Advances in post-model-selection inference

Along with improvements in computational power, variable selection has become one of the problems attracting the most effort. We (well.. experts) have made huge leaps in the realm of variable selection. Prediction being probably the most common objective. LASSO (Least Absolute Sum of Squares Operator) leading the way from the west (Stanford) with its many variations (Adaptive, Random, Relaxed, Fused, Grouped, Bayesian.. you name it), SCAD (Smoothly Clipped Absolute Deviation) catching up from the east (Princeton). With the good progress in that area, not secondary but has been given less attention -> Inference is now being worked out.

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PCA as regression

A way to think about principal component analysis is as a matrix approximation. We have a matrix X_{T \times P} and we want to get a ‘smaller’ matrix Z_{T \times K} with K<p. We want the new ‘smaller’ matrix to be close to the original despite its reduced dimension. Sometimes we say ‘such that Z capture the bulk of comovement in X. Big data technology is such that nowadays the number of cross sectional units (number of columns in X) P has grown to be very large compared to the sixties say. Now, with ‘google maps would like to use your current location’ and future ‘google fridge would like to access your amazon shopping list’, you can count on P growing exponentially, we are just getting started. A lot of effort goes into this line of research, and with great leaps.

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Bias vs. Consistency


Especially for undergraduate students but not just, the concepts of unbiasedness and consistency as well as the relation between these two are tough to get one’s head around. My aim here is to help with this. We start with a short explanation of the two concepts and follow with an illustration.

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Bootstrap Critisim (example)

In a previous post I underlined an inherent feature of the non-parametric Bootstrap, it’s heavy reliance on the (single) realization of the data. This feature is not a bad one per se, we just need to be aware of the limitations. From comments made on the other post regarding this, I gathered that a more concrete example can help push this point across.

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Detecting bubbles in real time

Recently, we hear a lot about a housing bubble forming in UK. Would be great if we would have a formal test for identifying a bubble evolving in real time, I am not familiar with any such test. However, we can still do something in order to help us gauge if what we are seeing is indeed a bubbly process, which is bound to end badly.

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Omitted Variable Bias

Frequently, we see the term ‘control variables’. The researcher introduces dozens of explanatory variables she has no interest in. This is done in order to avoid the so-called ‘Omitted Variable Bias’.

What is Omitted Variable Bias?

In general, OLS estimator has great properties, not the least important is the fact that for a finite number of observations you can faithfully retrieve the marginal effect of X on Y, that is E(\widehat{\beta}) = \beta. This is very much not the case when you have a variable that should be included in the model but is left out. As in my previous posts about Multicollinearity and heteroskedasticity, I only try to provide the intuition since you are probably familiar with the result itself.

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Bayesian vs. Frequentist in Practice

Rivers of ink have been spilled over the ‘Bayesian vs. Frequentist’ dispute. Most of us were trained as Frequentists. Probably because the computational power needed for Bayesian analysis was not around when the syllabus of your statistical/econometric courses was formed. In this age of tablets and fast internet connection, your training does not matter much, you can easily transform between the two approaches, engaging the right webpages/communities. I will not talk about the ideological differences between the two, or which approach is more appealing and why. Larry Wasserman already gave an excellent review.

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Understanding Multicollinearity

Roughly speaking, Multicollinearity occurs when two or more regressors are highly correlated. As with heteroskedasticity, students often know what does it mean, how to detect it and are taught how to cope with it, but not why is it so. From Wikipedia: “In this situation (Multicollinearity) the coefficient estimates may change erratically in response to small changes in the model or the data.” The Wikipedia entry continues to discuss detection, implications and remedies. Here I try to provide the intuition.

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