A question I'm qualified to answer. For context - I have a master's degree in Economics and made the break into a Data Science role a few years ago. The way I'd break down how you approach the job hunt is into three categories - Technical Skills, Data Science Skills, and Positioning.
Technical Skills - Most Data Science roles out there require knowledge of Python on SQL. R is rarely used in production settings so I'd be biased to ramping up on Python if you're not on that path already. I taught myself enough python to scrape by interviews using this specialization on Coursera specialization - https://www.coursera.org/specializations/data-science-python. Since you've already worked in STATA and R - it's just a matter of figuring out how the syntax works and this specialization is super hands-on so it gives you just what you need. I learnt SQL in a weekend - it sounds like an intimidating thing to do, but it's nothing you can't do if you got yourself through a masters degree in economics. Do all the tutorials on here (https://sqlzoo.net/wiki/SQL_Tutorial) and you can add to your resume in a week.
Data Science Skills - This is two-fold - ML algorithms and the ability to work smartly with data. I think the latter is a skill you already have if you are able to clean data and come up with a Fixed Effects model and make a thesis of it. In terms of ML algorithms - what I found most confidence-inspiring when I was trying to learn this stuff is starting from first principles. All the fancy buzz words in DS (Neural Networks, Regularization - you have it) can be mapped back to mathematical concepts that underpin a linear regression that you already have from Econometrics classes. For instance a Logistic Regression - the core building block of a Neural network for a classification task is the Probit and Logit models that you already have to have done in an Econometrics class. A lot of data scientists have no idea what they're doing from a mathematical standpoint when they are trying out DS algorithms and you have an advantage here. Long story short - find connections between what you already know and what is Machine Learning and learn all the buzz words. Trust me, you already know a lot of this stuff.
Positioning - As someone with a masters degree in Economics you already come with a lot of the skills required to be a good Data Scientist. You need to sell your skills in a way that is attractive to people in Industry. For instance - what most of Econometrics boils down to is finding smart ways to work with data and assumptions to arrive at the causal effect of an intervention on an outcome variable (Causal Inference). Take the fixed effects thesis for example - reframe it as working with observational data to arrive at a causal interpretation of an intervention. You need to speak industry language - not many people know what Fixed Effects is, so tell them what they want to hear. For example, you could sell yourself as someone who can work with observational data to prove causal interpretation of an intervention. Experimentation is another area where an Economics background is particularly helpful - talk about this. Natural experiments happen all the time and companies have data about it, talk about how you know what to do with a natural experiment. Econometrics + Machine Learning has a lot of research happening right now, Susan Athey at Stanford is at the forefront of this and her papers will give you good ideas as to how to blend the two disciplines.
Well, that was a long answer. As an Economist, you come with the right-thinking required to get a job in Data Science. Ramp up on technical skills - it should take you a month of dedicated studying. While prepping on the technical side start reaching out to people on LinkedIn - 'Economist + Data Science' in the search bar is a good place to start. In addition to online applications, talking to other people is a great way to get yourself a referral and jump the usual online application/rejection cycle (this is what worked for me in the end). As with most DS jobs you will learn most things on the job - so demonstrate a willingness in these conversations to want to ramp up on SWE skills which are required in the real world for sure
Hang in there - with a little more work and time I'm confident you'll be successful.