I often write about bootstrap (here an example and here a critique). I refer to it here as one of the most consequential advances in modern statistics. When I wrote that last post I was searching the web for a simple explanation to quickly show how useful bootstrap is, without boring the reader with the underlying math. Since I was not content with anything I could find, I decided to write it up, so here we go.
We are now collecting a lot of data. This is a good thing in general. But data collection and data storage capabilities have evolved fast. Much faster than statistical methods to go along with those voluminous numbers. We are still using good ole fashioned Fisherian statistics. Back then, when you had not too many observations, statistical significance actually meant something.
Few weeks back I gave a talk in the R/Finance 2016 conference, about forecast combinations in R. Here are the slides:
The laws of large numbers are the cornerstones of asymptotic theory. ‘Large numbers’ in this context does not refer to the value of the numbers we are dealing with, rather, it refers to a large number of repetitions (or trials, or experiments, or iterations). This post takes a stab at explaining the difference between the strong law of large numbers (SLLN) and the weak law of large numbers (WLLN). I think it is important, not amply clear to most, and I will need it as a reference in future posts.
Especially in economics/econometrics, modellers do not believe their models reflect reality as it is. No, the yield curve does NOT follow a three factor Nelson-Siegel model, the relation between a stock and its underlying factors is NOT linear, and volatility does NOT follow a Garch(1,1) process, nor Garch(?,?) for that matter. We simply look at the world, and try to find an apt description of what we see.
What is measurement error bias?
Errors-in-variables, or measurement error situation happens when your right hand side variable(s); your $x$ in a $y_t = \alpha + \beta x_t + \varepsilon_t$ model is measured with error. If $x$ represents the price of a liquid stock, then it is accurately measured because the trading is so frequent. But if $x$ is a volatility, well, it is not accurately measured. We simply don’t yet have the power to tame this variable variable.
Unlike the price itself, volatility estimates change with our choice of measurement method. Since no model is a perfect depiction of reality, we have a measurement error problem on our hands.
Ignoring measurement errors leads to biased estimates and, good God, inconsistent estimates.
GARCH models are very responsive in the sense that they allow the fit of the model to adjust rather quickly with incoming observations. However, this adjustment depends on the parameters of the model, and those may not be constant. Parameters’ estimation of a GARCH process is not as quick as those of say, simple regression, especially for a multivariate case. Because of that, I think, the literature on time-varying GARCH is not yet at its full speed. This post makes the point that there is a need for such a class of models. I demonstrate this by looking at the parameters of Threshold-GARCH model (aka GJR GARCH), before and after the 2008 crisis. In addition, you can learn how to make inference on GARCH parameters without relying on asymptotic normality, i.e. using bootstrap.
There are many problems with p-values, and I too have chipped in at times. I recently sat in a presentation of an excellent paper, to be submitted to the highest ranked journal in the field. The authors did not conceal their ruthless search for those mesmerizing asterisks indicating significance. I was curious to see many in the crowd are not aware of current history in the making regarding those asterisks.
The web is now swarming with thought-provoking discussions about the recent American Statistical Association (ASA) statement on p-values. Despite their sincere efforts, there are still a lot of back-and-forth over what they actually mean. Here is how I read it.
First things first.
What do we mean by sparse estimation?
Sparse – thinly scattered or distributed; not thick or dense.
We often see statements like “linear regression makes the assumption that the data is normally distributed”, “Data has no or little multicollinearity”, or other such blunders (you know who you are..).
Let’s set the whole thing straight.
Linear regression assumes nothing about your data
It has to be said. Linear regression does not even assume linearity for that matter, I argue. It is simply an estimator, a function. We don’t need to ask anything from a function.
Consider that linear regression has an additional somewhat esoteric, geometric interpretation. When we perform a linear regression you simply find the best possible, closest possible, linear projection we can. A linear combination in your X space that is as close as possible in a Euclidean sense (squared distance) to some other vector y.
That is IT! a simple geometric relation. No assumptions needed whatsoever.
You don’t ask anything from the average when you use it as an estimate for the mean do you? So why do that when you use regression? We only need to ask more if we do something more.
The term ‘curse of dimensionality’ is now standard in advanced statistical courses, and refers to the disproportional increase in data which is needed to allow only slightly more complex models. This is true in high-dimensional settings. Here is an illustration of the ‘Curse of dimensionality’ in action.
Some time during the 18th century the biologist and geologist Louis Agassiz said: “Every great scientific truth goes through three stages. First, people say it conflicts with the Bible. Next they say it has been discovered before. Lastly they say they always believed it”. Nowadays I am not sure about the Bible but yeah, it happens.
I express here my long-standing and long-lasting admiration for the following triplet of present-day great discoveries. The authors of all three papers had initially struggled to advance their ideas, which echos the quote above. Here they are, in no particular order.
In multivariate volatility forecasting (4), we saw how to create a covariance matrix which is driven by few principal components, rather than a complete set of tickers. The advantages of using such factor volatility models are plentiful.
What is tail dependence really? Say the market had a red day and saw a drawdown which belongs with the 5% worst days (from now on simply call it a drawdown):
One can ask what is now, given that the market is in the blue region, the probability of a a drawdown in a specific stock?