Greg Faletto

@gregoryfaletto.com

573 Followers 841 Following

Statistician, Data Scientist.

Documentation snippet for the fetwfe package titled 'Tidy output with broom', from the blog post at https://gregoryfaletto.com/2026/05/23/now-on-cran-fetwfe-version-1-10-0/. It demonstrates tidyverse integration using tidy(), glance(), and augment() functions, featuring a console summary table for ATT and cohorts 2 through 4.
Documentation snippet for the fetwfe package titled 'Experimental cluster-robust standard errors', from the blog post at https://gregoryfaletto.com/2026/05/23/now-on-cran-fetwfe-version-1-10-0/. It demonstrates using the se_type = 'cluster' option to calculate a unit-clustered sandwich estimator, complete with example R code and console output.
Documentation snippet for the fetwfe package titled 'Testing whether an effect is zero', from the blog post at https://gregoryfaletto.com/2026/05/23/now-on-cran-fetwfe-version-1-10-0/. It explains the 'selected' flag property using a console output table for 'result$catt_df' across Cohorts 2, 3, and 4.
Blog post excerpt (from https://gregoryfaletto.com/2026/05/23/now-on-cran-fetwfe-version-1-10-0/) for the fetwfe package titled 'Event-study estimates and plots'. It describes the eventStudy() and plot() functions, including an example output table showing event-time estimates, standard errors, confidence intervals, and p-values.
Screenshot from Faletto, Gregory. "Fused extended two-way fixed effects for difference-in-differences with staggered adoptions." arXiv preprint arXiv:2312.05985 (2023). Figure 1: A line graph visualizing connections between estimated marginal average treatment effect terms (tau_rt) across time t (horizontal axis, 1 to 6) and cohorts r (vertical axis, 2 to 5). Details of the grid and connections are described in the accompanying text. Figure 1: Visualization of which of the estimated marginal average treatment effect terms \hat{\tau}_{rt} from regression (1.5) (which estimate the average treatment effects \tau_{ATT}(r,t) from Equation 1.1) we penalize towards each other in the FETWFE penalty (5.1). In this setting, T = 6 and R = {2, ..., 5}. The horizontal axis depicts time and the vertical axis depicts cohorts. FETWFE works well under an assumption that the linked treatment effects tend to be close together, and at least some of them are exactly equal. See further details in Section 5.
Screenshot of an excerpt from Chapter 18 of: Hansen, Bruce. Econometrics. Princeton University Press, 2022. "See Exercise 18.1.) Thus the above regression is identical to the two-way fixed effects regression Y_it = theta D_it + u_i + v_t + epsilon_it (18.3) where u_i is a restaurant fixed effect and v_t is a time fixed effect. The simplest method to implement this"
Screenshot of a Tumblr post with a reply beneath it. The first post says: “My deepest darkest fantasy is that I collapse on the street and I am rushed to the hospital. They perform a bunch of tests and find out I am severely deficient in some kind of vitamin. Then I start taking the vitamin and I become the happiest cleverest person alive because all my problems were caused by this one deficiency.” The reply says: “Moreover, everyone gathers around to be tremendously compassionate and discreetly admiring: all this time, you lacked the Vitamin? And yet you persevered?” The humor comes from imagining a single simple explanation and cure for every struggle, along with receiving recognition for enduring it.
Line chart titled “Trump hits new approval low overall and on prices.” The chart shows Donald Trump’s net approval ratings from early 2025 through April 2026 across several issues, including inflation/cost of living, immigration, deportations, trade, health care, civil rights/democracy, and overall approval. All trend lines decline over time and remain below zero, indicating more disapproval than approval. The dark green “Inflation/cost of living” line falls the most sharply, dropping from about –3 in early 2025 to roughly –42 by April 2026, making it the lowest-rated issue. Other issue ratings cluster between about –10 and –25 near the end of the period. A tooltip highlights an inflation/cost of living rating of –26.1 on February 26, 2026. Source notes and methodology appear below the chart.
