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March 2022 (Revised July 2022)
Technical Note

Linear Regression

By: Iavor I. Bojinov, Michael Parzen and Paul Hamilton

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

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

Keywords: Linear Regression; Regression; Analysis; Forecasting and Prediction; Risk and Uncertainty; Theory; Compensation and Benefits; Mathematical Methods; Analytics and Data Science

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

Keywords: Econometric Models; Mathematical Methods

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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

Keywords: Mathematical Methods; Analytics and Data Science

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

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

Keywords: Mathematical Methods; Analytics and Data Science

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

Keywords: AI; Generative Ai; Generative Models; AI and Machine Learning; Innovation Strategy; Growth and Development Strategy; Business Model; Business Startups; Open Source Distribution; Competitive Advantage; Technology Industry; Israel

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

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

Keywords: Equilibrium; Sequential Trading; Econometric Models

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

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

Keywords: Consumer Behavior; Mathematical Methods; Credit Cards; Income

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

2024
Working Paper

The Crowdless Future? Generative AI and Creative Problem Solving

By: Léonard Boussioux, Jacqueline N. Lane, Miaomiao Zhang, Vladimir Jacimovic and Karim R. Lakhani

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

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

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

Keywords: Identification; Econometric Models; Analytics and Data Science; Theory

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

Keywords: Machine Learning Models; Recourse; Algorithm; Mathematical Methods

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

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1995
Chapter

Dynamic General Equilibrium Models with Imperfectly Competitive Product Markets

By: Julio J. Rotemberg and Michael Woodford

Keywords: Mathematical Methods; Competition; Markets

August 1976
Article

A Model of Economic Growth With Altruism Between Generations

By: Elon Kohlberg

Keywords: Economics; Growth and Development; Relationships

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June 2015
Supplement

Generating Higher Value at IBM (A): EPS Forecasting Model

By: Benjamin C. Esty and Scott Mayfield

This case analyzes IBM's financial performance and its capital allocation decisions over a 10-year period from 2004-2013, during which IBM returned more than $140B to shareholders through a combination of dividends and share repurchases. During this time, CEO Sam... View Details

Keywords: Dividends; Share Repurchases; Earnings Guidance; Financial Statement Analysis; Financial Ratios; Payout Policy; Earnings Per Share (EPS); Earnings Management; Change Management; Leadership; Transformation; Financial Strategy

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