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I am interested in deepening my Asymptotic Theory understanding. My current knowledge is that of a typical PhD student (from a decent University), say at the level of Green's textbook. Are there any good book(s) that you would recommend?

Much appreciated.

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    $\begingroup$ I thought economists believed in "In the long run, we are all dead". Where does asymptotics come in then ? $\endgroup$ – htrahdis Oct 22 '13 at 13:59
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    $\begingroup$ @htrahdis, that's the essence of Kolmogorov's 0/1 law. Either you become famous, like Adam Smith -- or rather like Trygve Haavelmo or Clive Granger or James Heckman if we are talking about econometrics -- or you perish. $\endgroup$ – StasK Oct 22 '13 at 14:43
  • $\begingroup$ related: Good econometric textbooks, Higher order asymptotics $\endgroup$ – StasK Oct 22 '13 at 14:46
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Ferguson's A Course in Large Sample Theory is the best concise introduction to the topic, and it is written in a nice didactic way of having an equivalent of a week's lecture course material in a chapter followed by a strong set of exercises. (Ferguson introduced GMM in 1968 under the name of the minimum $\chi^2$, and it is tucked in as one of the exercises in that book). Van der Vaart's Asymptotic Statistics, recommended by others, is great, too, but it's going off in weird directions (for an economist). Another relatively easy introduction to the first-order asymptotics is Lehmann's Elements of Large Sample Theory. I would argue though that you would get a better mileage out of a book like Smith & Young's Essentials of Statistical Inference, as it will teach you about how statisticians think (sufficiency, UMPT, Cramer-Rao bound, etc.).

Of course you won't find the odd econometric asymptotics such as unit roots or weak instruments. Few statisticians have heard of them, and these are wa-a-ay too exotic for them. However, you would definitely want to revisit these unusual papers to shake off the wrong belief that everything asymptotic is asymptotically normal at $\sqrt{n}$ rate (you can find disturbing counterexamples here and there, too).

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  • $\begingroup$ These will be very helpful for me. Thanks for the guidance. $\endgroup$ – Kirsty Oct 22 '13 at 12:43
  • $\begingroup$ For introduction to asymptotics of unit roots and time trends I suggest looking at JD Hamilton's Time Series analysis. I totaly forgot those. I would say though, that results about unit roots in particular albeit not heard by statisticians are very well known by mathematicians working with stochastic processes. Functional central limit theorem used by unit root literature was first mentioned by Donsker in 1952. Also interestingly that is one of directions van der Vaart book is going, namely empirical processes and their convergence. $\endgroup$ – mpiktas Oct 22 '13 at 13:39
  • $\begingroup$ This sounds pretty interesting, did not know that unit roots have such a history $\endgroup$ – Kirsty Oct 22 '13 at 13:41
  • $\begingroup$ For the empirical processes nice book is Vaart's and Wellner's "Weak Convergence and Empirical Processes". But this is really deep probabilistic stuff, which requires very good mathematical background. $\endgroup$ – mpiktas Oct 22 '13 at 13:42
  • $\begingroup$ @Kirsty, to be fair, not the unit roots have such a history, but weak convergence. Central limit theorem on which most of the econometrics is based is weak convergence of sequences of random vectors. Functional central limit theorem involves convergence of sequences of functions in nice spaces. Empirical processes generalise the notions of central limit theorem and law of large numbers to sequences (nets if you know what it is) of a very general random objects. $\endgroup$ – mpiktas Oct 22 '13 at 13:46
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Since you mention Greene's book, I assume you are interested in more in-depth understanding of asymptotic statistics. Then I can recommend A. van der Vaart's "Asymptotic statistics" and H. White's "Asymptotic theory for econometricians". Also J. Wooldridge's "Econometric Analysis of Cross Section and Panel Data" has nice chapters on asymptotic theory.

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  • $\begingroup$ Many thanks. I am much interested in getting hold of these books. $\endgroup$ – Kirsty Oct 22 '13 at 12:40
  • $\begingroup$ I second H.White's book as a reference book of results regarding asymptotic theory for least-squares and Instrumental variables estimators, (It has not been written with education in mind), because it is very clearly organized in covering gradually different stochastic set ups (deterministic regressors, non-determinsitic, i.i.d. non-identical, non-independent etc.). $\endgroup$ – Alecos Papadopoulos Oct 22 '13 at 13:15
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"Asymptotic theory for econometricians" by Halbert White. "Asymptotic Theory of Statistics and Probability", by Anirban DasGupta.

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  • $\begingroup$ The first book seems to be popular from the reviews, will definitely look into it. The second also sounds promising. Thanks. $\endgroup$ – Kirsty Oct 22 '13 at 12:45

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