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All HBS Web
(852)
- News (77)
- Research (635)
- Events (11)
- Multimedia (4)
- Faculty Publications (633)
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- January 1975
- Article
A Lattice Fixed-Point Theorem with Constraints
By: A. E. Roth
Keywords:
Mathematical Methods
Roth, A. E. "A Lattice Fixed-Point Theorem with Constraints." Bulletin of the American Mathematical Society 81 (January 1975): 136–138.
- September–October 2023
- Article
Interpretable Matrix Completion: A Discrete Optimization Approach
By: Dimitris Bertsimas and Michael Lingzhi Li
We consider the problem of matrix completion on an n × m matrix. We introduce the problem of interpretable matrix completion that aims to provide meaningful insights for the low-rank matrix using side information. We show that the problem can be...
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Keywords:
Mathematical Methods
Bertsimas, Dimitris, and Michael Lingzhi Li. "Interpretable Matrix Completion: A Discrete Optimization Approach." INFORMS Journal on Computing 35, no. 5 (September–October 2023): 952–965.
- 2022
- Article
Exploring Counterfactual Explanations Through the Lens of Adversarial Examples: A Theoretical and Empirical Analysis.
By: Martin Pawelczyk, Chirag Agarwal, Shalmali Joshi, Sohini Upadhyay and Himabindu Lakkaraju
As machine learning (ML) models become more widely deployed in high-stakes applications, counterfactual explanations have emerged as key tools for providing actionable model explanations in practice. Despite the growing popularity of counterfactual explanations, a...
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Keywords:
Machine Learning Models;
Counterfactual Explanations;
Adversarial Examples;
Mathematical Methods
Pawelczyk, Martin, Chirag Agarwal, Shalmali Joshi, Sohini Upadhyay, and Himabindu Lakkaraju. "Exploring Counterfactual Explanations Through the Lens of Adversarial Examples: A Theoretical and Empirical Analysis." Proceedings of the International Conference on Artificial Intelligence and Statistics (AISTATS) 25th (2022).
- November 2021
- Article
Panel Experiments and Dynamic Causal Effects: A Finite Population Perspective
By: Iavor Bojinov, Ashesh Rambachan and Neil Shephard
In panel experiments, we randomly assign units to different interventions, measuring their outcomes, and repeating the procedure in several periods. Using the potential outcomes framework, we define finite population dynamic causal effects that capture the relative...
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Keywords:
Panel Data;
Dynamic Causal Effects;
Potential Outcomes;
Finite Population;
Nonparametric;
Mathematical Methods
Bojinov, Iavor, Ashesh Rambachan, and Neil Shephard. "Panel Experiments and Dynamic Causal Effects: A Finite Population Perspective." Quantitative Economics 12, no. 4 (November 2021): 1171–1196.
- 2018
- Working Paper
Bundling Incentives in (Many-to-Many) Matching with Contracts
By: Jonathan Ma and Scott Duke Kominers
In many-to-many matching with contracts, the way in which contracts are specified can affect the set of stable equilibrium outcomes. Consequently, agents may be incentivized to modify the set of contracts upfront. We consider one simple way in which agents may do so:...
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Keywords:
Matching With Contracts;
Contract Design;
Bundling-proofness;
Substitutability;
Mathematical Methods
Ma, Jonathan, and Scott Duke Kominers. "Bundling Incentives in (Many-to-Many) Matching with Contracts." Harvard Business School Working Paper, No. 19-011, August 2018.
- 2011
- Chapter
Fundamental Data Anomalies
By: Ian D. Gow
Gow, Ian D. "Fundamental Data Anomalies." Chap. 5 in The Handbook of Equity Market Anomalies: Translating Market Inefficiencies into Effective Investment Strategies, edited by Len Zacks, 117–128. John Wiley & Sons, 2011.
- 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.
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Keywords:
Mathematical Methods
Eckstein, Jonathan. "Note on Linear Programming." Harvard Business School Background Note 191-085, November 1990. (Revised November 1992.)