Enshittification go up
Screenshot of a shopping app screen showing “4 of 4 Coupons Clipped” at the top. Four offers are listed vertically: * “Earn 4X Points” with Chevron/Texaco branding, marked “Clipped: Add 1 Total for Offer,” and a “Shop Deal” link. * “Earn 3X Points” with Waterfront Bistro branding, also marked “Clipped: Add 1 Total for Offer,” with a “Shop Deal” link. * “Earn 2X Points” with Signature Select branding, marked “Clipped: Add 1 Total for Offer,” with a “Shop Deal” link. * “Earn 2X Points” with FreshPass Perk branding, marked “Clipped: Add 1 Total for Offer,” with a “Shop Deal” link. Each offer includes a small tag icon on the right side.
Spanberger
Simple black-and-white comic drawing of a person standing beside a desktop computer. The person says, “Say ‘I am alive’.” The computer monitor displays the text “> I AM ALIVE.” The person responds, “oh my god.”
Screenshot of a tweet from user “kelsie (@iPad_Latino)” that reads: “Me: brutally murdered and found dumped on the side of the highway. Two 35yr old women with a podcast: ok murder muffins we got a real oopy goopy spoopy story for you today! Squarespace ad: ARE YOU LOOKING TO EXPAND Y”. Timestamp shown: 12:46 PM · Jun 6, 2022.
“Freak the fuck out and panic sell everything right now. It’s fucking over." — Warren Buffett
Quote from blog post with highlighting: "I think a useful starting point in these discussion is to say what you want out of the published literature. I don’t have a full answer, but one thing that I would like is for published research coefficients to be unbiased. If a published result has an effect of 1, then I would like to (correctly) believe that if I were to somehow average across all (published and unpublished) research conducted on this topic with similar setups then I would get something like 1. Ideally, this would also be informative of future replications of the research too." The highlighted portion is the sentence beginning “If a published result has an effect of 1 …” and ending “… then I would get something like 1.”
Meme of Michael Scott saying "I love inside jokes I hope to be a part of one someday" with the text replaced to say "I love deleting my account when I get 1000 followers I hope to get 1000 followers someday"
Excerpt from: Lehmann, Erich Leo, ed. Elements of large-sample theory. New York, NY: Springer New York, 1999. 4.1 Confidence intervals Rather than testing that a parameter ( \theta ) of interest has a specified value ( \theta_0 ) (or falls short of ( \theta_0 )), one will often want to estimate ( \theta ). Two approaches to the estimation problem are (i) estimation by confidence intervals with which we shall be concerned in the present section and (ii) point estimation which will be discussed in Section 2. Confidence intervals for ( \theta ) are random intervals [ \big(\underline{\theta}(X_1, \ldots, X_n), \overline{\theta}(X_1, \ldots, X_n)\big) ] 220  4. Estimation which have a guaranteed probability of containing the unknown parameter ( \theta ), i.e., for which [ P_\theta \big[ \underline{\theta}(X_1, \ldots, X_n) \le \theta \le \overline{\theta}(X_1, \ldots, X_n) \big] \ge \gamma \text{ for all } \theta ] for some preassigned confidence level ( \gamma ). Since excessively large coverage probability requires unnecessarily long intervals, we shall instead require that [ (4.1.1) \qquad \inf_\theta P_\theta \big[ \underline{\theta}(X_1, \ldots, X_n) \le \theta \le \overline{\theta}(X_1, \ldots, X_n) \big] = \gamma. ]
“Promotional image with the text ‘Watch the Alexa+ big game ad’ overlaid on a close-up of a person’s face in a blue-toned scene.”
Meme-style image from a movie scene. In a dark industrial setting, a muscular masked man stands with arms outstretched, blocking a rushing wall of water. Over his body is white text reading, ‘Soldiers with $693bn of financial backing and state of the art weapons from the Dept of Defense.’ To the right, a much smaller person in a bright pink bodysuit stands casually with arms open. Text next to them reads, ‘Me protecting my 650 sq ft second floor walk up studio from unconstitutional quartering.’
Source: https://bsky.app/profile/sfcpoll.bsky.social/post/3mbos452to22q Table showing results of a Washington Post poll conducted Jan. 3–4, 2026, among 1,004 U.S. adults (±3.5% margin of error). Question asks how much respondents have heard about the U.S. launching air strikes against Venezuela and capturing President Nicolás Maduro and his wife. Results: 60% have heard a great deal or a good amount (27% a great deal, 34% a good amount). 40% have heard a little or nothing (31% a little, 9% nothing).
