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.
The good thing about using open-source software is the community around it. There are very many R packages online, and recently CRAN package download logs were released. This means we can have a look at the number of downloads for each package, so to get a good feel for their relative popularity. I pulled the log files from the server and checked a few packages which are known to be related to machine learning. With this post you can see which are the community favorites, and get a feel for the R-software trend growth.
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.
Not sure why is that, but traditionally we consider 60% stocks and 40% bonds to be a good portfolio mix. One which strikes decent balance between risk and return. I don’t want to blubber here about the notion of risk. However, I do note that I feel uncomfortable interchanging risk with volatility as we most often do. I am not unhappy with volatility, I am unhappy with realized loss, that is decidedly not the same thing. Not to mention volatility does not have to be to the downside (though I just did).
Let’s take a look at this 60/40 mix more closely.
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.
This page shares some good comments readers make
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.
The top three for the year are:
Out-of-sample data snooping
Code for my yield curve forecasting paper
Review of a couple of books
I personally enjoyed the most writing a few words on ML estimation, and about those great statistical discoveries. Since the last post did not involve any code or images I initially thought it would be a breeze. I in fact spent twice the time I usually do, and it was all good fun.
In 2015 I wrote quite a bit about volatility and correlation. In 2016 I plan to learn more (so to write more) about portfolio construction.
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.