5 weeks ago we took a look at the rising volatility in the (US) equity markets via a time-series threshold model for the VIX. The estimate suggested we are crossing (or crossed) to the more volatile regime. Here, taking somewhat different Hidden Markov Model (HMM) approach we gather more corroboration (few online references at the bottom if you are not familiar with HMM models. The word hidden since the state is ‘invisible’).
In the previous post we reviewed a way to handle the problem of inference after model selection. I recently read another related paper which goes about this complicated issues from a different angle. The paper titled ‘A significance test for the lasso’ is a real step forward in this area. The authors develop the asymptotic distribution for the coefficients, accounting for the selection step. A description of the tough problem they successfully tackle can be found here.
The usual way to test if variable (say variable j) adds value to your regression is using the F-test. We once compute the regression excluding variable j, and once including variable j. Then we compare the sum of squared errors and we know what is the distribution of the statistic, it is F, or , depends on your initial assumptions, so F-test or -test. These are by far the most common tests to check if a variable should or should not be included. Problem arises if you search for variable j beforehand.
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.
A way to think about principal component analysis is as a matrix approximation. We have a matrix and we want to get a ‘smaller’ matrix with . 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.
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.
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.
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.
The title reads Bootstrap criticism, but in fact it should be Non-parametric bootstrap criticism. I am all in favour of Bootstrapping, but I point here to a major drawback.
We are all standing on the shoulders of giants. Bradley Efron is one such giant. With the invention of the bootstrap in 1979 and later with his very influential 2004 paper about the Least Angle Regression (and the accompanied software written in R).
I just finished reading An estimate of the science-wise false discovery rate and application to the top medical literature. The authors ask how many of what we read is scientific journals is actually incorrect, or false.
Overfitting is strongly related to variable selection. It is a common problem and a tough one, best explained by way of example.
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 . 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.
Few weeks back I simulated a model and made the point that in practice, the difference between Bayesian and Frequentist is not large. Here I apply the code to some real data; a model for Industrial Production (IP).
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.
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.