Screenshot of a tweet by owen cyclops (@owenbroadcast) that jokes about Hallmark movies: the bad guy in hallmark movies is a boyfriend who is like “uh no babe i cant drop everything + leave work this weekend im about to close a deal for ten million dollars that will set us up for life” and the good guy is a guy who is just standing there when she gets to her hometown The tweet is dated Nov 4, 2022.
Screenshot of a scholarly article page titled ‘ADVANCES.’ The page contains dense academic text discussing research on exceptional performance. A large middle paragraph is highlighted in light blue, emphasizing that early exceptional performers and later exceptional performers are usually different individuals. Examples note that top youth and adult performers in chess, education, and international athletics differ by nearly 90 percent over time. Highlighted text: "Early exceptional performers and later exceptional performers within a domain are rarely the same individuals but are largely discrete populations over time. For example, world top-10 youth chess players and later world top-10 adult chess players are nearly 90% different individuals across time. Top secondary students and later top university students are also nearly 90% different people. Likewise, international-level youth athletes and later international-level adult athletes are nearly 90% different individuals."
A screenshot of a social-media post showing the display name “SE Gyges,” the handle “@segyges.bsky.s…,” and a circular profile image with a gray geometric pattern. Below the name and handle, the post text reads: “i think thielism is good.”
**Alternative text:** A screenshot of a news article titled “Biden Administration Announces ‘Final’ Student Loan Debt Relief Approvals.” The byline reads: “By Maria Carrasco, NASFAA Staff Reporter.” The article explains that shortly before the Trump administration takes office, the Biden administration announced its final approvals of student loan forgiveness through the Income-Based Repayment (IBR) plan and borrower-defense claims. It states that the Department of Education approved $600 million in forgiveness for 4,550 borrowers under IBR and additional forgiveness for 4,100 former DeVry University students, with no total amount specified for that portion. The article concludes by noting that the administration has approved $188.8 billion in forgiveness for 5.3 million borrowers since taking office.
Source: https://hai.stanford.edu/ai-index/2025-ai-index-report/research-and-development A line chart shows the steep decline in inference prices for various large language models from 2022 to 2025. The vertical axis uses a log scale for price per million tokens, and the horizontal axis shows publication dates. Multiple colored lines represent benchmarks such as GPT-3.5-level MMLU, GPT-4-level HumanEval, PhD-level GPQA, and LMSYS Chatbot Arena Elo. All lines slope sharply downward over time. Models labeled GPT-3.5, GPT-4-0314, GPT-4o-2024-05, Claude-3.5-Sonnet-2024-06, Llama-3.1-Instruct-8B, DeepSeek-V3, Phi 4, and Gemini-1.5-Flash-8B appear along the trend. The chart illustrates dramatic cost reductions, from double-digit dollars per million tokens in 2022 to well under one dollar by late 2024.
A subway ad for Apex Technical School shows two panels. On the left, large text reads “Choose your trade at Apex,” with photos of two people working: one using a power tool and another leaning over an engine. On the right, a colorful panel displays the Apex Technical School logo, icons representing different trades, a list of benefits such as hands-on training and day/evening classes, the school’s address and phone number, and a “Career training” badge.
Andy Rooney
A data table titled “2017–2022 motor vehicle fatality data, Uber-related and United States” compares Uber-related fatality metrics with national metrics across three two-year periods: 2017–2018, 2019–2020, and 2021–2022. Columns list: Year; number of Uber-related fatalities; frequency of fatalities by number of trips; percentage of trips involving fatalities; Uber’s fatality rate per 100 million vehicle miles traveled; the national fatality rate per 100 million miles; and the rate change from the prior period for both Uber and the national figures. Values show Uber fatalities of 107, 101, and 153 in the three periods, with fatality frequencies ranging from roughly 1 in 22 million to 1 in 12 million trips. Uber’s fatality rate increases from 0.58 to 0.62 to 0.87 across periods, while national rates rise from 1.15 to 1.22 to 1.35. Rate-change columns show increases for later periods (e.g., +7% and +40% for Uber; +6% and +11% nationally), with dashes for the baseline period.
