The LLN is interesting as much for what it does not say as for what it does — O n August 24, 1966, a talented playwright by the name Tom Stoppard staged a play in Edinburgh, Scotland. The play had a curious title, “Rosencrantz and Guildenstern Are Dead.” Its central characters, Rosencrantz and Guildenstern, are childhood friends of Hamlet (of Shakespearean fame). The play opens with Guildenstern…

How we come to expect something, what it means to expect anything, and the math that gives rise to the meaning. — It was the summer of 1988 when I stepped onto a boat for the first time in my life. It was a passenger ferry from Dover, England to Calais, France. I didn’t know it then, but I was catching the tail end of the golden era of Channel crossings by…

When it comes to yelling “Surprise!”, the universe never gets tired. — In this article, we’ll take a contemplative walk across the land of chance. We’ll learn about random walks, about discrete and continuous random variables, and about their probability distributions. We’ll understand why it’s meaningless to assign a probability to a continuous random variable’s value. And in doing so, we’ll unearth…

How to fit a system of regression equations using Python and Statsmodels — In this article, we’ll walk through a tutorial on how to fit a system of linear regression models using the Generalized Least Squares (GLS) technique. We’ll use the term ‘regression model’ and ‘regression equation’ more or less interchangeably. This tutorial assumes that you already know the concepts and theory behind…

How to fit a mesh of inter-dependent regression models using techniques such as GLS. — In this article, we’ll study a very interesting topic in statistical modeling, namely, how to fit a system of inter-dependent regression models. At first glance, this topic may appear a bit esoteric, but systems of interconnected regression models pop up surprisingly often in real life. In the rest of this…

We’ll learn how to use the GLS estimator to fit a linear model on a real world data set — The Generalized Least Squares (GLS) estimator is an effective alternative to the Ordinary Least Squares (OLS) estimator for fitting linear models on data sets that exhibit heteroskedasticity (i.e., non-constant variance) and/or auto-correlation. In a previous article, we had detailed out the motivation for the GLS estimator and described how it…

A detailed look at how to fit a robust GLS model on heteroskedastic, auto-correlated data sets — Generalized Least Squares (GLS) estimation is a generalization of the Ordinary Least Squares (OLS) estimation technique. GLS is especially suitable for fitting linear models on data sets that exhibit heteroskedasticity (i.e., non-constant variance) and/or auto-correlation. …

How to use the White’s heteroskedasticity consistent estimator using Python and statsmodels — In this article, we shall learn how to employ the HC estimator to perform statistical inference that is robust to heteroskedasticity. This article is PART 2 of the following two part series: PART 1: Introducing White’s Heteroskedasticity Consistent Estimator PART 2: A tutorial on White’s Heteroskedasticity Consistent Estimator using Python and…

An introduction to the HC estimator, and its importance in building regression models in the face of heteroskedasticity — In this article, we’ll bring together two fundamental topics in statistical modeling, namely the covariance matrix and heteroskedasticity. Covariance matrices are the work horses of statistical inference. They are used for determining if regression coefficients are statistically significant (i.e. different from zero), and for constructing confidence intervals for each coefficient…