From convergence in distribution to convergence in probability

Posted: December 21, 2018 in Cross-Validated relay, Educational Material, stats.stackexchange relay
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When $\hat \theta \to_d D(\theta, v) \implies \hat\theta \to_p \theta$ ?

We know that in general, convergence in distribution does not imply convergence in probability. But for the case of most interest in econometrics, where we examine a sequence of estimators, convergence in distribution does imply convergence in probability to a constant, under two regularity conditions that are also satisfied in most cases of interest.

This post of mine in stats.stackexchange.com has the proof. Essentially, under these regularity conditions we are able to prove the even stronger result that convergence in distribution  implies  convergence in quadratic mean to a constant (which in turn implies convergence in probability to that constant).

Marginal and Joint Normality

Posted: December 9, 2018 in Educational Material
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Time and again I encounter people confused about marginal and joint Normality, and I could not find a single internet or book source that lists together the main cases of interest. So here they are:

1. Subject to the usual regularity conditions, the below hold if we are talking about asymptotic (limiting) marginal/joint Normality also.
2. If two random variables are not each marginally Normal, then they are certainly not jointly Normal.
3. If two random variables are not each marginally Normal, then their linear combinations are not marginally Normal.
4. If $X$ and $Y$ have Normal marginals and they are independent, then they are also jointly Normal. By the Cramér-Wold theorem, it then follows that all linear combinations of them (like their sum, difference, etc) are also marginally Normal. If we want to consider more than two random variables, then for the above to hold, they must be jointly independent, and not only pair-wise independent. If they are only pair-wise independent, then they may be jointly Normal, may be not.
5. If $X$ and $Y$ have Normal marginals but they are dependent, then it is not necessarily the case that they are jointly Normal. They may be, they may be not. It follows that a linear combination of them, may be Normal, may be not. It must be proven based on something more than marginal Normality. This is important to remember when one wants to show asymptotic Normality of a test statistic that is linearly composed by two random variables that are each asymptotically normal, but they are dependent without the dependence dying out asymptotically. This is the case, for example, in all “Hausman endogeneity tests” in econometrics, where the test statistic is the difference of two estimators that use the same data, and so are in general dependent. Even if each is asymptotically Normal, the asymptotic Normality of their difference does not necessarily follow. In his original paper in fact, Hausman (1978) explicitly assumed asymptotic normality of the test statistic, he did not prove it.
6. If $X$ and $Y$ have Normal marginals, are uncorrelated (i.e. their covariance is zero), but they have some higher order/”non-linear” form of dependence, than they are certainly NOT jointly normal (because in joint Normality, existence of dependence is always expressed also as non-zero covariance). Their linear combinations may be marginally Normal, may be not. Again, if a linear combination of them is the object of interest, its asymptotic normality must be proven based on something more than marginal Normality.

Why we should respect the privacy of the Taylor Remainder

Posted: May 4, 2016 in Educational Material
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We use Taylor expansions all the time to linearize functions, and sometimes 2nd-order expansions to examine variability. We conveniently ignore the Taylor Remainder, based on Taylor’s Theorem, and the question is : is this wise? The answer is yes. Download the following very short note to see why :

Taylor remainder privacy

The generalized Solow model with endogenous growth

Posted: December 13, 2015 in Educational Material, Grad Macro I Fall 2015-2016
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I apload here an exposition of the generalized Solow model of growth, i.e. the one including both physical and human capital, but with capabilities of sustained endogenous long-term growth. This came out of the Macro I graduate class in my University that I currently help. At the end, the pamphlet contains the discretization of the model and a Dynare script to simulate it.

Generalized Solow endogenous growth

The Blanchard-Weil Overlapping Generations model of growth (ENGLISH version)

Posted: November 18, 2015 in Educational Material, Grad Macro I Fall 2015-2016
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Here is  the ENGLISH version of the Blanchard-Weil Overlapping Generations model of growth. See the other post about what it contains.

It is identical to the Greek version, but a few pages shorter, because the English language has shorter words than the Greek language.

For my Macro I students : since this pamphlet lifts off your shoulders a great burden, I would suggest to compensate by giving special weight and inelastic labor to the following exercises contained therein: 1.1, 1.3, 2.3, 3.4, 4.1.

Overlapping Generations Blanchard Weil ENGLISH 15-11-2015

The Blanchard-Weil Overlapping Generations model of growth (GREEK version)

Posted: November 16, 2015 in Educational Material, Grad Macro I Fall 2015-2016
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I upload here the GREEK version of the most analytical educational take I have ever seen on the Blanchard-Weil Overlapping Generations model of growth. I have written it, by the way. The pamphlet contains an interesting extension, in that it calculates the implied, by the model, distribution of consumption and capital, something that also permits a deeper comparison with the representative household model.

I apologize to my non-Greek students and readers for not uploading at the same time the English version. I had in mind to just translate an older version of the pamphlet, but I ended up re-writing extensively its last parts, and this ate up the available time. I will, in a day or two, upload also the ENGLISH version.

Overlapping Generations Blanchard Weil GREEK 15-11-2015

Phase Diagrams Tutorial

Posted: November 13, 2015 in Educational Material, Grad Macro I Fall 2015-2016
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I upload here a Phase Diagram tutorial: how we construct it, and how it can be exploited to yield comparative statics results that may not be otherwise obtainable.

The tutorial is not comprehensive – it focuses on the basic growth models in economics that exhibit saddle-path stability.

Phase Diagrams Construction and Comparative Statics

Ramsey on the Greek crisis and depression

Posted: October 31, 2015 in Educational Material, Grad Macro I Fall 2015-2016
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UPDATE 13-11-2015: A new .pdf has been uploaded

I upload here an educational application of the standard Ramsey model of long-run growth, to show that it predicts correctly, in a qualitative sense, what has happened and is still happening in the Greek economy due to the recent crisis and current depression.

Even though the model is concerned with long-run growth rather than with short- and mid-term fluctuations, still it is important to see that by shifting our horizon-focus in economic analysis, we do not end up with incompatible conclusions.

Current Greek Depression and the Ramsey growth model 13-11-2015

Dynamic Stability for economic models

Posted: October 29, 2015 in Educational Material, Grad Macro I Fall 2015-2016
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I upload here an updated and considerably expanded  older pamphlet of mine, about dynamic stability in economic models, one- or two-dimensional. The standard results are combined and tabulated compactly to provide a coherent picture, while I also treat in detail the  (many) cases of “Saddle-path stability” in systems of difference equations. Saddle-path stability is a central concept in dynamic economics, being the mathematical concept that is consistent with dynamic adjustment that results from purposeful behavior, and can accommodate structural shifts.

Dynamic Stability for economic models

Tutorials on Linearization methods

Posted: October 24, 2015 in Educational Material, Grad Macro I Fall 2015-2016
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Two good tutorials:

The first is on the Taylor-series expansion of a function which is used for linearizing a non-linear function (1st-order Taylor expansion), and also, usually in a stochastic context, for “mean-variance” analysis (2nd-order, Taylor expansion). Among other things we use it to linearize non-linear differential and difference equations in order to study their stability properties.

Taylor Series tutorial

The second is about log-linearization, which is another linearizing technique often used in Economics, and especially in macroeconomic models.

Log-linearizing Guide