Meme of rapper Drake expressing preferences. In the top panel, Drake turns away with a disgusted expression, rejecting the label “DIFFERENCE-IN-DIFFERENCES.” In the bottom panel, he smiles and points approvingly at the label “DIFFERENCES.” The meme humorously suggests a preference for simple differences over the statistical method “difference-in-differences.” (it's a sarcastic meme, this table is just showing differences from a handful of cities. At the very least you'd want to look at difference-in-differences over a larger set of cities, and even that would be considered pretty crude by contemporary research standards.)
The image depicts a futuristic cityscape with sleek, modern architecture and advanced technology. Tall glass skyscrapers with curved designs rise in the background alongside domed and geometric structures made of reflective materials. Numerous flying vehicles move through the sky, while streamlined trains glide on elevated tracks. In the foreground, a person walks along a landscaped green area near a reflective pool, accompanied by a small robotic dog. The scene is bright and sunny, with a clean, high-tech atmosphere suggesting an advanced, utopian future.
Figure 1 in Faletto (2025) (https://arxiv.org/abs/2312.05985). The image shows a triangular grid plotted on a coordinate system, where the horizontal axis (labeled *t*) represents time and the vertical axis (labeled *r*) represents cohorts. The grid displays points connected by lines, forming a right triangle aligned along the diagonal from the lower left to the upper right. Each point is labeled with a symbol of the form **τ̂<sub>rt</sub>**, representing estimated marginal average treatment effect terms. The points appear as follows: * Along *r = 2*: τ̂₂₂, τ̂₂₃, τ̂₂₄, τ̂₂₅, τ̂₂₆ * Along *r = 3*: τ̂₃₃, τ̂₃₄, τ̂₃₅, τ̂₃₆ * Along *r = 4*: τ̂₄₄, τ̂₄₅, τ̂₄₆ * Along *r = 5*: τ̂₅₅, τ̂₅₆ Each horizontal line connects treatment effects for a given cohort across time, while the diagonal connects τ̂₂₂, τ̂₃₃, τ̂₄₄, and τ̂₅₅. The figure caption explains that this visualization illustrates which estimated treatment effect terms τ̂<sub>rt</sub> are penalized toward each other in the FETWFE model, where *T = 6* and *R = {2,…,5}*. The method assumes that linked treatment effects are close together or equal.
Difference-in-differences diagram. Pre-intervention means: treatment 50 (β₀+β₁), comparison 35 (β₀). Post-intervention: comparison 55 (β₀+β₂); treatment observed 85 (β₀+β₁+β₂+β₃). A dashed green line shows the treatment counterfactual at 70 (β₀+β₁+β₂). Gray lines connect pre to post for observed trends; x-axis labeled “Pre-intervention” and “Post-intervention.”
The image shows the title and citation information from the top of an academic journal article. Text reads: **JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS** **2021, VOL. 30, NO. 2, 503–517** *[https://doi.org/10.1080/10618600.2020.1831930](https://doi.org/10.1080/10618600.2020.1831930)* Below that, the article title appears in large bold blue text: **Local Linear Forests** The author line lists: Rina Friedberg¹, Julie Tibshirani², Susan Athey³, and Stefan Wager³ — with superscript letters **a**, **b**, and **c** in blue following the names.
“To avoid this issue, we cast forests as a type of adaptive locally weighted estimators that first use a forest to calculate a weighted set of neighbors for each test point 𝑥 x, and then solve a plug-in version of the estimating equation (1) using these neighbors. Section 2.1 gives a detailed treatment of this perspective.”
Screenshot of a Wikipedia article section discussing Senator Joe Manchin’s opposition to extending the child tax credit. The text explains that Senate Democrats supported the enhanced credit, citing reductions in child poverty and food insecurity. Manchin demanded stricter work requirements and criticized reliance on temporary funding. He rejected claims that he wanted to eliminate the credit, calling them false, but expressed concerns about misuse of funds. The Census Bureau is cited as reporting that most families used the payments for essentials like food, clothing, utilities, and education. Numerous reference citations in blue are interspersed throughout the text.
