I am attempting to read around the theory of probability theory from the ground up, coming from a background of economics I have little experience in set/measure theory, whilst I am not new to statistics and econometrics the rigorous treatment of the topics have been difficult to follow. For reference, I have been using 'Probability, Statistics, and Econometrics - Oliver Linton' to read on the subject. What would be the best way to approach this? I ask this question as this has been recommended in preparation for my MSc in Economics by a lecturer.
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4$\begingroup$ The contents of that book suggests that you may not want much of a measure-theory basis beyond reading section 3.1, wondering what and why they are talking about sigma-algebras and Borel sets, and then promptly ignoring it for the rest of the course; something similar may happen with Lebesgue measure. Set theory requirements may not extend much beyond Venn diagrams, intersections, unions and complements (hence Borel sets). In terms of difficulty, I would have thought the different forms of convergence may be the hardest to get your head round, and important for asymptotic results. $\endgroup$– HenryJun 30, 2020 at 20:26
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$\begingroup$ Have you head some course on proof-based real analysis? $\endgroup$– Michael GreineckerJun 30, 2020 at 21:14
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$\begingroup$ I have not had any experience in real analysis, however, having glanced through another set of required reading there is a section that introduces real analysis and has some more advanced topics. I plan to go through this first before attempting to tackle probability theory. $\endgroup$– user23720Jun 30, 2020 at 22:05
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$\begingroup$ @Henry I think that approach seems more reasonable - I've just been thrown off a bit with the terminology given that before today I had not heard of this in any probability theory, I will likely go back to my mathematics reading before tackling the probability theory material. $\endgroup$– user23720Jun 30, 2020 at 22:08
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2 Answers
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If you want a good book with emphasis on rigorous then An Introduction to Probability: Theory and Application, by William Feller is good source.
The book starts completely from the first principles and covers also a lot of applications in statistics. Arguably the book is more suited to graduate as it has a steep curve - the content difficulty increases rapidly, but I think that it can be suitable even at Msc level - especially if you are doing more research oriented Msc.
If you are would be interested in something more focused on econometrics then a primer in econometric theory by John Stachurski is good source although it does not cover probability so broadly as the above source.
answered Jun 30, 2020 at 20:29
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1$\begingroup$ Feller is quite old school. It is also (IIRC) restricted to countable probability spaces, where measure theoretic considerations are trivial---in that aspect it's not informative, but perhaps that is by design and substituted by more probabilistic aspects. $\endgroup$– MichaelJul 2, 2020 at 14:06
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1$\begingroup$ @Michael Feller has two volumes; the second volume does introduce and use the general measure-theoretic machinery. $\endgroup$ Jul 2, 2020 at 19:38
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The best introduction to measure-theoretic probability for economics is probably:
Chapter 7 Measure Theory and Integration, Recursive Methods in Economic Dynamics by Stokey and Lucas.
Its presentation is along mathematical lines but with judicious (and numerous) omissions for economics audience.
For example, a measure space is defined but no real concrete construction, such as the Lebesgue measure on $\mathbb{R}^n$, is presented. Such omission would be basic and unforgivable for a mathematical audience but optional for many economists. On the other hand, results like Monotone Class Lemma is included so the reader gets a less superficial impression of measure theory.
It also proceed at a slow pace. (Monotone Class Lemma, which would be a single lemma in mathematics texts, occupies an entire section.)
In about 30 pages, you get "almost all" the measure-theoretic treatment of probability and integration you need for basic economics and econometrics.
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$\begingroup$ That book doesn't really teach probability. No LLNs or CLT, no martingales, no 0-1 laws, not even a definition of independence. $\endgroup$ Jul 2, 2020 at 19:50
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$\begingroup$ @MichaelGreinecker In my defense, I did say "introduction" and "numerious omissions". Different bar for different audience. Strong LLN (ergodic/martingale/etc) is probably a reach, and arguably not needed, for many economists anyway, while weak LLN is trivial under L^2 moment assumptions (measure theory not really needed). Definition of independence is also easily deduced. Notions like weak convergence, and CLT, yes, one would have to get from elsewhere. (Didn't know about Feller vol 2, thanks for pointing it out.) $\endgroup$– MichaelJul 2, 2020 at 20:20
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$\begingroup$ Also, martingales are discussed in the presentation of Stokey and Lucas (if not very explicitly), as the dynamic programming principle can be re-stated as the martingale optimality principle---the value function is a supermartingale along feasible paths and martingale along optimal path. $\endgroup$– MichaelJul 2, 2020 at 20:31
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$\begingroup$ I think the book teaches what it teaches reasonably well, but it is not really a book about probability and for someone whose interest is in statistics and econometrics probably not the first place to go to. The Strong LLN for iid random variables with finite fourth moments is relatively easy to prove. $\endgroup$ Jul 2, 2020 at 20:33