Catrina Posted March 30, 2014 Share Posted March 30, 2014 (edited) Before I start, I would like to say that I am very curious to hear the opinions of others, particularly current graduate students, on this issue. I have debated for a while about whether to post this given that I am setting myself up for a lot of criticism, and if I am wrong I run the risk of misleading people, but given a that this has come up in several recent discussions, I think that this is a point that should be made. Mathematics is a language that is used widely in economics, and it is therefore crucial that economists have a solid understanding of mathematical concepts and familiarity with the language of mathematics. In particular, I think that is crucial for everyone to have a solid understanding of calculus and linear algebra, and exposure to proof-based coursework. I would go further and say that, at least for top-50 applicants, everyone should take a real analysis course, both to gain familiarity with the concepts that will be used in first-year coursework and for signalling purposes. However, while I am aware that Mankiw advises students to take as many math classes as they can stomach (Greg Mankiw's Blog: Advice for Aspiring Economists), I'm not convinced this is advisable as a general rule. I have seen far too many cases here, including my own, of people attempting to take too many upper-level math classes in a short period of time in order to fit in "as much math as they can stomach", or simply taking a level of math beyond which they are capable of doing well at. Attempting to take a level of math far beyond what is required for a PhD program or expected of incoming students has significant costs, and as far as I can tell, little benefit. Personally, I really enjoy learning math so I did get some consumption benefit from taking courses like functional analysis, but my admissions would have likely been better if I had stopped earlier. For those who don't like math that much, getting a masters in math, or even just taking tons of upper-level undergraduate math, in an attempt to get into a PhD program sounds like a really terrible idea. Based on my own thoughts and numerous discussions that I have had with various people, here are my thoughts on the optimal amount of math preparation: 1. If you want to get into a top-5, maybe even top-10, program and have a realistic chance of doing so, maybe you really should take as much math as you can stomach. 2. If you want to be a pure theorist, you really should take as much math as you can stomach. Applied theorists can get away with somewhat less, particularly at lower ranked schools. I met a number of applied theorists at visit days with only a background in real analysis (sometimes even self-taught), but I would still think that it would be much better to take math beyond real analysis I for any type of theory. 3. If you don't want to do theory, and aren't aiming for the top-20, I don't see any real reason to take math much beyond real analysis, unless perhaps you have been doing really well at it and really enjoy it. The typical Maryland student seems to have taken one or two semesters of RA. At the schools that I visited, quite a number of students hadn't taken RA at all. I would still recommend taking RA given its use in economics and signalling value, but it clearly isn't an admissions requirement in the top-50. The amount of math that I had taken was far more than that of any other applicant or grad student that I met at visit days who wasn't studying theory. 4. If you haven't done well in math, taking a bunch of upper-level courses or getting a math or stat masters in an attempt to correct your profile is a really bad idea. I have seen several profile evaluations this week from people attempting to do this. As someone who has taken graduate math, I can say that the graduate math was in a totally different league than even upper-level undergraduate math. If you didn't do well in basic undergraduate math, you will almost certainly do worse in graduate math. Instead, if you are in that situation, consider taking one or two courses at the same level or a slightly higher level than those that you did badly in. If you got a B in real analysis 1, trying to take, and getting an A in, RA 2 is probably a good idea, while applying to a math masters or signing up for graduate math probably isn't. What does everyone else think? Edited March 30, 2014 by Catrina 1 Quote Link to comment Share on other sites More sharing options...
huaxue09 Posted March 30, 2014 Share Posted March 30, 2014 Really well said! UC Irvine has an option to get an additional MS in Stat in their PhD program, and I felt that's really good and probably would do the similar thing myself. Quote Link to comment Share on other sites More sharing options...
PhDPlease Posted March 30, 2014 Share Posted March 30, 2014 I think one thing to add to this is that it might depend where you went to undergrad. For example someone who went to MIT and took math through 1 semester of Analysis at MIT might be considered a very strong candidate (as the admissions staff would be confident that the math classes at MIT were sufficiently rigorous). I think that sometimes for someone who went to a school that is not considered especially mathematically rigorous, a few extra math classes can help out if of course one does well in them (but of course, you are taking a risk if you aren't confident you'll do well). I recall a few years back that there was someone on this forum who took tons of extra math at a non-top school earning A's and got into very top programs. Otherwise I agree with many of your points. Quote Link to comment Share on other sites More sharing options...
