If you're interested in econometrics, I highly recommend checking out Marc Bellemare's "Metrics Mondays" blog posts, which are full of useful, pragmatic advice for applying econometric methods to real-world data: http://marcfbellemare.com/wordpress/metrics-mondays.
If you're coming from an ML-focused approach to statistics, studying econometrics can be an interesting change of pace, because the focus is totally different. ML practicioners tend to be focused on prediction, while econometricians tend to focus on causal inference - utilizing pseudo-experimental variation within the data to estimate causal effects between variables. This turns out to be really hard to do correctly, and learning the pitfalls can make it easy to identify potential weaknesses in other research.
Most econometric work has historically been done in Stata, although it seems like both R and Python have been increasing in prominence a bit recently.
When I was in school around 2010 or so, a lot of the younger econ grad students were primarily interested in R. I don't think Stata's going away any time soon, but it might not be completely dominant for that much longer.
A lot of people I know at various departments are switching their undergrad stats/econometrics classes from Stata to R. That's the beginning of the end of Stata.
That matters, but I don't think that's happening until all of the big graduate-level metrics textbooks get R versions. And even then, at least a few papers are going to run into trouble with older reviewers who are used to seeing work done in Stata and don't trust anything else.
Yes, and it's also non-trivial to write R code that matches your textbook's answer if your textbook used Stata. You have to do things like look up which specific variant of the sandwich estimator Stata uses for robust standard errors, so you can tell R to match that.
In Stata's defense: It helps that Stata is actually really good for the "running regressions" part. In particular, it gets robust standard errors right without much extra work in complex cases that would require a lot of additional code in Python or R.
R wins easily for data visualization and scripting, though. It's also much better as a skill you can "take with you". If you end up working in industry, you may not be able to expense a Stata license, but you'll almost certainly be able to use R (although maybe not RStudio).
You don't even want to expense a Stata license. Stata is the worst thing I've ever had to use. Maybe as a person who can't program it makes sense, but as a professional developer almost everything about Stata is non-intuitive, confusing, and stupid. Also the only thing to go on is their stupid pdf manual. Finding real people on the internet who actually use it is almost impossible.
If you're coming from an ML-focused approach to statistics, studying econometrics can be an interesting change of pace, because the focus is totally different. ML practicioners tend to be focused on prediction, while econometricians tend to focus on causal inference - utilizing pseudo-experimental variation within the data to estimate causal effects between variables. This turns out to be really hard to do correctly, and learning the pitfalls can make it easy to identify potential weaknesses in other research.
Most econometric work has historically been done in Stata, although it seems like both R and Python have been increasing in prominence a bit recently.