Headline styled as a debate introduction. It reads: "Point/Counterpoint: This War Will Destabilize The Entire Mideast Region And Set Off A Global Shockwave Of Anti-Americanism vs. No It Won’t." It's from this 2003 Onion article about the Iraq war https://theonion.com/this-war-will-destabilize-the-entire-mideast-region-and-1819594296/
A man sitting in a chair with his legs stretched out, holding a drink can and a cigarette in one hand while pointing with the other. He appears to be reacting strongly to something off-screen.
A screenshot of a Twitter post embedded in an article. The article text above the post discusses persuasion, noting that persuadable people are often bystanders who rarely participate in online debates but may be influenced by what they see. It asks whether readers will leave such bystanders exposed only to certain arguments. The Twitter post is from an account called *The Transformed Wife (@godlywomanhood)* and reads: “Men’s role is to provide. Women’s role is to nurture. Our society has lost SO much (especially the children) because most women no longer nurture. They would rather provide. 😟” The post shows engagement metrics: 63 comments, 67 retweets, 210 likes, 6 bookmarks, and a share icon.
Screenshot of the ChatGPT interface showing the "Alt Text Writer 5" GPT menu open. The dropdown menu lists options including Model, New chat, About, Edit GPT, Hide from sidebar, Copy link, Review GPT, and Report GPT. The "Review GPT" selection is highlighted. On the right, the Alt Text Writer page displays its name, creator "Gregory M Faletto," and description explaining it generates W3C-compliant alt text when an image is uploaded.
A grid of official portrait photographs of U.S. presidents from 1929 to the present, each labeled with name and years in office. Top row: Herbert Hoover (1929–1933), Franklin D. Roosevelt (1933–1945), Harry S. Truman (1945–1953), Dwight D. Eisenhower (1953–1961). Second row: John F. Kennedy (1961–1963), Lyndon B. Johnson (1963–1969), Richard Nixon (1969–1974), Gerald R. Ford (1974–1977). Third row: Jimmy Carter (1977–1981), Ronald Reagan (1981–1989), George H. W. Bush (1989–1993), Bill Clinton (1993–2001). Bottom row: George W. Bush (2001–2009), Barack Obama (2009–2017), Donald Trump (2017–2021), Joe Biden (2021–present).
A cartoon scene showing Squidward from behind as he peers through six horizontal window blinds. Outside, SpongeBob SquarePants and Patrick Star stand on a light-colored surface, arms raised and mid-jump, as if dancing or cheering.
A dark-mode screenshot of the ChatGPT model-selection dropdown. At the top it reads “ChatGPT o3” with a downward arrow. Under a “Models” heading are four options: GPT-4o – Great for most tasks o3 – Uses advanced reasoning (checked) o4-mini – Fastest at advanced reasoning o4-mini-high – Great at coding and visual reasoning Below them is an “Other models >” button, which is expanded to reveal two more entries: GPT-4.1 – Great for quick coding and analysis GPT-4.1-mini – Faster for everyday tasks
Screenshot of R code and console output showing the use of the `fetwfe` package to estimate a Fused Extended Two-Way Fixed Effects model on panel data and print a summary that includes an overall average treatment effect (ATT), cohort-specific treatment effects (CATT) for three cohorts, and model details (number of units, time periods, treated cohorts, covariates, features, selected size, and tuning parameter). --- The image is divided into two main sections: 1. **R code:** * `library(fetwfe)` * A call to `fetwfe()` with arguments: * `pdata = pdata` (panel dataset) * `time_var = "time"` (time variable) * `unit_var = "unit"` (unit identifier) * `treatment = "treated"` (treatment dummy indicator) * `response = "response"` (response variable) * A call to `summary(result)` to display the estimation results. 2. **Console output (Summary of Fused Extended Two-Way Fixed Effects):** * **Overall ATT:** 29.4540 (SE = 5.6503, 95% CI = \[18.3796, 40.5283]) * **CATT (preview) table:** * Cohort 11: Estimated TE = 13.93837, SE = 0.50308, 95% CI = \[12.95236, 14.92438] * Cohort 12: Estimated TE = 19.65173, SE = 0.41869, 95% CI = \[18.83111, 20.47235] * Cohort 7: Estimated TE = 56.13475, SE = 0.37109, 95% CI = \[55.40744, 56.86207] * **Model Details:** * Units (N): 30 * Time periods (T): 20 * Treated cohorts (R): 3 * Covariates (d): 0 * Features (p): 55 * Selected size: 17 * Lambda\*: 0.8412 This image illustrates both the code needed to run the estimator and the key numerical results of the fixed-effects analysis.