Catrina Posted March 30, 2014 Author Share Posted March 30, 2014 That is a good point. If one is coming from a weaker undergrad and is aiming for a top-20 or maybe even top-30, taking extra math can help. If you are aiming for a top-50, it would still probably be overkill. Quote Link to comment Share on other sites More sharing options...
Econhead Posted March 30, 2014 Share Posted March 30, 2014 I agree with what you have indicated. Although Q GRE is well correlated with GPA and rank, some individuals suffer on such tests for several reasons. Given a particular placement from your post above, a candidate who receives a lower Q GRE score would likely benefit from taking more math classes (assuming earning A's) than an otherwise identical candidate whom earned a nearly perfect Q GRE. That is, I agree that more math often will not help, but at the margin with all else equal, RA2, FA, or Topology could make up for a somewhat lower Q GRE. (I am by no means emphasizing this will make up for a 150.) Quote Link to comment Share on other sites More sharing options...
superduper Posted March 30, 2014 Share Posted March 30, 2014 Catrina and I followed similar paths until last semester, so I know where she's coming from and I can confirm and second most of what she said. I'm not a big fan of the term "Real Analysis", because it means very different things at different schools. In terms of the math necessary to get you through the first year without major problems, I can say with 95% confidence that Baby Rudin is enough. However, when I say it's enough, I mean knowing "every proof of every chapter" and being able to do most of the exercises. Being familiar with it is not enough. The other thing that is really important, and this only comes with time and practice, is rigor. I once had an micro theory RA who spoke really bad English lecture me on the importance of writing rigorous proofs. In order to become a good researcher, you MUST be consistent and rigorous in your work. And one of the reasons people make such a big deal of Real Analysis is that a good course in the subject will teach you how to be consistent and rigorous. Unfortunately for me, the first two courses I took in analysis were not "good", and I realized it the hard way later on. I agree with what she said about the risk of taking master's courses but I'd like to point out that a master's in math and a master's in stat are not the same thing. I'm pretty confident that a master's in stat is not as bad as in math, since I'm in a stat program, but it may be just as useful. I know for a fact that I'm not smart enough to take graduate level math, but adcoms don't know that and all they'll see in a couple of years is a lot of good grades in mathematical statistics and applied stat courses. The other thing is that if you do poorly in a master's in stat then you will know for sure you won't be a good economist (if you ever manage to get a PhD in econ) but you'll still have an extremely valuable degree in statistics which can guarantee you a good job somewhere in the blue marble. My advice for someone with a poor undergrad record is that if you have taken probability and stat and done reasonably well, then maybe getting a master's in stat before applying for a PhD in econ may be a good option for you. The worst thing that can happened is landing an 80k job at some startup and becoming a millionaire when it's sold to Facebook a couple of years later (and this would happen at least 5 years before getting tenure at an econ department). Quote Link to comment Share on other sites More sharing options...
Catrina Posted March 30, 2014 Author Share Posted March 30, 2014 I agree with what you have indicated. Although Q GRE is well correlated with GPA and rank, some individuals suffer on such tests for several reasons. Given a particular placement from your post above, a candidate who receives a lower Q GRE score would likely benefit from taking more math classes (assuming earning A's) than an otherwise identical candidate whom earned a nearly perfect Q GRE. That is, I agree that more math often will not help, but at the margin with all else equal, RA2, FA, or Topology could make up for a somewhat lower Q GRE. (I am by no means emphasizing this will make up for a 150.) I would think that for the vast majority of people, improving their GRE scores would be considerably easier than taking something like functional analysis or topology. Quote Link to comment Share on other sites More sharing options...
Econhead Posted March 30, 2014 Share Posted March 30, 2014 I agree, I have just net some very smart people that suck at standardized tests (test anxiety). I just thought it might be worth mentioning for conpleteness. Quote Link to comment Share on other sites More sharing options...
