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Thread: Math prep for econ PhD

  1. #11
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    Re: Math prep for econ PhD

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    Quote Originally Posted by vulcan View Post
    A couple of the courses seem oddly similar to this one at LSE:
    MA212 Further Mathematical Methods

    You even use the same book!
    Quote Originally Posted by eBopBob View Post
    Similarly, the mathematical analysis appears very similar to:

    MA203 Real Analysis

    MA203 Real Analysis
    That's because LSE is the one providing the academic direction. That's why we're using the same books and following a very similar syllabus.

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    Re: Math prep for econ PhD

    The math require for economics is extensive and varied, so "sufficient" is difficult since you will always need to pick up some odds and ends on the way. That said, this looks like an excellent set of courses to prepare: calculus, linear algebra, and proofs/analysis are the core of graduate economics. A couple quick thoughts:

    First, it looks like "further calculus" is comparable to some "calculus 2" courses in the US, given its focus on series and integrals. Have you taken multivariate calculus previously? I would imagine linear algebra might require it, but if not, this is probably the most important missing element.

    Second, have you taken any probability/statistics or econometrics? If not, these would be the next most important missing elements.

    Finally, while often covered in economics classes, one area that is a bit more applied, but useful to either take a course on or do some self-studying with would be optimization. This is a lower priority, but definitely would be another area to consider working on a bit once higher priority items have been covered.

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    Re: Math prep for econ PhD

    Quote Originally Posted by win View Post
    The math require for economics is extensive and varied, so "sufficient" is difficult since you will always need to pick up some odds and ends on the way. That said, this looks like an excellent set of courses to prepare: calculus, linear algebra, and proofs/analysis are the core of graduate economics. A couple quick thoughts:

    First, it looks like "further calculus" is comparable to some "calculus 2" courses in the US, given its focus on series and integrals. Have you taken multivariate calculus previously? I would imagine linear algebra might require it, but if not, this is probably the most important missing element.

    Second, have you taken any probability/statistics or econometrics? If not, these would be the next most important missing elements.

    Finally, while often covered in economics classes, one area that is a bit more applied, but useful to either take a course on or do some self-studying with would be optimization. This is a lower priority, but definitely would be another area to consider working on a bit once higher priority items have been covered.
    The further calc course deals with mutlivariate calculus.

    I should be adequately prepared on the statistics & econometrics front since I have gone through the courses here and have gotten my As for them. A recent development in my masters programme allows me to switch over to an MRes and take PhD Econometrics (ref book: Hayashi 2000) so an A in that course will definitely help allay some concerns on that front. I'm currently auditing a metrics class using Wooldridge to prepare myself for next year.

    Duly noted on the part about optimisation. I'll be auditing a Mathematical Econs class that deals with optimisation, starting next week. The course synopsis is as follows:

    Constrained Optimisation
    Topics include: Definitions of a feasible set and of a solution, sufficient conditions for the existence of a solution, maximum value function, shadow prices, Lagrangian and Kuhn Tucker necessity and sufficiency theorems with applications in economics, for example General Equilibrium theory, Arrow-Debreu securities and arbitrage.

    Intertemporal optimisation
    Bellman approach. Euler equations. Stationary infinite horizon problems. Continuous time dynamic optimisation (optimal control). Applications, such as habit formation, Ramsey-Kass-Coopmans model, Tobin’s q, capital taxation in an open economy, are considered.

    Tools for optimal control: ODEs.

    These are studied in detail and include linear 2nd order equations, phase portraits, solving linear systems, steady states and their stability.

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    Re: Math prep for econ PhD

    Also, can anyone recommend a good book to purchase that covers the calculus aspect and will also be adequate for the abstract math/analysis course I'm taking?

    2) Further Calculus

    - Limit of a function of one variable, continuity.
    - Riemann integral, Fundamental Theorem of Calculus.
    - Improper Integrals, Test for convergence.
    - Double Integrals.
    - Dominated convergence.
    - Laplace Transforms.

    3) Abstract Mathematics

    - Mathematical statements, proof, logic and sets
    - Natural numbers and proof by induction
    - Functions and counting
    - Equivalence relations and integers
    - Divisibility and prime numbers
    - Congruence and modular arithmetic
    - Rational, real and complex numbers
    - Supremum and infimum
    - Sequences and limits
    - Limits of functions and continuity
    - Groups
    - Subgroups
    - Homomorphisms and Lagrange's Theorem


    4) Advanced Mathematical Analysis
    - series of real numbers
    - series and sequences in n-dimensional real space Rn
    - limits and continuity of functions mapping between Rn and Rm
    - differentiation (Maxima, minima and the derivative, Rolle's Theorem and Mean Value Theorem)
    - the topology of Rn
    - metric spaces
    - uniform convergence of sequences of functions.
    While I have numerous pdf-versions of the textbooks (like Apostol's, Spivak's, Tao, Rudin, etc), I strongly prefer a physical book for revision. Obviously, Rudin is far too advanced for me at this juncture. Since I'm financially constrained, can anyone recommend 1 book that covers most of the material in the 3 courses quoted above?

    Thanks a lot!

