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- March 2022 (Revised January 2025)
- Technical Note
Linear Regression
This note provides an overview of linear regression for an introductory data science course. It begins with a discussion of correlation, and explains why correlation does not necessarily imply causation. The note then describes the method of least squares, and how to... View Details
Keywords: Data Science; Linear Regression; Mathematical Modeling; Mathematical Methods; Analytics and Data Science
Bojinov, Iavor I., Michael Parzen, and Paul Hamilton. "Linear Regression." Harvard Business School Technical Note 622-100, March 2022. (Revised January 2025.)
- June 2021
- Technical Note
Introduction to Linear Regression
By: Michael Parzen and Paul Hamilton
This technical note introduces (from an applied point of view) the theory and application of simple and multiple linear regression. The motivation for the model is introduced, as well as how to interpret the summary output with regard to prediction and statistical... View Details
- 2022
- Working Paper
A Linear Panel Model with Heterogeneous Coefficients and Variation in Exposure
By: Jesse M. Shapiro and Liyang Sun
Linear panel models featuring unit and time fixed effects appear in many areas of empirical economics. An active literature studies the interpretation of the ordinary least squares estimator of the model, commonly called the two-way fixed effects (TWFE) estimator, in... View Details
Shapiro, Jesse M., and Liyang Sun. "A Linear Panel Model with Heterogeneous Coefficients and Variation in Exposure." NBER Working Paper Series, No. 29976, April 2022.
- 2024
- Working Paper
Anytime-Valid Inference in Linear Models and Regression-Adjusted Causal Inference
By: Michael Lindon, Dae Woong Ham, Martin Tingley and Iavor I. Bojinov
Linear regression adjustment is commonly used to analyze randomized controlled experiments due to its efficiency and robustness against model misspecification. Current testing and interval estimation procedures leverage the asymptotic distribution of such estimators to... View Details
Lindon, Michael, Dae Woong Ham, Martin Tingley, and Iavor I. Bojinov. "Anytime-Valid Inference in Linear Models and Regression-Adjusted Causal Inference." Harvard Business School Working Paper, No. 24-060, March 2024.
- November 1990 (Revised November 1992)
- Background Note
Note on Linear Programming
Gives an elementary introduction to formulating linear programming models and interpreting linear programmings output. Other aspects of linear programming are discussed briefly. View Details
Keywords: Mathematical Methods
Eckstein, Jonathan. "Note on Linear Programming." Harvard Business School Background Note 191-085, November 1990. (Revised November 1992.)
- May 2020
- Article
Scalable Holistic Linear Regression
By: Dimitris Bertsimas and Michael Lingzhi Li
We propose a new scalable algorithm for holistic linear regression building on Bertsimas & King (2016). Specifically, we develop new theory to model significance and multicollinearity as lazy constraints rather than checking the conditions iteratively. The resulting... View Details
Bertsimas, Dimitris, and Michael Lingzhi Li. "Scalable Holistic Linear Regression." Operations Research Letters 48, no. 3 (May 2020): 203–208.
- May 2024
- Teaching Note
AI21 Labs in 2023: Strategy for Generative AI
By: David Yoffie
Teaching Note for HBS Case 724-383. The case has 3 important teaching purposes: First, what are the advantages and disadvantages of imitation? (e.g., Should AI21 imitate OpenAI with a chatbot?) Second, what are the advantages and disadvantages of keeping new technology... View Details
- 2018
- Working Paper
Bayesian Ensembles of Binary-Event Forecasts: When Is It Appropriate to Extremize or Anti-Extremize?
By: Kenneth C. Lichtendahl Jr., Yael Grushka-Cockayne, Victor Richmond R. Jose and Robert L. Winkler
Many organizations face critical decisions that rely on forecasts of binary events. In these situations, organizations often gather forecasts from multiple experts or models and average those forecasts to produce a single aggregate forecast. Because the average... View Details
Keywords: Forecast Aggregation; Linear Opinion Pool; Generalized Additive Model; Generalized Linear Model; Stacking.; Forecasting and Prediction
Lichtendahl, Kenneth C., Jr., Yael Grushka-Cockayne, Victor Richmond R. Jose, and Robert L. Winkler. "Bayesian Ensembles of Binary-Event Forecasts: When Is It Appropriate to Extremize or Anti-Extremize?" Harvard Business School Working Paper, No. 19-041, October 2018.
- September–October 2024
- Article
The Crowdless Future? Generative AI and Creative Problem-Solving
The rapid advances in generative artificial intelligence (AI) open up attractive opportunities for creative problem-solving through human-guided AI partnerships. To explore this potential, we initiated a crowdsourcing challenge focused on sustainable, circular economy... View Details
Keywords: Large Language Models; Generative Ai; Crowdsourcing; AI and Machine Learning; Creativity; Technological Innovation
Boussioux, Léonard, Jacqueline N. Lane, Miaomiao Zhang, Vladimir Jacimovic, and Karim R. Lakhani. "The Crowdless Future? Generative AI and Creative Problem-Solving." Organization Science 35, no. 5 (September–October 2024): 1589–1607.
- Article
Temporary General Equilibrium in a Sequential Trading Model with Spot and Futures Transactions
By: Jerry R. Green
The existence of an equilibrium is proven for a two-period model in which there are spot transactions and futures transactions in the first period and spot markets in the second period. Prices at that date are viewed with subjective uncertainty by all traders. This... View Details
Green, Jerry R. "Temporary General Equilibrium in a Sequential Trading Model with Spot and Futures Transactions." Econometrica 41, no. 6 (November 1973): 1103–1123.