Catrina Posted March 30, 2014 Author Share Posted March 30, 2014 Catrina and I followed similar paths until last semester, so I know where she's coming from and I can confirm and second most of what she said. I'm not a big fan of the term "Real Analysis", because it means very different things at different schools. In terms of the math necessary to get you through the first year without major problems, I can say with 95% confidence that Baby Rudin is enough. However, when I say it's enough, I mean knowing "every proof of every chapter" and being able to do most of the exercises. Being familiar with it is not enough. The other thing that is really important, and this only comes with time and practice, is rigor. I once had an micro theory RA who spoke really bad English lecture me on the importance of writing rigorous proofs. In order to become a good researcher, you MUST be consistent and rigorous in your work. And one of the reasons people make such a big deal of Real Analysis is that a good course in the subject will teach you how to be consistent and rigorous. Unfortunately for me, the first two courses I took in analysis were not "good", and I realized it the hard way later on. I agree with what she said about the risk of taking master's courses but I'd like to point out that a master's in math and a master's in stat are not the same thing. I'm pretty confident that a master's in stat is not as bad as in math, since I'm in a stat program, but it may be just as useful. I know for a fact that I'm not smart enough to take graduate level math, but adcoms don't know that and all they'll see in a couple of years is a lot of good grades in mathematical statistics and applied stat courses. The other thing is that if you do poorly in a master's in stat then you will know for sure you won't be a good economist (if you ever manage to get a PhD in econ) but you'll still have an extremely valuable degree in statistics which can guarantee you a good job somewhere in the blue marble. My advice for someone with a poor undergrad record is that if you have taken probability and stat and done reasonably well, then maybe getting a master's in stat before applying for a PhD in econ may be a good option for you. The worst thing that can happened is landing an 80k job at some startup and becoming a millionaire when it's sold to Facebook a couple of years later (and this would happen at least 5 years before getting tenure at an econ department). I do agree that one should carefully select the best real analysis professor/course available. As you (Superduper) know, I had two semesters of RA with an excellent professor and that definitely helped. I never realized that the statistics graduate courses were that much easier than the mathematics ones. I had heard that the theory sequence in your stat masters program was quite difficult, but maybe the term "difficult" is relative. I don't agree with your statement that " if you do poorly in a master's in stat then you will know for sure you won't be a good economist." I don't think that being able to do well in a measure-theory based probability course is necessary in order to do good applied research. Quote Link to comment Share on other sites More sharing options...
setsanto Posted March 30, 2014 Share Posted March 30, 2014 I'm not a big fan of the term "Real Analysis", because it means very different things at different schools. In terms of the math necessary to get you through the first year without major problems, I can say with 95% confidence that Baby Rudin is enough. However, when I say it's enough, I mean knowing "every proof of every chapter" and being able to do most of the exercises. Being familiar with it is not enough. I'm going to have to disagree strongly with this part. All I have taken thus far in terms of proof-writing is Real Analysis I at my not particularly rigorous undergraduate institution. The class covered about Chapters 1-4 of Baby Rudin. I did not perform incredibly well in the class, barely scraping by with an A-. Nonetheless I have been able to follow along quite well with PhD Micro II at that same institution. Granted, my school is only ranked around the level of Arizona or GWU, but it is still a PhD class. I have no doubt that being intimately familiar with every chapter in Baby Rudin is very important for very highly ranked schools, but for the VAST majority of applicants I would say that such a requirement is too restrictive. Quote Link to comment Share on other sites More sharing options...
coffeehouse Posted March 30, 2014 Share Posted March 30, 2014 Some reason I don't believe anyone needs to know 95% of Baby Rudin. I have a friends in pure math (going on to PhD in Mathematics at top 10) who probably does not know 95% of the proofs in Baby Rudin. I have attended visit days, and the graduate students have told me that in some top 10 schools, analysis is not required beforehand, you will learn it in Math Camp. I doubt anyone learns 95% of Baby Rudin in a few weeks of Math Camp. Quote Link to comment Share on other sites More sharing options...
Catrina Posted March 30, 2014 Author Share Posted March 30, 2014 I also agree that the 95% of baby Rudin part is far too high. While I do think that is a great exaggeration, I think that Superduper's general point that the rigor of the real analysis class taken matters. I don't think that the amount of chapters covered matters that much, but rather the level of rigor at which it is taught. Quote Link to comment Share on other sites More sharing options...
Catrina Posted March 30, 2014 Author Share Posted March 30, 2014 Some reason I don't believe anyone needs to know 95% of Baby Rudin. I have a friends in pure math (going on to PhD in Mathematics at top 10) who probably does not know 95% of the proofs in Baby Rudin. I have attended visit days, and the graduate students have told me that in some top 10 schools, analysis is not required beforehand, you will learn it in Math Camp. I doubt anyone learns 95% of Baby Rudin in a few weeks of Math Camp. I attended Maryland's math camp, and the specific concepts from RA that were needed were covered. However, math camp is intended to be a review, and I would think that anyone without a decent proof-based course like RA would have struggled in the coursework. IMO, the key here isn't the specific concepts as much as familiarity with proofs, although exposure to the concepts in RA before math camp would be very, very helpful. Quote Link to comment Share on other sites More sharing options...