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    Re: Math prep for econ PhD

    If Rudin is too challenging at the moment I recommend you check out Introduction to Real Analysis by Robert G. Bartle and Donald R. Sherbert. I believe a pdf version is floating around on the internet. So you could have a chance to examine it before you purchase it. It's a lot wiser to start with a simpler intro analysis book and build the foundation then looking for a book that encompasses the most.

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    Re: Math prep for econ PhD

    Quote Originally Posted by Double Jump View Post
    If Rudin is too challenging at the moment I recommend you check out Introduction to Real Analysis by Robert G. Bartle and Donald R. Sherbert. I believe a pdf version is floating around on the internet. So you could have a chance to examine it before you purchase it. It's a lot wiser to start with a simpler intro analysis book and build the foundation then looking for a book that encompasses the most.
    I found the pdf version online. I'll delve deeper into it. Thanks a lot.

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    Re: Math prep for econ PhD

    Quote Originally Posted by tutonic View Post
    The further calc course deals with mutlivariate calculus.

    I should be adequately prepared on the statistics & econometrics front since I have gone through the courses here and have gotten my As for them. A recent development in my masters programme allows me to switch over to an MRes and take PhD Econometrics (ref book: Hayashi 2000) so an A in that course will definitely help allay some concerns on that front. I'm currently auditing a metrics class using Wooldridge to prepare myself for next year.

    Duly noted on the part about optimisation. I'll be auditing a Mathematical Econs class that deals with optimisation, starting next week. The course synopsis is as follows:

    Constrained Optimisation
    Topics include: Definitions of a feasible set and of a solution, sufficient conditions for the existence of a solution, maximum value function, shadow prices, Lagrangian and Kuhn Tucker necessity and sufficiency theorems with applications in economics, for example General Equilibrium theory, Arrow-Debreu securities and arbitrage.

    Intertemporal optimisation
    Bellman approach. Euler equations. Stationary infinite horizon problems. Continuous time dynamic optimisation (optimal control). Applications, such as habit formation, Ramsey-Kass-Coopmans model, Tobinís q, capital taxation in an open economy, are considered.

    Tools for optimal control: ODEs.

    These are studied in detail and include linear 2nd order equations, phase portraits, solving linear systems, steady states and their stability.
    In my opinion, given all the courses you've mentioned, you should be very well prepared. While there is always more you can learn, I would suggest focusing more on making sure to learn the topics in your classes deeply and thoroguhly as opposed to trying to cover more subjects. Even at the expense of other topics. For example, learning the material in these courses you initially listed should take priority over auditing an optimization course, which is a nice bonus but something that you can do with relative ease if you really understand the math.

    As for books suggestions, Rudin's Principles is the obvious choice, but I think most people's first encounter is a little intimidating given its sparing prose, lack of examples, and so on. I think once you are in class you will find that less the case, but I think two useful bridge books are Understanding Analysis by Abbott, which is a gentler introduction which I think is more accessibly written. It wont replace rudin, but it would probably be an easier introduction to the most important topics.

    Another one, which is not specifically analysis, but I think is very helpful is Thomas Sibley's Foundations of Mathematics. I have not come across an electronic version (if you do, let me know), but it is really a very accessible introduction to a lot of the topics that rudin either takes for granted or moves through quickly but are fundamental later on. For example, it spends a fair about of time explaining the basics of set theory, of how to write proofs, the foundations of functions and relations.

    Anyways, it sounds like you are well covered. Good luck with your courses.

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    Re: Math prep for econ PhD

    Quote Originally Posted by win View Post
    In my opinion, given all the courses you've mentioned, you should be very well prepared. While there is always more you can learn, I would suggest focusing more on making sure to learn the topics in your classes deeply and thoroguhly as opposed to trying to cover more subjects. Even at the expense of other topics. For example, learning the material in these courses you initially listed should take priority over auditing an optimization course, which is a nice bonus but something that you can do with relative ease if you really understand the math.

    As for books suggestions, Rudin's Principles is the obvious choice, but I think most people's first encounter is a little intimidating given its sparing prose, lack of examples, and so on. I think once you are in class you will find that less the case, but I think two useful bridge books are Understanding Analysis by Abbott, which is a gentler introduction which I think is more accessibly written. It wont replace rudin, but it would probably be an easier introduction to the most important topics.

    Another one, which is not specifically analysis, but I think is very helpful is Thomas Sibley's Foundations of Mathematics. I have not come across an electronic version (if you do, let me know), but it is really a very accessible introduction to a lot of the topics that rudin either takes for granted or moves through quickly but are fundamental later on. For example, it spends a fair about of time explaining the basics of set theory, of how to write proofs, the foundations of functions and relations.

    Anyways, it sounds like you are well covered. Good luck with your courses.
    I concur that Rudin is too advanced for me right now. To be quite frank, I can't make any sense of what I see in there. I'll look into Abbot. It was actually one of the shortlisted ones

    So you're suggesting I hold off on auditing the mathematical econs/optimisation course? The reason I am of the mind to do it is because the course is a relatively easy A. So another A to add to my collection will increase my overall profile, right?

    Another reason is that doing 'applied' math in the form of mathematical econs course will make the year much more pleasurable, as compared to doing just pure math courses. So, taking it will add a factor of fun, for lack of a better term.

    Lastly, with the mathematical econs course, I'll only be attending 9 classes a week. True, it's considerably higher than the usual 4 classes but it seems quite manageable.

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