Math is an integral part of economics, especially so at the masters and particularly at the PhD level. Mathematical analysis is used in all areas of economics to construct and understand models and statistics is used in econometrics and empirical analysis, which is a ubiquitous in many areas of economics. Good familiarity with mathematical techniques and concepts is thus crucial for any student that wants to take economics seriously, particularly at the more advanced level. Students coming from a less mathematically enriched background oftentimes feel overwhelmed with the level of math they have to digest in a short period of time. That was the case of myself when I left the University of Nottingham to join Barcelona GSE for my masters. What seemed a mountain of a task, turned out to be not as hard as I had originally imagined and in the process I gathered a list of very helpful online references:

• Various universities post the notes of their respective math camp (the intensive mathematical preparation that takes place in the summer before PhD starts).
  • UC Berkeley Economics math camp lecture notes and lecture slides.
  • Yale Economics math camp lecture notes.
  • UC Santa Barbara Economics math camp lecture notes.
  • Also check out “An Introduction to Mathematical Reasoning” Part1, Part2, Part3, by Peter Eccles of the University of Manchester.
• For graduate economics I strongly recommend the mathematical appendix of “Microeconomic theory” by Mas-Colell, Whinston and Green which is an Econ-Math preparatory course in itself. Complement this with the appendices of Krishna’s “Auction theory” and Xavier Sala-i-Martin’s “Economic Growth“.
• For introduction to real analysis I recommend “Introduction to Real Analysis” by William Trench.
• For topology check out “Topology without tears” by Sidney Morris.
• For Dynamic programming “Introduction to Dynamic Programming Applied to Economics” by Paulo Brito.

Econometrics is the statistical branch of economists and as such it is perhaps the most useful toolset for any economist. It deals with the theoretical development and application of statistical methods to the analysis of observational data, oftentimes attempting to estimate causal relationships through mathematical modeling. Econometricians and applied economists are interested in issues of estimator unbiasedness, efficiency and consistency. 

Angrist and Pischke’s Mostly Harmless Econometrics is without a doubt a great resource for empirical work. Their second book, Mastering Metrics is intended for beginners / less mathematically oriented audiences and is also a great read.

If you’re interested in using R for applied econometrics, check out this guide here.

Program evaluation and causal inference is a an area of econometrics that deals specifically with identifying causal relationships between variables, sometimes through the design of randomized clinical trials, natural experiments or other observational datasets.

Introduction / Overview of Program Evaluation literature:

• For a great overview of the most widely used techniques in program evaluation check out the lecture notes by Alberto Abadie and Matias Cattaneo.
• Guido Imbens is one of the leading researchers in this area of econometrics. Take a look at his syllabus and class website from Duke. His book “Causal Inference for Statistics, Social, and Biomedical Sciences” is a classic in this literature.
• Manuel Arellano’s slides are another great resource. 
• A good introduction to the Rubin Causal Model. 
• NBER notes from Wooldridge and Imbens.

Randomized experiments:

• A very well written methods paper by two of the leaders in the field Guido Imbens and Susan Athey.
• Another nice overview paper by Paul Rosenbaum with great references.

Difference-in-Differences (DiD):

• A good introduction to the estimator here.
• Another Wooldridge/Imbens lecture note.

Regression Discontinuity Designs (RDD):

• Yet another Wooldridge/Imbens lecture note.
• A great and comprehensive “user guide” to RDD in economics. 
• Slides fort applying RDD in STATA
• R package for analyzing RDD.

Instrumental Variables (IV):

• An easy and intuitive way to think about IV in this blog post.
• A very nice overview paper by Angrist & Krueger.
• A somewhat more technical overview by Daniel McFadden.

Matching & Propensity Score Approaches:

• Using propensity score to estimate ATE slides.
• A very intuitive resource for applying some of these approaches in STATA are these lecture notes by Grotta and Bellocco.

Synthetic Control Methods:

• A good overview of Synthetic Control methods.
• Check out this very interesting article for an application.
• “Synth” package for R, Python and STATA.

Imputation methods for missing data:

• Gabriele Durrant’s notes cover a wide range of such methods. In these notes he also provides resources for implementing these methods on various statistical packages.
• Chapter 25 on Missing-data imputation from Andrew Gelman’s “Data Analysis Using Regression and Multilevel/Hierarchical Models” is available online.

