https://www.econometrics.blog/email-feed.html Opening paragraphs only. This feed exists so Buttondown can build the subscriber email; readers want /index.xml. quarto-1.10.18 Fri, 14 Aug 2026 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/econometrics-puzzler-4-rescaling-the-reduced-form/ and a single valid instrument . You are given the estimated slope coefficient from the reduced form regression of on along with its standard error . You are also given the estimated slope coefficient from the first stage regression of on . The IV estimate equals . Does its standard error equal ? ]]> econometrics causal inference puzzler https://www.econometrics.blog/post/econometrics-puzzler-4-rescaling-the-reduced-form/ Fri, 14 Aug 2026 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/econometrics-puzzler-3-ive-got-the-power/ in your head. No paper, no pencils, no normal tables, and no statistical software packages are allowed! ]]> statistics puzzler https://www.econometrics.blog/post/econometrics-puzzler-3-ive-got-the-power/ Thu, 06 Aug 2026 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/complex-step-differentiation/ of a real-valued function at some real value . So here’s a fun fact that you may not know. If is a small positive number and can be evaluated at a complex argument, then where is the imaginary unit and denotes the imaginary part of a complex number. This unexpected but highly accurate approximation is called complex step differentiation. The method dates back to Lyness and Moler (1967); Squire and Trapp (1998) give a concise modern exposition. ]]> computing https://www.econometrics.blog/post/complex-step-differentiation/ Sun, 26 Apr 2026 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/makeover-for-econometrics-blog/ econometrics.blog got a long-overdue update from blogdown to Quarto. Thanks to Claude Code, the transition was seamless: all existing links are preserved, along with the Utterances comment sections. ]]> meta https://www.econometrics.blog/post/makeover-for-econometrics-blog/ Thu, 09 Apr 2026 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/chris-sims-rip/ helicopter tour paper on this blog a while back. Some of my other favorites are unpublished notes or slides from his website, many of them with a philosophical bent. Thinking about instrumental variables is a paper I read in grad school that really clarified why things can go so badly wrong in IV estimation. I read Understanding Non-Bayesians for the first time a couple of years ago and wished I had read it sooner. It articulates a view of Bayesian econometrics that I find particularly compelling. Robins-Wasserman, Round N and Sharp Econometrics have also shaped the approach I’m taking in some recent work with Laura Liu. ]]> meta https://www.econometrics.blog/post/chris-sims-rip/ Sun, 15 Mar 2026 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/overlapping-confidence-intervals-part-ii/ Overlapping Confidence Intervals I asked what we can learn from the overlap, or lack thereof, between confidence intervals for two population means constructed using independent samples. To recap: if the individual confidence intervals for groups A and B do not overlap, there must be a statistically significant difference between the population means for the two groups. In other words, the interval for the difference of means will not include zero. If the individual intervals do overlap, on the other hand, anything goes. The interval for the difference of means may or may not include zero. Indeed, it’s even possible for the individual intervals for both A and B to include zero while the interval for their difference does not! ]]> statistics https://www.econometrics.blog/post/overlapping-confidence-intervals-part-ii/ Sat, 22 Nov 2025 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/overlapping-confidence-intervals/ flawed. ]]> statistics https://www.econometrics.blog/post/overlapping-confidence-intervals/ Sat, 15 Nov 2025 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/a-good-instrument-is-a-bad-control-part-ii/ 1 A lively discussion of bad controls ensued, during which I offered one of my favorite examples: a good instrument is a bad control. To summarize that earlier post: including a valid instrumental variable as a control variable can only amplify the bias on the coefficient for our endogenous regressor of interest. When used as a control, the instrument “soaks up” the good (exogenous) variation in the endogenous regressor, leaving behind only the bad (endogenous) variation. This is the opposite of what happens in an instrumental variables regression, where we use the instrument to extract only the good variation in the endogenous regressor. More generally, a “bad control” is a covariate that we shouldn’t adjust for when using a selection-on-observables approach to causal inference. ]]> econometrics causal inference https://www.econometrics.blog/post/a-good-instrument-is-a-bad-control-part-ii/ Thu, 28 Aug 2025 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/two-fwl-theorems-for-the-price-of-one/ Yule’s Rule, more commonly known as the “Frisch-Waugh-Lovell (FWL) theorem”, shows how to calculate the regression slope coefficient for one predictor by carrying out additional “auxiliary” regressions that adjust for all other predictors. You’ve probably encountered this result if you’ve studied introductory econometrics. But it may surprise you to learn that there are actually two variants of the FWL theorem, each with its pros and cons. Today we’ll take a look at the less familiar version and then circle back to understand what makes the more familiar one a textbook staple. ]]> econometrics https://www.econometrics.blog/post/two-fwl-theorems-for-the-price-of-one/ Thu, 14 Aug 2025 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/econometrics-puzzler-2-fitting-a-regression-with-fitted-values/ second regression of the outcome variable on the fitted values, what will I get? For extra credit: how will the R-squared from the second regression compare to that from the first regression? ]]> econometrics puzzler https://www.econometrics.blog/post/econometrics-puzzler-2-fitting-a-regression-with-fitted-values/ Thu, 24 Jul 2025 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/econometrics-puzzler-1-to-instrument-or-not/ Econometrics Puzzler, a new series of shorter posts that will test and strengthen your econometric intuition.