- July 2023 (Revised July 2023)
- Background Note
Generative AI Value Chain
By: Andy Wu and Matt Higgins
Generative AI refers to a type of artificial intelligence (AI) that can create new content (e.g., text, image, or audio) in response to a prompt from a user. ChatGPT, Bard, and Claude are examples of text generating AIs, and DALL-E, Midjourney, and Stable Diffusion are... View Details
Keywords: AI; Artificial Intelligence; Model; Hardware; Data Centers; AI and Machine Learning; Applications and Software; Analytics and Data Science; Value
Wu, Andy, and Matt Higgins. "Generative AI Value Chain." Harvard Business School Background Note 724-355, July 2023. (Revised July 2023.)
- 1980
- Article
Consumer Impulse Purchase and Credit Card Usage: An Empirical Examination Using the Log Linear Model
By: Rohit Deshpandé and S. Krishnan
Most of the work in impulse purchase behavior has investigated the association of socioeconomic variables and unplanned purchases with equivocal results. This paper examines the interrelationship between impulse purchases, credit card usage, cost of items bought, and... View Details
- January 2008 (Revised November 2009)
- Case
Linear Air: Creating the Air Taxi Industry
Linear Air is an air taxi start-up established to take advantage of the emergence of Very Light Jets, which incorporate new technology that cuts jet operating costs by about 40%. Air taxis could make use of the 5400 smaller regional airports throughout the US,... View Details
Keywords: Business Model; Business Startups; Entrepreneurship; Disruptive Innovation; Product Launch; Industry Structures; Competition; Air Transportation Industry
Tripsas, Mary, Davin Chow, Adam Prewett, and Kevin Yttre. "Linear Air: Creating the Air Taxi Industry." Harvard Business School Case 808-107, January 2008. (Revised November 2009.)
- 2024
- Working Paper
The Crowdless Future? Generative AI and Creative Problem Solving
The rapid advances in generative artificial intelligence (AI) open up attractive opportunities for creative problem-solving through human-guided AI partnerships. To explore this potential, we initiated a crowdsourcing challenge focused on sustainable, circular economy... View Details
Keywords: Large Language Models; Crowdsourcing; Generative Ai; Creative Problem-solving; Organizational Search; AI-in-the-loop; Prompt Engineering; AI and Machine Learning; Innovation and Invention
Boussioux, Léonard, Jacqueline N. Lane, Miaomiao Zhang, Vladimir Jacimovic, and Karim R. Lakhani. "The Crowdless Future? Generative AI and Creative Problem Solving." Harvard Business School Working Paper, No. 24-005, July 2023. (Revised July 2024.)
- March 1999 (Revised February 2000)
- Case
Patient Care Delivery Model at the Massachusetts General Hospital, The
By: Amy C. Edmondson, Richard M.J. Bohmer and Emily Heaphy
Examines the implementation of a new patient care delivery model at Massachusetts General Hospital. Uses clinical and financial data to examine different choices for staffing non-physician health care professionals and to understand the challenges of managing change... View Details
Keywords: Change Management; Service Delivery; Health Care and Treatment; Health Industry; Massachusetts
Edmondson, Amy C., Richard M.J. Bohmer, and Emily Heaphy. "Patient Care Delivery Model at the Massachusetts General Hospital, The." Harvard Business School Case 699-154, March 1999. (Revised February 2000.)
- Article
Learning Models for Actionable Recourse
By: Alexis Ross, Himabindu Lakkaraju and Osbert Bastani
As machine learning models are increasingly deployed in high-stakes domains such as legal and financial decision-making, there has been growing interest in post-hoc methods for generating counterfactual explanations. Such explanations provide individuals adversely... View Details
Ross, Alexis, Himabindu Lakkaraju, and Osbert Bastani. "Learning Models for Actionable Recourse." Advances in Neural Information Processing Systems (NeurIPS) 34 (2021).
- 2020
- Working Paper
A General Theory of Identification
By: Iavor Bojinov and Guillaume Basse
What does it mean to say that a quantity is identifiable from the data? Statisticians seem to agree
on a definition in the context of parametric statistical models — roughly, a parameter θ in a model
P = {Pθ : θ ∈ Θ} is identifiable if the mapping θ 7→ Pθ is injective.... View Details
Bojinov, Iavor, and Guillaume Basse. "A General Theory of Identification." Harvard Business School Working Paper, No. 20-086, February 2020.
- July 2012
- Article
Discrete Choice Cannot Generate Demand That Is Additively Separable in Own Price
By: Sonia Jaffe and Scott Duke Kominers
We show that in a unit demand discrete choice framework with at least three goods, demand cannot be additively separable in own price. This result sharpens the analogous result of Jaffe and Weyl (2010) in the case of linear demand and has implications for testing of... View Details
Keywords: Discrete Choice; Unit Demand; Separable Demand; Linear Demand; Demand and Consumers; Market Design; Mathematical Methods; Economics
Jaffe, Sonia, and Scott Duke Kominers. "Discrete Choice Cannot Generate Demand That Is Additively Separable in Own Price." Economics Letters 116, no. 1 (July 2012): 129–132.
- 2024
- Working Paper
Scaling Core Earnings Measurement with Large Language Models
By: Matthew Shaffer and Charles CY Wang
We study the application of large language models (LLMs) to the estimation of core earnings, i.e., a firm's persistent profitability from its core business activities. This construct is central to investors' assessments of economic performance and valuations. However,... View Details
Keywords: Large Language Models; AI and Machine Learning; Accounting; Profit; Corporate Disclosure; Analytics and Data Science; Measurement and Metrics
Shaffer, Matthew, and Charles CY Wang. "Scaling Core Earnings Measurement with Large Language Models." Working Paper, November 2024.
- 1995
- Chapter
Dynamic General Equilibrium Models with Imperfectly Competitive Product Markets
By: Julio J. Rotemberg and Michael Woodford