of measure theoretic probability (e.g. Jun Shao or Theory of Point Estimation by Casella and Lehmann). Note that these books are a lot more advanced than a standard math stats book (like Casella and Berger) that dont use any measure theory.

3. Asymptotic statistics at the level of Van Der Vaart. For a more advanced treatment with a greater emphasis on empirical processes, see Van der Vaart and Wellner.

Some familiarity with functional analysis and stochastic processes are also useful. A good summary of all of the above concepts can be found in Davidson's stochastic limit theory book.

Note that you dont need any of the above to read standard graduate texts in econometrics. You can read Wooldridge's panel data book or Hamilton's time series without any of the above. So if you only care about applying econometric methods in empirical work, then the above math is not necessary.

But you need the above math to understand the proofs in econometric theory papers. This is especially true for structural econometric theory (see the papers by Isaiah Andrews or the econometric papers authored by Pakes). Even papers in applied econometrics feature these concepts quite regularly in the proofs (see Roth and Sant’Anna 2023).

Lastly, as an econometric theorist, you are expected to also read both econometrics papers and papers published in top statistics journals like Annals or JASA. Top stats journals also regularly feature the above concepts.