One of the most annoying criticisms economists face are charges that humans can't be rational, so models that assume it can't possibly be right. Such an objection is valid - virtually no economist thinks that modeling agents as rational is without its faults, but as the standard argument in the philosophy of science goes, the model … Continue reading Rationality from Irrationality: A close reading of Hart and Mas-Colell

The last post on Black Scholes got quite long, so ideally, I'd like to post some short addendums to flesh out some additional thoughts related to the post. Today, I'd like to provide a fully worked out solution to the following exercise: Exercise 6: Use the following facts about Brownian motion to prove the infinite … Continue reading Black Scholes Addendum 1: Indecisive Monkeys

I will preface this post by saying that the idea is entirely unoriginal. The main goal we will be building towards here is an intuitive understanding of the Black-Scholes equation. But this is a grossly overdone topic by now. Wikipedia, as well as countless other sources all offer their own intuitive derivations. My first time really … Continue reading Delta Hedging, Stochastic Calculus, and Black-Scholes

Today, I share yet another problem that I encountered in real life (slightly modified), but which would make an excellent addition to a problem set. Exercise 1 (Inverse propensity weighted regression): You want to test the effect of treatment 1 on an individual's responsiveness to treatment 2. You therefore run a series of experiments where … Continue reading Propensity Score Perplexities

Over the past series of posts, I have described the standard theory for proving consistency in a large class of parametric models. While this is useful for showing that the estimators we dream up are actually doing the right thing, it's still a bit unsatisfactory: so far, for any fixed dataset, we don't have a … Continue reading Asymptotic Normality in Parametric Models

Last weekend, I posited a fairly applied question that highlighted precisely what sort of a causal effect instruments identify. Today, I wanted to share a more theoretical question that is useful for thinking through the algebra of linear IV. For the sake of notational convenience here, we assume that every random variable is mean 0 … Continue reading Putting the “metrics” in “econometrics”

To see why the estimator I defined was an IV, note that for binary $latex X$, we have that $latex \bar Y_1 - \bar Y_0$ is the OLS slope coefficient for the regression $latex Y = \alpha + \beta X + \varepsilon$. Then using the normal equations, we have that $latex \frac{\overline{\log D_1} - \overline{\log … Continue reading Putting the “econ” back in “econometrics”: solutions

If I ever teach an applied econometrics course, I would want to assign the following problem (although I might present it slightly differently than I do below to make it more "fair"). I think it helps focus attention on the really subtle points about what IV estimators are actually identifying. Imagine that United Airlines wants … Continue reading Putting the “econ” back into “econometrics”

You may have heard of Hilbert's paradox of the Grand Hotel, but have you heard of its shady past? Hilbert's Hotel once had deep connections to the local mafia. What was the nature of their business? Money laundering. How did they do it? That will be the subject of today's post. When the mafia wanted … Continue reading High-stakes High-jinks at Hilbert’s Hotel

An important result in analysis is the Arzelà–Ascoli theorem. It provides a fairly simple way to check compactness in the space $latex C(X)$ of continuous functions on compact Hausdorff spaces $latex X$ where $latex C(X)$ is endowed with the uniform norm $latex ||f|| = \sup_{x\in X} f(x)$. It can be used, for example, in the … Continue reading An Addendum on Compactness