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- March 2022 (Revised July 2022)
- 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 July 2022.)
- 14 Aug 2017
- Conference Presentation
A Convex Framework for Fair Regression
By: Richard Berk, Hoda Heidari, Shahin Jabbari, Matthew Joseph, Michael J. Kearns, Jamie Morgenstern, Seth Neel and Aaron Roth
We introduce a flexible family of fairness regularizers for (linear and logistic) regression problems. These regularizers all enjoy convexity, permitting fast optimization, and they span the range from notions of group fairness to strong individual fairness. By varying... View Details
Berk, Richard, Hoda Heidari, Shahin Jabbari, Matthew Joseph, Michael J. Kearns, Jamie Morgenstern, Seth Neel, and Aaron Roth. "A Convex Framework for Fair Regression." Paper presented at the 4th Workshop on Fairness, Accountability, and Transparency in Machine Learning, Special Interest Group on Knowledge Discovery and Data Mining (SIGKDD), August 14, 2017.
- March 1993 (Revised May 1994)
- Background Note
Multiplicative Regression Models
Schleifer, Arthur, Jr. "Multiplicative Regression Models." Harvard Business School Background Note 893-013, March 1993. (Revised May 1994.)
- 2006
- Chapter
Advanced Regression Models
By: Raghuram Iyengar and Sunil Gupta
Keywords: Mathematical Methods
- 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
- July 1985 (Revised May 1988)
- Background Note
Multiplicative Regression Models
Schleifer, Arthur, Jr. "Multiplicative Regression Models." Harvard Business School Background Note 186-031, July 1985. (Revised May 1988.)
- 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.
- 2022
- Working Paper
Slowly Varying Regression under Sparsity
By: Dimitris Bertsimas, Vassilis Digalakis Jr, Michael Lingzhi Li and Omar Skali Lami
We consider the problem of parameter estimation in slowly varying regression models with sparsity constraints. We formulate the problem as a mixed integer optimization problem and demonstrate that it can be reformulated exactly as a binary convex optimization problem... View Details
Keywords: Mathematical Methods
Bertsimas, Dimitris, Vassilis Digalakis Jr, Michael Lingzhi Li, and Omar Skali Lami. "Slowly Varying Regression under Sparsity." Working Paper, September 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.
- 2022
- Article
Nonparametric Subset Scanning for Detection of Heteroscedasticity
By: Charles R. Doss and Edward McFowland III
We propose Heteroscedastic Subset Scan (HSS), a novel method for identifying covariates that are responsible for violations of the homoscedasticity assumption in regression settings. Viewing the problem as one of anomalous pattern detection, we use subset scanning... View Details
Doss, Charles R., and Edward McFowland III. "Nonparametric Subset Scanning for Detection of Heteroscedasticity." Journal of Computational and Graphical Statistics 31, no. 3 (2022): 813–823.
- 2022
- Working Paper
Machine Learning Models for Prediction of Scope 3 Carbon Emissions
By: George Serafeim and Gladys Vélez Caicedo
For most organizations, the vast amount of carbon emissions occur in their supply chain and in the post-sale processing, usage, and end of life treatment of a product, collectively labelled scope 3 emissions. In this paper, we train machine learning algorithms on 15... View Details
Keywords: Carbon Emissions; Climate Change; Environment; Carbon Accounting; Machine Learning; Artificial Intelligence; Digital; Data Science; Environmental Sustainability; Environmental Management; Environmental Accounting
Serafeim, George, and Gladys Vélez Caicedo. "Machine Learning Models for Prediction of Scope 3 Carbon Emissions." Harvard Business School Working Paper, No. 22-080, June 2022.