Banach Tarsky Posted March 30, 2014 Share Posted March 30, 2014 it is not about "knowing" the proofs, it is about understanding techniques and ideas. I think maths is not required, Mankiw's point stands (a bit less politically correctly paraphrased) maths courses are simply a form of long intelligence tests Quote Link to comment Share on other sites More sharing options...
superduper Posted March 30, 2014 Share Posted March 30, 2014 I have to stop with my tendency to exaggerate my statements to make a point. I never realized that the statistics graduate courses were that much easier than the mathematics ones. They are not. But they are different things. A hard stat course is not the same as a hard math course, that's what I meant. I don't think that being able to do well in a measure-theory based probability course is necessary in order to do good applied research. I didn't say that. Most of a stat master's is applied courses, and what I meant to say is that if you can't do well in those courses then I really doubt you can do good research. Apparently I also have to be more rigorous in my writing lol. Quote Link to comment Share on other sites More sharing options...
Catrina Posted March 30, 2014 Author Share Posted March 30, 2014 I have to stop with my tendency to exaggerate my statements to make a point. They are not. But they are different things. A hard stat course is not the same as a hard math course, that's what I meant. I didn't say that. Most of a stat master's is applied courses, and what I meant to say is that if you can't do well in those courses then I really doubt you can do good research. Apparently I also have to be more rigorous in my writing lol. Okay, that makes sense. I would agree that someone who couldn't do well in the applied stats courses would be less likely to be a good applied researcher. However, because the stat masters also involves theoretical courses, someone choosing to go that route runs the risk of doing poorly in one of the theoretical classes and therefore getting rejected from PhD programs that they may have gotten into otherwise. Not only that, someone who did poorly in undergraduate math would be likely to do poorly in such a course, and may well be better off not attempting that route and instead applying to lower ranked economics PhD programs. Quote Link to comment Share on other sites More sharing options...
superduper Posted March 30, 2014 Share Posted March 30, 2014 Some reason I don't believe anyone needs to know 95% of Baby Rudin. I have a friends in pure math (going on to PhD in Mathematics at top 10) who probably does not know 95% of the proofs in Baby Rudin. I didn't say you have to know 95% of Baby Rudin. I said that Baby Rudin was enough and what I meant by it was that you don't need topology, functional analysis etc. Catrina said Real Analysis was enough and I just wanted to set a more clear standard for what Real Analysis could mean. Also, I thought the quote-unquote 'know every proof of every chapter' was an obvious exaggeration, but I guess I'm wrong. I do think being only familiar with it won't be enough in higher ranked institutions. This opinion comes from taking micro at a top-25 with a crazy theorist and seeing half the class struggle with it. Quote Link to comment Share on other sites More sharing options...
superduper Posted March 30, 2014 Share Posted March 30, 2014 However, because the stat masters also involves theoretical courses, someone choosing to go that route runs the risk of doing poorly in one of the theoretical classes and therefore getting rejected from PhD programs that they may have gotten into otherwise. Not only that, someone who did poorly in undergraduate math would be likely to do poorly in such a course, and may well be better off not attempting that route and instead applying to lower ranked economics PhD programs. Exactly!! And my point was that if you go that route and end up performing poorly in those courses and get rejected everywhere, at least you have a master's and can go to industry and run lots of regressions. Going to a low ranked PhD program is probably lead you to that route anyways. Quote Link to comment Share on other sites More sharing options...
PhDPlease Posted March 30, 2014 Share Posted March 30, 2014 I would think that for the vast majority of people, improving their GRE scores would be considerably easier than taking something like functional analysis or topology. I agree. In the upper-level math classes I took, the final exam determined the majority of your grade. Also there were usually people who knew the material but couldn't work as quickly as others who ended up not finishing the exam and being upset that they hadn't done well despite understanding the material. So I'm not really sure that someone who didn't do well on the GRE due to test anxiety or not working well under time pressure would have done any better in these classes (since they also involved a high-stakes test determining most of your grade and had a number of people who couldn't finish in the allotted time despite understanding the material). Of course I am sure there are always exceptions and this might not be true of every person/class. Quote Link to comment Share on other sites More sharing options...
Catrina Posted March 30, 2014 Author Share Posted March 30, 2014 Exactly!! And my point was that if you go that route and end up performing poorly in those courses and get rejected everywhere, at least you have a master's and can go to industry and run lots of regressions. Going to a low ranked PhD program is probably lead you to that route anyways. I guess it depends on your interests. If you really like economics, you would probably prefer going to a low-ranked PhD where you can do applied research and teach at a regional state school/LAC than working in industry running the regressions that they tell you to run. Quote Link to comment Share on other sites More sharing options...