An image showing a screenshot of R code in a monospaced font. The code reads: ``` coefs <- genCoefs(R = 3, T = 4, d = 2, density = 0.1, eff_size = 2, seed = 2025) result_piped <- coefs |> simulateData(N = 60, sig_eps_sq = 1, sig_eps_c_sq = 1) |> fetwfeWithSimulatedData() cat("Estimated Overall ATT from piped workflow:", result_piped$att_hat, "\n") #> Estimated Overall ATT from piped workflow: -0.7045089 true_tes_piped <- coefs |> getTes() # Print the true overall treatment effect cat("True Overall ATT:", true_tes_piped$att_true, "\n") #> True Overall ATT: -0.6666667 ``` The code first generates coefficient values with `genCoefs`, then pipes them through `simulateData` and `fetwfeWithSimulatedData`, printing an estimated average treatment‐on‐the‐treated (ATT) of approximately –0.7045. It then uses `getTes()` on the same coefficients to retrieve and print the true ATT of –0.6667.
A screenshot of an R console showing code and output for an extended two-way fixed effects analysis. The code loads the `fetwfe` and `did` packages, loads the `mpdta` dataset, and transforms it into `pdata` with `attgtToFetwfeDf`, specifying outcome `lemp`, time variable `year`, unit identifier `countyreal`, treatment onset `first.treat`, and covariate `lpop`. Then it runs `etwfe(pdata, time_var="time_var", unit_var="unit_var", treatment="treatment", response="response", covs="lpop")`. Below is the “Extended Two-Way Fixed Effects Results”: * **Overall Average Treatment Effect (ATT):** – Estimate: –0.0452 – Std. Error: 0.0145 – 95 % CI: \[–0.0736, –0.0167] * **Cohort Average Treatment Effects (CATT):** | Cohort | Estimated TE | SE | 95 % CI low | 95 % CI high | | ------ | ------------ | ---------- | ----------- | ------------ | | 2004 | –0.08762696 | 0.03555885 | –0.15732102 | –0.01793290 | | 2006 | –0.02127833 | 0.02128938 | –0.06300475 | 0.02044809 | | 2007 | –0.04595453 | 0.01633327 | –0.07796715 | –0.01394190 | * **Model Details:** – Units (N): 500 – Time periods (T): 5 – Treated cohorts (R): 3 – Covariates (d): 1 – Features (p): 29
Screenshot of a dark-themed macOS Terminal window running “Gemini.” Across the top, large ASCII-art letters spell “GEMINI” in a pastel gradient from blue (left) through purple to pink (right). Beneath, white text on black lists four “Tips for getting started,” such as asking questions, being specific, creating a GEMINI.md file, and using /help. The prompt then shows a user command instructing Gemini to load a PDF on Oracle efficient variable selection and a question.txt file for feedback. A bordered status box follows, with a green checkmark and the message “ReadManyFiles Will attempt to read and concatenate files using patterns…,” indicating the files are being processed from the user’s Documents directory.
A dark-styled formatting sidebar showing three top tabs—Format (selected), Animate, Document—and, within Format, three subtabs: Style, Text, and Arrange (blue highlight on Arrange). At the bottom are “Position” fields with X = 75 pt and Y = 73 pt. These can be used to precisely position an object in Keynote. Remaining image: Under “Arrange” are four layer buttons: “Back” and “Front” on the left, “Backward” and “Forward” on the right. Below is an “Align” dropdown (set to Align) and a disabled “Distribute” control. Further down are “Size” fields with Width = 575 pt and Height = 229 pt, and an unchecked “Constrain proportions” box.
