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.

# Category: Blog

## Most popular posts – 2016

Another year. Looking at my google analytics reports I can’t help but wonder how is it that I am so bad in predicting which posts would catch audience attention. Anyhow, top three for 2016 are:

– On the 60/40 portfolio mix

– The case for Regime-Switching GARCH

– Most popular machine learning R packages

And my personal favorites:

– ASA statement on p-values

– Why bad trading strategies may perform well? Mathematical explanation

It is also an opportunity to say **thank you**, and to wish you a happy and productive 2017.

## 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:

## R tips and tricks – Faster Loops

## Insert or bind?

This is the first in a series of planned posts, sharing some R tips and tricks. I hope to cover topics which are not easily found elsewhere. This post has to do with loops in R. There are two ways to save values when looping:

1. You can predefine a vector and fill it, or

2. you can recursively bind the values.

Which one is faster?

## Optimism of the Training Error Rate

We all use models. We all continuously working to improve and validate our models. Constant effort is made trying to estimate: *how good our model actually is?*

A general term for this estimate is *error rate*. Low error rate is better than high error rate, it means our model is more accurate.

## 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.

## Good coding practices – part 2

## Introduction

In part 1 of *Good coding practices* we considered how best to code for someone else, may it be a colleague who is coming from Excel environment and is unfamiliar with scripting, a collaborator, a client or the future-you, the you few months from now. In this second part, I give some of my thoughts on how best to write functions, the do’s and dont’s.

## 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.

## Good coding practices – part 1

## Introduction

At work, I recently spent a lot of time coding for someone else, and like anything else you do, there is much to learn from it. It also got me thinking about scripting, and how best to go about it. To me it seems that the new working generation mostly tries to escape from working with Excel, but “let’s not kid ourselves: the most widely used piece of software for statistics is Excel” (Brian D. Ripley). this quote is 15 years old almost, but Excel still has a strong hold on the industry.

Here I discuss few good coding practices. Coding for someone else is not to be taken literally here. ‘Someone else’ is not necessarily a colleague, it could just as easily be the “future you”, the you reading your code six months from now (if you are lucky to get responsive referees). Did it never happened to you that your past-self was unduly cruel to your future-self? that you went back to some old code snippets and dearly regretted not adding few comments here and there? Of course it did.

Unlike the usual metric on which “good” is usually measured by when it comes to coding: *good = efficient*, here the metric would be different: *good = friendly*. They call this literate programming. There is a fairly deep discussion about this paradigm by John D. cook (follow what he has to say if you are not yet doing it, there is something for everyone).

## Why statistical bootstrap

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.

## Human significance, economic significance and statistical significance

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.

## Forecast combinations in R

Few weeks back I gave a talk in the R/Finance 2016 conference, about forecast combinations in R. Here are the slides:

## Laws of large numbers

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.