PhDPlease Posted March 30, 2014 Share Posted March 30, 2014 Also agreed that whether or not RA1 is sufficient depends on the particular course and the particular individual. In my experience there are some math classes that are taught very poorly so the class average on the exam ends up being very low, and then a huge curve is applied, so it's possible that you ended up with an okay grade due to the curve but didn't actually master the material. There are also classes where the professor puts questions on the exam that are extremely similar to the homework questions so people can remember how to solve questions from the homeworks but not really develop a comfort in solving new questions (that are not extremely similar to the homework questions) but still get a good grade in the class. In the two previous examples someone might receive an acceptable grade for admissions purposes but not have developed adequate comfort with proofs to be really prepared for the program (again depending on the individual's ability... for one person a single mediocrely-taught proofs course might be sufficient preparation but for someone else who learns a little slower several better-taught proofs courses might be beneficial in terms of actually succeeding in the program). I think something to think about in terms of whether you are personally prepared is whether if you see a proof from Analysis that you haven't studied before, if you read through the relevant part of the text and steps to the proof, can you understand it within a reasonable amount of time? If so, I think that is a good sign, as the ability to go through theoretical material you haven't seen before and understand it within a reasonable amount of time (given that the classes will move relatively quickly) is important. Quote Link to comment Share on other sites More sharing options...
mcsokrates Posted March 30, 2014 Share Posted March 30, 2014 There are three (recursive) problems potential grad students face: getting into a program, passing the qualifying exams, and doing well on the job market. The optimal level of math is different for each problem (especially if you end up doing primarily empirical work). Unless doing math problems gives you unreasonably large consumption value, the best strategy is to take the minimum amount of math that will solve the three problems. For most people, that's the standard calc sequence, linear algebra, RA, and a fair number of stats classes. Basically, you really only need enough to get through the first year and to be able to understand the proofs in papers you might read. Quote Link to comment Share on other sites More sharing options...
Catrina Posted March 31, 2014 Author Share Posted March 31, 2014 There are three (recursive) problems potential grad students face: getting into a program, passing the qualifying exams, and doing well on the job market. The optimal level of math is different for each problem (especially if you end up doing primarily empirical work). Unless doing math problems gives you unreasonably large consumption value, the best strategy is to take the minimum amount of math that will solve the three problems. For most people, that's the standard calc sequence, linear algebra, RA, and a fair number of stats classes. Basically, you really only need enough to get through the first year and to be able to understand the proofs in papers you might read. That is exactly the point that I wanted to make, and you made it in a much more clear and concise manner than I did. I think the issue here is that many people miscalculate the amount of math needed for steps one and two (getting into the program, passing comps). This is complicated by the fact that the math required for both varies by considerably be school. For example, the micro sequence appears to be quite different at two of schools that I visited, both in the same range of the US News rankings. At one of the two schools, current students told me that problem sets and exams are primarily proofs, while students at the other school said that problem sets and exams have very few proofs and that real analysis really isn't necessary. (If anyone needs to know which school is which, you can PM me). Therefore, to be safe, I would recommend that a student applying to the top-50 have real analysis to prepare for first year coursework, even if it isn't necessarily required for admission. However, it seems pretty clear to me that, at least outside the top-20, more math isn't usually required for either #1 or #2, and is only required for #3 if you want to be a theorist. However, if someone is really strong at math and has other deficiencies, more math may help with #1 too. Quote Link to comment Share on other sites More sharing options...
yankeefan Posted March 31, 2014 Share Posted March 31, 2014 The unfortunate thing is that two weeks from now when this thread slips to the second page, someone is going to start a new thread that says "Need help deciding which math classes to take". Quote Link to comment Share on other sites More sharing options...
Catrina Posted March 31, 2014 Author Share Posted March 31, 2014 The unfortunate thing is that two weeks from now when this thread slips to the second page, someone is going to start a new thread that says "Need help deciding which math classes to take". Worse yet, there will be more people with a bunch of Bs in math asking if they should do a math masters. My reasoning for creating this thread is to stop people from repeating my mistake, and I just hope that this thread doesn't fade away too quickly (or that people will do a search). I can say with near certainty that I would have gotten better admissions results if I hadn't overloaded my schedule with math. Quote Link to comment Share on other sites More sharing options...
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