Microeconomics deals with the theory of individual and group decision making and lays down the foundations of most other areas of economics. A standard course in microeconomics goes through the consumer demand theory, the theory of production, market structures like perfect competition, monopoly and enters into game theoretic situations of oligopolistic competition. Information economics, one of my areas of research is part of microeconomic theory. A great non-technical resource to get started is this online book recommended by MITOpenCourseWare. In that respect I highly recommend MITOpenCourseWare online economics courses. Also try this free online book. Andreu Mas Colell, Michael Whinston and Jerry Green’s “Microeconomic theory” is without a doubt the gold-standard in graduate microeconomics. I advise reading every page of that book, including the appendix! I also recommend supplementing that with Ariel Rubinstein’s “Lecture Notes in Microeconomic Theory“.

Game theory studies the strategic interactions between rational self-interested agents who compete to maximize some individual (or group) objective. It is a very useful tool to think about a variety of real-world situations ranging from firm competition, labor markets, military defense, auctions, insurance and many more. For beginners I would recommend Gibbon’s ”A primer in game theory”. It’s easy to read and introduces the most important concepts in game theory. For more advanced students, an immensely useful resource is Ariel Rubinstein’s web-page where you can download his books and other resources. Another good book is Roger Myerson’s “Game Theory: Analysis of Conflict“. 

Game theory has many branches. Evolutionary game theory takes insights from evolutionary biology and studies the equilibria that may arise when interactions are modeled in a dynamic and usually infinite-horizon setting. It involves features like fitness, generational survival and mutation. Behavioral game theory is the branch of game theory which uses psychological explanations to rationalize new equilibrium behavior outcomes of various strategic interactions. Perhaps the most well-known resource in this field is Colin Camerer’s “Behavioral Game Theory: Experiments in Strategic Interaction“. Finally algorithmic game theory, the intersection of computer science and economics, takes a more computational perspective on the solution of various games, oftentimes coming up with solutions that are either easier to describe or that where impossible to get with standard optimization techniques. Check out this “Algorithmic game theory” by Nisan, Tardos, Roughgarden and Vazirani and the book “Multiagent Systems” by Kevin Leyton-Brown and Yoav Shoham. I also recommend checking out their online courses on coursera.org here and here. Read this article for how game theory and AI were combined to tackle real world problems.

Behavioral economics is an interdisciplinary field of research in economics which incorporates insights from psychology (and sometimes sociology) to help explain economic behavior that is not well captured by the rational paradigm of standard economic models. Experimental economics is a methodology in which experimental subjects are invited to participate in studies conducted in a controlled laboratory setting usually through computer terminals. Participants are incentivized for their performance usually using money. This methodology is usually employed to extract behavioral insights and for that reason the two field are often thought of as one. Behavioral and experimental economics is my main field of research. It is my long term goal to one day write my own book about behavioral economics. For now please see my behavioral economics resources page here.

Computational economics is an interdisciplinary research field across economics, computer science and management science. Computational economics uses computational methods to solve complex problems in economics and management. Agent-based computational economics (ACE) is a sub-discipline of computational economics . Read this paper for a great introduction and overview of the subject.

• Vincent Conitzer of Duke provides the materials of two classes on computational economics online here and here.
• I suggest checking out the work done by the Santa Fe Institute in understanding complex systems. If you want to get started with agent-based modeling I highly recommend their online course here, which I found very interesting.
• Thomas Sargent and John Stachurski ‘s website presenting their lectures on quantitative economic modeling on various topics includes a host of useful resources.
• Also check out some of my own agent-based models here.

Macroeconomics looks the evolution of the economy as a whole both in the long term and in short term fluctuations, examining trends and relationships between macroeconomic variables like economic growth, unemployment, inflation, interest rates and others. International trade, a subset of macroeconomics also examines the effects of trade and the links between economies. More traditional development economics models examining the determinants of growth in the long run have their roots in macroeconomics. Various macroeconomic models are based on the aggregation of individual decision-making models developed in microeconomics. Find below some freely available resources for macroeconomics:

• For an easy-to-follow introduction to macroeconomics I recommend Harvard Business School’s online course here.
• One of the most important books in growth by Robert Barro and Xavier Sala-i-Martin “Economic Growth” is available through Thomas Piketty’s website where you will find a host of useful resources on the topic.
• A useful introduction to the Neoclassical Growth Model is available from Stanford Online
• A classic macroeconomics book for the graduate level (Lars Ljungqvist and Thomas Sargent “Recursive Macroeconomic Theory“) is available through NYU Stern’s website.