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Help Thread - Preparing for First Year Math and Stats

Background:

Apps are out, so I want to focus on preparing myself for the the first year of Econ programs, rather than stressing about which one I may or may not get into. For reference, I have been out of school for a couple of years working an industry job, and haven't had the opportunity to do much actual math since leaving undergrad. I'm worried that I will be behind at the start of programs if I don't refresh my math knowledge over the coming 7-8 months, even with a pre-program "boot camp".

Let's assume for exposition that I have forgotten essentially everything Math and Statistics-related, including even some topics from single-variable Calculus. Some topics require only quick review, others more extensive (e.g. I spent 30 minutes today remembering how to do implicit differentiation and getting myself back to "exam ready" for that topic, but probably would want a few weeks to go back through my Real Analysis course).

My current plan is to spend a small amount of time reviewing basic (single- and multi-variable) calculus, then moving on to a fuller review of more complex topics (bounded optimization, Rudin, ODE, etc.).
Questions for the forum:

What Math and Stat topics should I focus most on to prepare myself for the first year of Econ PhD?

How would you go about self-studying to "get back up to speed" on the topics from 1? What are your favorite textbooks? Post textbooks or online resources to check out (e.g. Khan Academy), study strategies, etc.

For Prob/Stats specifically, what is most important to review? I've been "reg monkeying" for 2+ years, so I'm confident on a lot of Metrics stuff, but have forgotten/grown rusty with more basic topics (I'm sure I would not do well on a "Combinations and Permutations" test, for example).

Please also feel free to riff off of these questions however you wish. Any advice is highly appreciated!

Thanks for any and all help, and best wishes for a stress-free application season!

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Re: Help Thread - Preparing for First Year Math and Stats

1. Mathematical tools which will be used in your first year classes include single/multivariable calculus, differential/difference equations, linear algebra, probability theory, mathematical statistics, a bit of analysis/topology, and proof techniques. Keep in mind that you do not necessarily need to learn ALL of this material before you enter your program, though you will be very well prepared if you do.

2. Flip through some books, maybe do some practice problems and create some outline sheets. Any calculus text will do, Sheldon Axler's linear algebra text can be found online (but perhaps it's a little advanced). Grinstead & Snell, Grimmet & Welsh, Wackerly et al. (Mathematical Statistics) are options for Probability & Statistics. Rosenlicht and Rudin are good for analysis.

3. Honestly I'd work through/skim Wackerly's Mathematical Statistics book, which covers both probability and statistics.

4. Simon & Blume is also a good resource that includes a lot of pertinent material, with more brief and concise treatments that you may be looking for.

Re: Help Thread - Preparing for First Year Math and Stats

All of this is super helpful! Thanks so much for taking the time to read and respond.

Follow up: for reviewing single/multivariable calculus, I get the feeling that the types of problems a lot of math texts spend lots of time on might not come up for PhD econ. One extreme example would be using trig substitutions to solve integrals; a more reasonable example would be memorizing all of the trig or hyperbolic derivatives again.

So, when reviewing Calc, do you think it would be important to catch every little detail? Or OK to simply have a solid grasp of the basics, acknowledging that I might need to go to WolframAlpha for a complex integral in the future?

If the more the latter than the former, is there a good Calc for Economists book that would skip over the unnecessary stuff? Or a resource somewhere that says what the "basics" are, for econ at least?

Re: Help Thread - Preparing for First Year Math and Stats

Sure, there may be some elements of Calculus which you may not use in your first year (e.g. trig substitutions).

Personally speaking, I just have a different perspective on the usefulness of math - I love pure math and think it's a beautiful discipline, so usefulness honestly does not mean a whole lot to me. But I do understand it means something in an Economics PhD program.

I would say just understanding how to take derivatives, integrals, chain rule, optimization (constrained / unconstrained), simple differential equations, and perhaps even indefinite integrals for statistical purposes will definitely pay off. You won't necessarily be using Stoke's/Green's/Divergence Theorems. You might use some stuff from infinite series in probability/statistics but I'm not too sure.

I guess a solid grasp of the essentials is most important. You can learn as you go, or use WolframAlpha for homework assignments. As for a book, maybe Simon & Blume, I'm not sure. I personally wouldn't want to learn Calculus from S&B but it may work better for you. There is a very concise appendix in MWG you might want to check out as well, but there are obviously things in that section which you will not have encountered before.