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Structural Estimation of the Effect of Out-of-Stocks

Structural Estimation of the Effect of Out-of-Stocks. Andr é s Musalem Duke U. (Fuqua) Marcelo Olivares Columbia U. (CBS) Eric T. Bradlow U. of Pennsylvania (Wharton) Christian Terwiesch U. of Pennsylvania (Wharton) Daniel Corsten IE Business School. Agenda.

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Structural Estimation of the Effect of Out-of-Stocks

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  1. Structural Estimation of the Effect of Out-of-Stocks Andrés Musalem Duke U. (Fuqua) Marcelo Olivares Columbia U. (CBS) Eric T. Bradlow U. of Pennsylvania (Wharton) Christian Terwiesch U. of Pennsylvania (Wharton) Daniel Corsten IE Business School

  2. Agenda • Motivation & Managerial issues • Contribution • Model & Methodology • Empirical Results • Managerial Implications • Conclusions • Big picture

  3. Motivation

  4. Managerial Issues: • What fraction of consumers were exposed to an out-of-stock (OOS)? • How many choose not to buy? (money left on the table) • How many choose to buy another product? • Can we reduce lost sales? • What is the impact of these policies on the retailer’s profits? • Can OOS’s lead to misleading demand estimates? (assortment planning, inventory decisions)

  5. …Motivation • Dealing with OOS’s: • Operations Management: • Tools for assortment and inventory management (e.g., Mahajan and van Ryzin 2001) given a choice model. • Marketing: • Most applications of demand estimation in the marketing literature ignore out-of-stocks (OOS) • But…

  6. …Motivation • Marketing: • Assume: • 0 sales => no availability • Positive sales => availability (e.g., ACV weighted distribution) • Anupindi, Dada and Gupta (1998): • Vending Machines Application / EM • Jointly model sales and availability • One-Stage Substitution assumption. • Kalyanam et al. (2007): • COM-Poisson, reduced-form model of substitution, categorical variables. • Bruno and Vilcassim (2008) extension of BLP: • ACV as a proxy for product availability • P(OOS Brand A) independent of OOS for Brand B. • Zero sales issues (slow-moving items). • Conlon and Mortimer (2007): • EM method becomes more difficult to implement as the # of products simultaneously OOS increases.

  7. Contribution: What’s new? • Joint model of sales and availability consistent with utility maximization (structural demand model) • No restrictive assumptions about availability (e.g., OOS independence) • No restrictive assumptions about substitution (e.g., one-stage substitution) • Multiple stores / relatively large number of SKUs • Heterogeneity: Observed (different stores) / Unobserved (within stores) • Products characteristics: categorical and continuous • Simple expressions to estimate lost sales / evaluate policies to mitigate the consequences of OOS’s.

  8. Modeling the impact of OOS: • A simple way to capture the effect of an OOS (reduced-form): • If an OOS is observed in period t: f(Salesjt)=Xjt’+ OOSjt+jt • However, it is important to determine when the product became out-of-stock. • Why? Mktg Variables OOS dummy variable

  9. Example: • Available information: • N= total number of customers=20. • SA= number of customers buying A = 10. • SB= number of customers buying B =3. • IA= inventory at the beginning and the end of the period for brand A: 100. • IB= inventory at the beginning and the end of the period for brand B: 52.

  10. Example: • Available information: • N= total number of customers=20. • SA= number of customers buying A = 10. • SB= number of customers buying B =3. • IA= inventory at the beginning and the end of the period for brand A: 100. • IB= inventory at the beginning and the end of the period for brand B: 52.

  11. Demand Model: • Multinomial Logit Model with heterogeneous customers. marketing variables demand shock availability indicator product choice market consumer period

  12. Demand Model: • Multinomial Logit Model with heterogeneous customers. • Heterogeneity: marketing variables demand shock availability indicator product choice market consumer period demographics

  13. Estimation: • If availability and individual choices were observed (aijtm) => standard methods • Solution: data augmentation conditional on aggregate data (followingChen & Yang 2007; Musalem, Bradlow & Raju 2007, 2008) Key elements: • Use aggregate data to formulate constraints on the unobserved individual behavior. • Define a mechanism to sample availability & choices from their posterior distribution.