¹ Here’s the format: I’ll pose a question that requires only introductory econometrics knowledge, but has an unexpected answer. The idea is for you to ponder the question before reading my solution. Many of these questions are based on common misconceptions that come up year-after-year in my econometrics teaching. I hope you’ll find them both challenging and enlightening. Today we’ll revisit everyone’s favorite example: Angrist & Krueger’s 1991 paper on the returns to education.² ]]> econometrics causal inference puzzler https://www.econometrics.blog/post/econometrics-puzzler-1-to-instrument-or-not/ Sun, 13 Jul 2025 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/not-quite-the-james-stein-estimator/ “James-Stein Estimator”. You’ve probably learned in your introductory econometrics course that ordinary least squares (OLS) is the best linear unbiased estimator (BLUE) in a linear regression model under the Gauss-Markov assumptions. The stipulations “linear” and “unbiased” are crucial here. If we remove them, it’s possible to do better–maybe even much better–than OLS.¹ Stein’s paradox is a famous example of this phenomenon, one that created much consternation among statisticians and fellow-travelers when it was first pointed out by Charles Stein in the mid-1950s. The example is interesting in its own right, but also has deep connections to ideas in Bayesian inference and machine learning making it much more than a mere curiosity. ]]> statistics https://www.econometrics.blog/post/not-quite-the-james-stein-estimator/ Sat, 10 Aug 2024 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/how-to-do-regression-adjustment/ regression adjustment, a widely-used approach to causal inference under the selection-on-observables assumption. While regression adjustment is simple in theory, the finer points of how and when to apply it in practice are much more subtle. One of these finer points is how to tell whether a particular covariate is a “good control” that will help us learn the causal effect of interest or a “bad control” that will only make things worse.¹ Another, and the topic of today’s post, is how to actually implement regression adjustment after we’ve decided which covariates to adjust for. ]]> causal inference https://www.econometrics.blog/post/how-to-do-regression-adjustment/ Fri, 02 Aug 2024 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/is-it-better-to-improve-sensitivity-or-specificity/ 1 Imagine that you’re developing a diagnostic test for a disease. The test is very simple: it either comes back positive or negative. You have a choice between slightly increasing either your test’s sensitivity or its specificity. If your goal is to maximize the positive predictive value (PPV) of your test, i.e. the probability that a patient has the disease given that the test comes back positive, which test characteristic should you choose to improve? ]]> statistics teaching puzzler https://www.econometrics.blog/post/is-it-better-to-improve-sensitivity-or-specificity/ Thu, 25 Jul 2024 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/how-to-read-an-econometrics-paper/ teaching https://www.econometrics.blog/post/how-to-read-an-econometrics-paper/ Sat, 20 Jul 2024 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/sims-and-uhlig-1991-replication/ Sims & Uhlig (1991) contrasting Bayesian and Frequentist inferences for a unit root. In the post I’ll focus on explaining and implementing the authors’ simulation design. In the reading group session (and possibly a future post) we’ll talk more about the paper’s implications for the Bayesian-Frequentist debate and relate it to more recent work by Mueller & Norets (2016). We’ll also be joined by special guest Frank Schorfheide who will help guide us through the recent literature on Bayesian approaches to VARs, including Giannone et al (2015) and (2019). If you’re an Oxford student or staff member, you can sign up for the reading group here. Otherwise, send me an email and I’ll add you manually. ]]> time series https://www.econometrics.blog/post/sims-and-uhlig-1991-replication/ Mon, 15 Jul 2024 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/the-return-of-econometrics-blog/ meta https://www.econometrics.blog/post/the-return-of-econometrics-blog/ Sun, 14 Jul 2024 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/a-good-instrument-is-a-bad-control/ both an endogenous regressor and a valid instrument for that regressor? I hadn’t thought about this question until 2018, when one of my undergraduate students asked it during class. If memory serves, my off-the-cuff answer left much to be desired.¹ Five years later I’m finally ready to give a fully satisfactory answer; better late than never I suppose! ]]> econometrics causal inference https://www.econometrics.blog/post/a-good-instrument-is-a-bad-control/ Thu, 29 Jun 2023 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/the-r-formula-cheatsheet/ 1 This post is a quick reference covering all of the symbols that have a “special” meaning inside of an R formula: ~, +, ., -, 1, :, *, ^, and I(). You may never use some of these in practice, but it’s nice to know that they exist. It was many years before I realized that I could simply type y ~ x * z instead of the lengthier y ~ x + z + x:z, for example. While R formulas crop up in a variety of places, they are probably most familiar as the first argument of lm(). For this reason, my verbal explanations assume a simple linear regression setting in which we hope to predict y using a number of regressors x, z, and w. ]]> computing https://www.econometrics.blog/post/the-r-formula-cheatsheet/ Wed, 19 Apr 2023 00:00:00 GMT Francis J. DiTraglia https://www.econometrics.blog/post/random-variables-cheatsheet/ updated pdf on github. You can fork the repository here. If you spot any errors or want to suggest any additions, feel free to raise an issue or send me a pull request. ]]> statistics https://www.econometrics.blog/post/random-variables-cheatsheet/ Sat, 07 Jan 2023 00:00:00 GMT