- 2009
- Article
Modeling Expert Opinions on Food Healthfulness: A Nutrition Metric
By: Jolie M. Martin, John Beshears, Katherine L. Milkman, Max H. Bazerman and Lisa Sutherland
Research over the last several decades indicates the failure of existing nutritional labels to substantially improve the healthiness of consumers' food and beverage choices. The difficulty for policy-makers is to encapsulate a wide body of scientific knowledge in a... View Details
Keywords: Judgments; Food; Nutrition; Labels; Knowledge Use and Leverage; Demand and Consumers; Measurement and Metrics; Mathematical Methods
Martin, Jolie M., John Beshears, Katherine L. Milkman, Max H. Bazerman, and Lisa Sutherland. "Modeling Expert Opinions on Food Healthfulness: A Nutrition Metric." Journal of the American Dietetic Association 109, no. 6 (June 2009): 1088–1091.
- Article
Applying Random Coefficient Models to Strategy Research: Identifying and Exploring Firm Heterogeneous Effects
By: Juan Alcácer, Wilbur Chung, Ashton Hawk and Gonçalo Pacheco-de-Almeida
Strategy aims at understanding the differential effects of firms’ actions on performance. However, standard regression models estimate only the average effects of these actions across firms. Our paper discusses how random coefficient models (RCMs) may generate new... View Details
Alcácer, Juan, Wilbur Chung, Ashton Hawk, and Gonçalo Pacheco-de-Almeida. "Applying Random Coefficient Models to Strategy Research: Identifying and Exploring Firm Heterogeneous Effects." Strategy Science 3, no. 3 (September 2018): 481–553.
- September 2009
- Article
A Detailed Analysis of the Reduction Mammaplasty Learning Curve: A Statistical Process Model for Approaching Surgical Performance Improvement
By: Matthew Carty MD, Rodney Chan, Robert S. Huckman, Daniel C. Snow and Dennis Orgill
Background: The increased focus on quality and efficiency improvement within academic surgery has met with variable success among plastic surgeons. Traditional surgical performance metrics, such as morbidity and mortality, are insufficient to improve the... View Details
Keywords: Experience and Expertise; Health Care and Treatment; Medical Specialties; Outcome or Result; Performance Efficiency; Performance Improvement
Carty, Matthew, MD, Rodney Chan, Robert S. Huckman, Daniel C. Snow, and Dennis Orgill. "A Detailed Analysis of the Reduction Mammaplasty Learning Curve: A Statistical Process Model for Approaching Surgical Performance Improvement." Plastic and Reconstructive Surgery 124, no. 3 (September 2009): 706–714.
- 2018
- Working Paper
Forecasting Airport Transfer Passenger Flow Using Real-Time Data and Machine Learning
By: Xiaojia Guo, Yael Grushka-Cockayne and Bert De Reyck
Problem definition: In collaboration with Heathrow Airport, we develop a predictive system that generates quantile forecasts of transfer passengers’ connection times. Sampling from the distribution of individual passengers’ connection times, the system also produces... View Details
Keywords: Quantile Forecasts; Regression Tree; Copula; Passenger Flow Management; Data-driven Operations; Forecasting and Prediction; Data and Data Sets
Guo, Xiaojia, Yael Grushka-Cockayne, and Bert De Reyck. "Forecasting Airport Transfer Passenger Flow Using Real-Time Data and Machine Learning." Harvard Business School Working Paper, No. 19-040, October 2018.
- May 2010
- Article
Bye Bye Bundles: The Unbundling of Music in Digital Channels
By: Anita Elberse
Fueled by digital distribution, unbundling is prevalent in many information industries. What is the effect of this unbundling on sales? And what bundle characteristics drive this effect? I empirically examine these questions in the context of the music industry, using... View Details
Keywords: Unbundling; Bundling; Digital Distribution; System-of-equations Modeling; Sales; Distribution Channels; Framework; Mathematical Methods; Revenue; Reputation; Internet and the Web; System; E-commerce; Information Industry; Music Industry
Elberse, Anita. "Bye Bye Bundles: The Unbundling of Music in Digital Channels." Journal of Marketing 74, no. 3 (May 2010): 107–123.