- August 2005 (Revised April 2008)
- Teaching Note
GuestFirst Hotel (A): Statistics Review with Data Desk (TN)
By: Frances X. Frei
Presents an overview of the statistical analysis covered in the case discussion.
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Keywords:
Mathematical Methods
- 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.
- May 1983
- Article
The Fragility of Econometric Reporting
By: Edward Leamer and Dutch Leonard
Keywords:
Mathematical Methods
Leamer, Edward, and Dutch Leonard. "The Fragility of Econometric Reporting." Review of Economics and Statistics 65, no. 2 (May 1983).
- 2004
- Other Unpublished Work
Long-Horizon Mean-Variance Analysis: A User Guide
By: Luis M. Viceira and John Y. Campbell
Keywords:
Mathematical Methods
Viceira, Luis M., and John Y. Campbell. "Long-Horizon Mean-Variance Analysis: A User Guide." September 2004.
- April 1966
- Article
A Two-Stage Forecasting Model: Exponential Smoothing and Multiple Regression
By: D. B. Crane and James R. Crotty
Keywords:
Mathematical Methods
Crane, D. B., and James R. Crotty. "A Two-Stage Forecasting Model: Exponential Smoothing and Multiple Regression." Management Science 13, no. 8 (April 1966).
- November 1977 (Revised February 1979)
- Background Note
Growth Analysis
By: David F. Hawkins
Keywords:
Mathematical Methods
Hawkins, David F. "Growth Analysis." Harvard Business School Background Note 178-113, November 1977. (Revised February 1979.)
- Article
The Shapley Value as a von Neumann-Morgenstern Utility
By: A. E. Roth
Roth, A. E. "The Shapley Value as a von Neumann-Morgenstern Utility." Econometrica 45, no. 3 (April 1977): 657–664.
- Article
Portfolio Value-at-Risk Optimization for Asymmetrically Distributed Asset Returns
By: Joel Goh, Kian Guan Lim, Melvyn Sim and Weina Zhang
We propose a new approach to portfolio optimization by separating asset return distributions into positive and negative half-spaces. The approach minimizes a newly-defined Partitioned Value-at-Risk (PVaR) risk measure by using half-space statistical information. Using...
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Goh, Joel, Kian Guan Lim, Melvyn Sim, and Weina Zhang. "Portfolio Value-at-Risk Optimization for Asymmetrically Distributed Asset Returns." European Journal of Operational Research 221, no. 2 (September 1, 2012): 397–406.
- July 1985 (Revised March 1994)
- Background Note
Exposure and Hedging
By: David E. Bell
Describes the concept of exposure; the dependence of a goal on an uncertain external event. Describes in detail how hedges may be constructed to eliminate exposure, including the algebra of cross-hedging and hedge ratios. The relevance of regression analysis is...
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Bell, David E. "Exposure and Hedging." Harvard Business School Background Note 186-036, July 1985. (Revised March 1994.)
- 1976
- Book
Lies, Damn Lies and Statistics: The Manipulation of Public Opinion in America
Wheeler, Michael A. Lies, Damn Lies and Statistics: The Manipulation of Public Opinion in America. New York, NY: W. W. Norton & Company, 1976.
- 18 Sep 2019
- Working Paper Summaries
Using Models to Persuade
Keywords:
by Joshua Schwartzstein and Adi Sunderam
- September 1993 (Revised August 2011)
- Exercise
ExtendSim® Simulation Exercises in Process Analysis (A)
By: Roy D. Shapiro
First set of exercises meant to be used with ExtendSim, a simulation system created by Imagine That, Inc. of San Jose, California. These exercises allow students to investigate the impact of variable processing times on the performance of simple in-line processes....
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Shapiro, Roy D. "ExtendSim® Simulation Exercises in Process Analysis (A)." Harvard Business School Exercise 694-039, September 1993. (Revised August 2011.)