A screenshot of a journal article header. At the top is a light gray label “JOURNAL ARTICLE.” Below it is the title in large bold type: Double/debiased machine learning for treatment and structural parameters To the right of the title is a blue button labeled “Get access.” Under the title are the authors, each name in blue and separated by commas: Victor Chernozhukov, Denis Chetverikov, Mert Demirer, Esther Duflo, Christian Hansen, Whitney Newey, James Robins Below the authors is the citation line in italics and regular font: The Econometrics Journal, Volume 21, Issue 1, 1 February 2018, Pages C1–C68, https://doi.org/10.1111/ectj.12097 At the bottom left is “Published: 16 January 2018” and at the bottom right is a dropdown label “Article history ▼.”
A screenshot with white text on a dark background showing the heading and formula for the AIPW estimator. It reads: “The augmented inverse propensity weighting (AIPW) estimator for the average treatment effect is ψ̂_AIPW = (1/n) ∑ⁿ_{i=1} [   m̂₁(X_i) – m̂₀(X_i)   + W_i·(Y_i – m̂₁(X_i)) / e(X_i)   – (1–W_i)·(Y_i – m̂₀(X_i)) / (1–e(X_i)) ], where m̂₁(X_i) is a model for the treated potential outcome given covariates X_i, m̂₀(X_i) is the predicted untreated potential outcome, and e(X_i) := P(W_i = 1 | X_i) is the propensity score.
A photograph of a textbook excerpt presenting Theorem 7.2 and its interpretation in a regression-for-experiments context. The content reads: Results similar to Theorem 7.1 can be obtained for this case. The least squares estimator 𝜏 ^ o l s τ ^ ols is consistent for the average treatment effect 𝜏 𝑠 𝑝 τ sp ​ , and inference can be based on least squares methods. Theorem 7.2 Suppose we conduct a completely randomized experiment in a random sample from a super-population. Then (i) 𝜏 ∗ = 𝜏 𝑠 𝑝 , τ ∗ =τ sp ​ , and (ii) 𝑁   ( 𝜏 ^ o l s − 𝜏 𝑠 𝑝 ) → 𝑑 𝑁  ⁣ ( 0 ,    𝐸 𝑠 𝑝 [ ( 𝑊 𝑖 − 𝑝 ) 2   ( 𝑌 𝑖 𝑜 𝑏 𝑠 − 𝛼 ∗ − 𝜏 𝑠 𝑝   𝑊 𝑖 − 𝑋 𝑖 𝛽 ∗ − 𝑊 𝑖   ( 𝑋 𝑖 − 𝜇 𝑋 )   𝛾 ∗ ) 2 ] 𝑝 2   ( 1 − 𝑝 ) 2 ) . N ​ ( τ ^ ols −τ sp ​ ) d ​ N(0, p 2 (1−p) 2 E sp ​ [(W i ​ −p) 2 (Y i obs ​ −α ∗ −τ sp ​ W i ​ −X i ​ β ∗ −W i ​ (X i ​ −μ X ​ )γ ∗ ) 2 ] ​ ).
A two‐page excerpt from a statistics textbook showing a linear regression model with treatment–covariate interactions and its least‐squares and population parameter estimates. On the first page the observed‐outcome model is written as Yᵢᵒᵇˢ = α + τ·Wᵢ + Xᵢ β + Wᵢ·(Xᵢ – X̄) γ + εᵢ, followed by the definition of the ordinary‐least‐squares estimators (α⁽ᵒˡˢ⁾, τ⁽ᵒˡˢ⁾, β⁽ᵒˡˢ⁾, γ⁽ᵒˡˢ⁾) as the values minimizing ∑ᵢ₌₁ᴺ [Yᵢᵒᵇˢ – α – τ·Wᵢ – Xᵢ β – Wᵢ·(Xᵢ – X̄) γ]². On the opposite page, titled “Regression Methods for Completely Randomized Experiments,” the corresponding population parameters (α*, τ*, β*, γ*) are defined analogously as the minimizers of Eₛₚ[ (Yᵢᵒᵇˢ – α – τ·Wᵢ – Xᵢ β – Wᵢ·(Xᵢ – μₓ) γ)² ].
Dr. Steve Brule