In a previous post, I discussed parametric M-estimators, and proved a consistency result for a large class of such estimators in finite dimensional settings. A lot of the heavy lifting in the proof was done by a compactness assumption. In this post, I will run through some of the basic theory about compact sets that … Continue reading Compactness and its statistical uses

Last time, we discussed laws of large numbers when sampling a fixed random variable from an i.i.d. distribution. From this, we were able to derive the method of moments estimator. In the method of moments, we have some $latex \beta$ that is a function of some moments. Formally, $latex \beta = \beta(M_1,M_2,\ldots,M_k)$ where $latex M_j = … Continue reading Uniform Laws of Large Numbers, M-Estimators

In a previous post, I introduced some basic concepts in probability theory. Here, I will derive a few powerful results that follow. Recall from last time that I introduced the definitions of convergence in probability and convergence almost surely. Intuitively, both of these definitions capture the idea that we want to consider cases when a sequence … Continue reading Laws of Large Numbers, Method of Moments

The first time I learned about the inner workings of the lfe package in R, I was told to read a paper by mathematician Israel Halperin called "The product of projection operators" (good luck finding it though. The only reason why I was able to access it is because I was sent a pdf at … Continue reading LFExplained

I take a break from my usual pure theory to present something a little more applied. Suppose you estimate the OLS model $latex Y = \alpha + \beta_d D + \beta_x X + \beta_i D \cdot X + \varepsilon$ where $latex D$ is a dummy variable and $latex X$ is continuous. For conceptual simplicity, suppose … Continue reading Interactions are hard.

A popular musing about math is to ask whether an advanced alien civilization would have the same mathematics as us. A common answer is that while the language and symbols would almost certainly be different, the most important underlying concepts would be isomorphic. A quip I had after my first undergrad statistics class is that … Continue reading Some probability theory

Roughly speaking, I can classify difficulties in doing math into two categories (understanding that any such exercise is necessarily somewhat arbitrary). The first is that the precision of mathematical language requires a clarity of thought that is unnecessary and even impractical for everyday life. This is why reading through math textbooks is difficult, even when the … Continue reading Introduction post/How did anyone ever discover the central limit theorem?