Miscellaneous

In class yesterday we reviewed a few key concepts, which I'm going to revisit here for posterity. Also, here is a link to a nice monograph on regression discontinuity designs (RDD). Please read chapters 1 and 2 before class tomorrow (a total length of 17 pages). The first agenda item is to consider in more…

Gaussian mixture model

In class on Wednesday I briefly described a bivariate mixture model and I "live-coded" an R implementation of a Gibbs sampler for performing posterior inference. In this post I'm just going to revisit these same points. A Gaussian mixture model has a density function that is a weighted sum of Gaussian density functions, with weights…

Instrumental variables

Links! Notes on IV. R demonstration script. The past three class sessions we (by which I mean "I") have fumbled around trying to describe instrumental variables --- the basic strategy, the details of how and when this strategy will work, and some basic formal derivations. I've finally settled on a presentation that I like reasonably…

Monte Carlo methods

In the past few classes we have taken a look at Monte Carlo methods, which are computational techniques for doing statistics instead of doing calculus. That is, instead of calculating definite integrals, we instead sample from an appropriate probability distribution and then take sample averages. The guiding expression is just this: $latex \frac{1}{M} \sum_{m =…

Resource and references

OK, while I take my sweet time getting the summary posts up, here are links to the various papers/chapters we have talked about in class so far (or will talk about soon). Sections 7.4 and 7.5 from Imbens and Rubin's book spells out the conditions under which a linear regression model will estimate an ATE.…

Putting graphs on a blog is hard

This is a test: I actually think that looks pretty good, but it wasn't especially convenient...thanks for being patient while I sort this out.

Exercise solutions

In class recently we sketched out solutions to the first set of exercises. Below we write down those details so you can go over it more slowly at your convenience. As usual, pipe up if you see something incorrect. Let's first show that the mean is the optimal action under mean squared error. We start…

Bernoulli and Poisson models

Today in class we covered the Beta-Binomial model. That is, we considered a model where for $latex i = 1, \dots, n&s=1$, $latex Y_i \sim \mbox{Bernoulli}(\theta)&s=1$ independently and $latex \theta \sim \mbox{Beta}(a,b)&s=1$. For this model, the total number of successes, or $latex S = \sum_i Y_i&s=1$, is a sufficient statistic for the parameter $latex \theta&s=1$…

Jan 17

Today in class we: Described the commonly invoked SUTVA condition, reviewed the Morgan and Rubin paper, and saw some simple examples of adjustments for confounding by blocking on observable features. Here again is a link to the Morgan and Rubin paper, and here is a link to the R script we will take a look…

Potential outcomes

The potential outcome formalism of Donald Rubin and Jerzy Neyman is a key development in modern causal inference. One of our textbooks has a pretty good list of scholarly references. See also this blog post of Gelman's for some interesting discussion about the intellectual history of potential outcomes in economics. The assigned paper by Holland defines the fundamental problem of causal…