Many years ago, when I was still trying to beat the market, I used to pair-trade. In principle it is quite straightforward to estimate the correlation between two stocks. The estimator for beta is very important since it determines how much you should long the one and how much you should short the other, in order to remain market-neutral. In practice it is indeed very easy to estimate, but I remember I never felt genuinely comfortable with the results. Not only because of instability over time, but also because the Ordinary Least Squares (OLS from here on) estimator is theoretically justified based on few text-book assumptions, most of which are improper in practice. In addition, the OLS estimator it is very sensitive to outliers. There are other good alternatives. I have described couple of alternatives here and here. Here below is another alternative, provoked by a recent paper titled *Adaptive Huber Regression*.

# Category: Risk

## Visualizing Tail Risk

Tail risk conventionally refers to the risk of a large and sharp draw down of the portfolio. How large is subjective and depends on how you define what is a tail.

A lot of research is directed towards having a good estimate of the tail risk. Some fairly new research also now indicates that investors perceive tail risk to be a stand-alone risk to be compensated for, rather than bundled together with the usual variability of the portfolio. So this risk now gets even more attention.

## Outliers and Loss Functions

## A few words about outliers

In statistics, outliers are as thorny topic as it gets. Is it legitimate to treat the observations seen during global financial crisis as outliers? or are those simply a feature of the system, and as such are integral part of a very fat tail distribution?

## Density Confidence Interval

Density estimation belongs with the literature of non-parametric statistics. Using simple bootstrapping techniques we can obtain confidence intervals (CI) for the whole density curve. Here is a quick and easy way to obtain CI’s for different risk measures (VaR, expected shortfall) and using what follows, you can answer all kind of relevant questions.

## Trim your mean

The mean is arguably the most commonly used measure for central tendency, no no, don’t fall asleep! important point ahead.

We routinely compute the average as an estimate for the mean. All else constant, how much return should we expect the S&P 500 to deliver over some period? the average of past returns is a good answer. The average is the Maximum Likelihood (ML) estimate under Gaussianity. The average is a private case of least square minimization (a regression with no explanatory variables). It is a good answer. BUT:

## On Central Moments

Sometimes I read academic literature, and often times those papers contain some proofs. I usually gloss over some innocent-looking assumptions on moments’ existence, invariably popping before derivations of theorems or lemmas. Here is one among countless examples, actually taken from Making and Evaluating Point Forecasts:

## Modeling Tail Behavior with EVT

## Extreme Value Theory (EVT) and Heavy tails

Extreme Value Theory (EVT) is busy with understanding the behavior of the distribution, in the extremes. The extreme determine the average, not the reverse. If you understand the extreme, the average follows. But, getting the extreme right is extremely difficult. By construction, you have very few data points. By way of contradiction, if you have many data points then it is not the extreme you are dealing with.

## Multivariate Volatility Forecast Evaluation

The evaluation of volatility models is gracefully complicated by the fact that, unlike other time series, even the realization is not observable. Two researchers would never disagree about what was yesterday’s stock price, but they can easily disagree about what was yesterday’s stock volatility. Because we don’t observe volatility directly, each of us uses own proxy of choice. There are many ways to skin this cat (more on volatility proxy here).

In a previous post Univariate volatility forecast evaluation we considered common ways in which we can evaluate how good is our volatility model, dealing with one time-series at a time. But how do we evaluate, or compare two models in a multivariate settings, with two covariance *matrices*?

## Why bad trading strategies may perform well? Mathematical explanation

You probably know that even a trading strategy which is actually no different from a random walk (RW henceforth) can perform very well. Perhaps you chalk it up to short-run volatility. But in fact there is a deeper reason for this to happen, in force. If you insist on using and continuously testing a RW strategy, you will find, at some point **with certainty**, that it has significant outperformance.

This post explains why is that.

## Multivariate volatility forecasting, part 6 – sparse estimation

First things first.

## What do we mean by *sparse estimation*?

Sparse – thinly scattered or distributed; not thick or dense.

## Curse of dimensionality part 1: Value at Risk

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.

## Multivariate volatility forecasting (5), Orthogonal GARCH

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.

## Correlation and correlation structure (3), estimate tail dependence using regression

## Multivariate volatility forecasting (4), factor models

To be instructive, I always use very few tickers to describe how a method works (and this tutorial is no different). Most of the time is spent on methods that we can easily scale up. Even if exemplified using only say 3 tickers, a more realistic 100 or 500 is not an obstacle. But, is it really necessary to model the volatility of each ticker individually? No.

If we want to forecast the covariance matrix of all components in the Russell 2000 index we don’t leave much on the table if we model only a *few underlying factors*, much less than 2000.

Volatility factor models are one of those rare cases where the appeal is both theoretical and empirical. The idea is to create a few principal components and, under the reasonable assumption that they drive the bulk of comovement in the data, model those few components only.

## Multivariate volatility forecasting (3), Exponentially weighted model

Broadly speaking, complex models can achieve great predictive accuracy. Nonetheless, a winner in a kaggle competition is required only to attach a code for the replication of the winning result. She is not required to teach anyone the built-in elements of his model which gives the specific edge over other competitors. In a corporation settings your manager and his manager and so forth MUST feel comfortable with the underlying model. Mumbling something like “This artificial-neural-network is obtained by using a grid search over a range of parameters and connection weights where the architecture itself is fixed beforehand…”, forget it!