  14. Simulating Sequence of Choices • Constraints: choice indicator sales Choices inventory faced by customer i initial inventory Constraints Inventory product availability indicator Product Availability

  15. Out-of-Stocks (OOS) • Available information: • N= total number of customers=20. • SA= number of customers buying A = 10. • SB= number of customers buying B =3. • IA= inventory at the beginning and the end of the period for brand A: 100. • IB= inventory at the beginning and the end of the period for brand B: 52.

  16. Out-of-Stocks (OOS) • Available information: • N= total number of customers=20. • SA= number of customers buying A = 10. • SB= number of customers buying B =3. • IA= inventory at the beginning and the end of the period for brand A: 100. • IB= inventory at the beginning and the end of the period for brand B: 52.

  17. Estimation Gibbs Sampling: • The choices of the consumers in a given pair are swapped according to the following full-conditional probability: choices in new sequence product availability based on new sequence

  18. Estimation: Initial Values: Sequence of Choices, Availability and Demand Parameters Gibbs Sampler: Individual Choices & Availability Hyper Parameters Demand Shocks MCMC Simulation Individual Parameters

  19. Numerical Example: • Choice Set: J=10 products + no-purchase. • Markets: M=12 markets • Utility function: • Covariates: • X1-X3: dummy variables (2 brands, purchase option) • X4: continuous variable~N(2,1) • Preferences in each market ~ N( ,): • =diag( 0, 0, 0.8, 2) • jtm~N(0,0.5)

  20. …Numerical Example • Two models: • Ignoring OOS (Benchmark): all products are available all the time • Full model: jointly modeling demand and availability

  21. First Case: OOS=29% mean of pref. coefficients interaction with z2 heterogeneity var()

  22. Second Case: OOS=1.3% mean of pref. coefficients interaction with z2 heterogeneity var()

  23. Simulation Study: 50 replications Summary statistics for the posterior mean for each model across 50 replications. mean of pref. coefficients interaction with z2 heterogeneity var()

  24. Estimating Lost Sales: • Let A*: Set of all products • Let Ai: Set of missing products • Probability of a given consumer having chosen one of the missing alternatives had it been available:

  25. Estimating Lost Sales: • Lost Sales: MCMC draws

  26. Data Set: • M=6 stores from a major retailer in Spain • J=24 SKUs (shampoo) • T=15 days • Sales and price data for each SKU in each day and periodic inventory data • Demographics (income)

  27. Summary Statistics

  28. Empirical Results:

  29. Empirical Results:

  30. Estimating Lost Purchases: Store 1 Store 2 Store 3 Store 4 Store 5 Store 6

  31. % Lost Sales vs. OOS incidence 30% % Lost Sales 9.5% Number of OOS products

  32. Dynamic Pricing: Sales Improvement • Lost sales reduction after a temporary price promotion: • It’s not equal to the anticipated change in sales! • Instead, it’s equal to the fraction of consumers who meet the following 3 requirements: • Did not buy any products • Would have purchased a product had all alternatives been available • Would purchase one of the available alternatives if a discount is offered.

  33. Lost Sales Reduction • Market 5, Day 3 (p=-20%): • 10 Missing products: 4 (Timotei), 9 (Other), 10-13 (Pantene), 14 (Other), 18-19 (H&S), 23 (Cabello Sano)

  34. Lost Sales Reduction • Market 2, Day 15 (p=-20%): • Only 1 missing product: SKU 15 (Pantene)

  35. Conclusions: • Bayesian methods / data augmentation enable us to jointly model choices and product availability w/o restrictive assumptions on: • Joint probability of out-of-stocks / substitution • Key: use available information to formulate constraints on unobserved individual data: • Constraints and Data Augmentation • As a byproduct, we obtain simple expressions to: • Estimate the magnitude of lost sales • Assess effectiveness of policies aimed at mitigating the costs of OOS’s • Several extensions are possible

  36. Big Picture: • Many situations in which we don’t observe individual behavior, but we may have some aggregate or limited information. • Key: use aggregate data to formulate constraints on the unobserved individual behavior. • Dependent variables: Choices • Independent variables: Coupon promotions • Shopping Environment: Out-of-stocks • Other applications: Shopping paths

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