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- All HBS Web
(297)
- News (9)
- Research (259)
- Events (1)
- Multimedia (1)
- Faculty Publications (181)
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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
- 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.
- 2023
- Article
Exploiting Discovered Regression Discontinuities to Debias Conditioned-on-observable Estimators
By: Benjamin Jakubowski, Siram Somanchi, Edward McFowland III and Daniel B. Neill
Regression discontinuity (RD) designs are widely used to estimate causal effects in the absence of a randomized experiment. However, standard approaches to RD analysis face two significant limitations. First, they require a priori knowledge of discontinuities in... View Details
Jakubowski, Benjamin, Siram Somanchi, Edward McFowland III, and Daniel B. Neill. "Exploiting Discovered Regression Discontinuities to Debias Conditioned-on-observable Estimators." Journal of Machine Learning Research 24, no. 133 (2023): 1–57.
- February 1991 (Revised February 1993)
- Background Note
Regression Analysis
By: David E. Bell
Provides a relatively simple introduction to multivariate regression analysis. View Details
Keywords: Mathematical Methods
Bell, David E. "Regression Analysis." Harvard Business School Background Note 191-117, February 1991. (Revised February 1993.)
- May 2022
- Exercise
Regression Exercises
By: David E. Bell
Bell, David E. "Regression Exercises." Harvard Business School Exercise 522-098, May 2022.
- 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.
- October 1993 (Revised August 1996)
- Background Note
Forecasting with Regression Analysis
Provides an example of regression in one of its most important roles. Relating probabilistic forecasts based on past data to decision analysis. View Details
Keywords: Management Analysis, Tools, and Techniques; Analytics and Data Science; Forecasting and Prediction; Mathematical Methods
Schleifer, Arthur, Jr. "Forecasting with Regression Analysis." Harvard Business School Background Note 894-007, October 1993. (Revised August 1996.)
- 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.)
- 2006
- Chapter
Advanced Regression Models
By: Raghuram Iyengar and Sunil Gupta
Keywords: Mathematical Methods
- February 2005 (Revised March 2005)
- Background Note
Simple Regression Mathematics
By: Frances X. Frei and Dennis Campbell
Describes the underlying mathematics of regression. View Details
Keywords: Mathematical Methods
Frei, Frances X., and Dennis Campbell. "Simple Regression Mathematics." Harvard Business School Background Note 605-061, February 2005. (Revised March 2005.)
- 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.)
- November 1988 (Revised June 1989)
- Case
Introduction to Regression Analysis with Lotus 1-2-3 and Regress
By: David E. Bell
Bell, David E. "Introduction to Regression Analysis with Lotus 1-2-3 and Regress." Harvard Business School Case 189-110, November 1988. (Revised June 1989.)
- 2023
- Working Paper
PRIMO: Private Regression in Multiple Outcomes
By: Seth Neel
We introduce a new differentially private regression setting we call Private Regression in Multiple Outcomes (PRIMO), inspired the common situation where a data analyst wants to perform a set of l regressions while preserving privacy, where the covariates... View Details
Neel, Seth. "PRIMO: Private Regression in Multiple Outcomes." Working Paper, March 2023.
- September 2010 (Revised January 2011)
- Background Note
Using Regression Analysis to Estimate Time Equations
This note presents a simple way to estimate time equations using regression analysis in Excel. The note quickly outlines regression analysis, then presents a real-life case example from the natural gas industry that students can use to gain experience developing and... View Details
Martinez-Jerez, Francisco de Asis, and Ariel Andres Blumenkranc. "Using Regression Analysis to Estimate Time Equations." Harvard Business School Background Note 111-001, September 2010. (Revised January 2011.)
- Forthcoming
- Article
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
Bertsimas, Dimitris, Vassilis Digalakis Jr, Michael Lingzhi Li, and Omar Skali Lami. "Slowly Varying Regression Under Sparsity." Operations Research (forthcoming). (Pre-published online March 27, 2024.)
- 2001
- Other Teaching and Training Material
Interaction Terms in Regression
By: William B. Simpson and Kimball Lewis
Keywords: Mathematical Methods
Simpson, William B., and Kimball Lewis. "Interaction Terms in Regression." 2001. Electronic.
- 2005
- Working Paper
Pseudo Market Timing and Predictive Regressions
By: Malcolm Baker, Ryan Taliaferro and Jeffrey Wurgler
A number of studies claim that aggregate managerial decision variables, such as aggregate equity issuance, have power to predict stock or bond market returns. Recent research argues that these results may be driven by an aggregate time-series version of Schultz's... View Details
Keywords: Managerial Roles; Equity; Market Timing; Financial Instruments; Investment Return; Mathematical Methods
Baker, Malcolm, Ryan Taliaferro, and Jeffrey Wurgler. "Pseudo Market Timing and Predictive Regressions." NBER Working Paper Series, No. 10823, January 2005. (First Draft in 2004.)
- September 2010
- Supplement
Using Regression Analysis to Estimate Time Equations (CW)
By: Francisco de Asis Martinez-Jerez
This note presents a simple way to estimate time equations using regression analysis in Excel. The note quickly outlines regression analysis, then presents a real-life case example from the natural gas industry that students can use to gain experience developing and... View Details