- 2020
- Working Paper
Cutting the Gordian Knot of Employee Health Care Benefits and Costs: A Corporate Model Built on Employee Choice
By: Regina E. Herzlinger and Barak D. Richman
The U.S. employer-based health insurance tax exclusion created a system of employer-sponsored insurance (ESI) with limited insurance choices and transparency that may lock employed households into health plans that are costlier or different from those they prefer to... View Details
Keywords: After-tax Income; Consumer-driven Health Care; Health Care Costs; Health Insurance; Income Inequality; Tax Policy; Health Care and Treatment; Cost; Insurance; Employees; Income; Taxation; Policy; United States
Herzlinger, Regina E., and Barak D. Richman. "Cutting the Gordian Knot of Employee Health Care Benefits and Costs: A Corporate Model Built on Employee Choice." Duke Law School Public Law & Legal Theory Series, No. 2020-4, December 2019. (Revised January 2021.)
- October 2014 (Revised August 2018)
- Case
Caesars Entertainment
By: Janice H. Hammond and Aldo Sesia
This case describes the introduction of a regression analysis model for forecasting guest arrivals to Caesars Palace hotel in Las Vegas, Nevada. The company will use the forecast to staff the front desk in the hotel. The staff is unionized and the company has little... View Details
Keywords: Forecasting; Staffing; Gaming; Gaming Industry; Hotel Industry; Decision Making; Forecasting and Prediction; Human Resources; Selection and Staffing; Entertainment; Games, Gaming, and Gambling; Operations; Service Delivery; Service Operations; Accommodations Industry; Travel Industry; Tourism Industry; Food and Beverage Industry; Las Vegas
Hammond, Janice H., and Aldo Sesia. "Caesars Entertainment." Harvard Business School Case 615-031, October 2014. (Revised August 2018.)
- August 2018 (Revised September 2018)
- Supplement
Predicting Purchasing Behavior at PriceMart (B)
By: Srikant M. Datar and Caitlin N. Bowler
Supplements the (A) case. In this case, Wehunt and Morse are concerned about the logistic regression model overfitting to the training data, so they explore two methods for reducing the sensitivity of the model to the data by regularizing the coefficients of the... View Details
Keywords: Data Science; Analytics and Data Science; Analysis; Customers; Household; Forecasting and Prediction
Datar, Srikant M., and Caitlin N. Bowler. "Predicting Purchasing Behavior at PriceMart (B)." Harvard Business School Supplement 119-026, August 2018. (Revised September 2018.)
- August 2023 (Revised January 2024)
- Supplement
Arla Foods: Data-Driven Decarbonization (B)
By: Michael Parzen, Michael W. Toffel, Susan Pinckney and Amram Migdal
The case describes Arla’s history, in particular its climate change mitigation efforts, and how it implemented a price incentive system to motivate individual farms to implement scope 1 greenhouse gas emissions mitigation measures and receive a higher milk price. The... View Details
Keywords: Dairy Industry; Earnings Management; Environmental Accounting; Animal-Based Agribusiness; Mergers and Acquisitions; Decisions; Voting; Climate Change; Environmental Regulation; Environmental Sustainability; Green Technology; Pollution; Moral Sensibility; Values and Beliefs; Financial Strategy; Price; Profit; Revenue; Food; Geopolitical Units; Cross-Cultural and Cross-Border Issues; Global Strategy; Cooperative Ownership; Performance Efficiency; Performance Evaluation; Problems and Challenges; Natural Environment; Science-Based Business; Business Strategy; Commercial Banking; Cooperation; Corporate Strategy; Motivation and Incentives; Food and Beverage Industry; Agriculture and Agribusiness Industry; Europe; United Kingdom; European Union; Denmark; Sweden; Luxembourg; Belgium
Parzen, Michael, Michael W. Toffel, Susan Pinckney, and Amram Migdal. "Arla Foods: Data-Driven Decarbonization (B)." Harvard Business School Supplement 624-036, August 2023. (Revised January 2024.)