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. 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) Managerial Issues:

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? 1.Joint model of sales and availability consistent with utility maximization (structural demand model) 2.No restrictive assumptions about availability (e.g., OOS independence) 3.No restrictive assumptions about substitution (e.g., one-stage substitution) 4.Multiple stores / relatively large number of SKUs 5.Heterogeneity: Observed (different stores) / Unobserved (within stores) 6.Products characteristics: categorical and continuous 7.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(Sales jt )=X jt ’  +  OOS jt +  jt –However, it is important to determine when the product became out-of-stock. – Why? Mktg Variables OOS dummy variable

9. consumerchoicebeg inv Abeg inv Boos Aoos B 1A105no 2A95 3A85 4B75 5A74 6O64 7A64 8A54 9A44 10O34no 11A34no 12A24no 13A14no 14O04yesno 15B04yesno 16O03yesno 17O03yesno 18B03yesno 19O02yesno N=20O02yesno Available information: N= total number of customers=20. S A = number of customers buying A = 10. S B = number of customers buying B =3. I A = inventory at the beginning and the end of the period for brand A: 10  0. I B = inventory at the beginning and the end of the period for brand B: 5  2. Example:

10. Available information: N= total number of customers=20. S A = number of customers buying A = 10. S B = number of customers buying B =3. I A = inventory at the beginning and the end of the period for brand A: 10  0. I B = inventory at the beginning and the end of the period for brand B: 5  2. consumerchoicebeg inv Abeg inv Boos Aoos B 1A105no 2A95 3A85 4B75 5A74 6O64 7A64 8A54 9A44 10O34no 11A34no 12A24no 13O14no 14O14no 15B14no 16O13no 17O13no 18B13no 19O12no N=20A12no

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

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

13. Estimation: If availability and individual choices were observed (a ijtm ) => standard methods Solution: data augmentation conditional on aggregate data ( following Chen & Yang 2007; Musalem, Bradlow & Raju 2007, 2008 ) Key elements: 1.Use aggregate data to formulate constraints on the unobserved individual behavior. 2.Define a mechanism to sample availability & choices from their posterior distribution.

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

15. consumerchoicebeg inv Abeg inv B 1-a iA 1-a iB 1A105no 2A95 3A85 4B75 5A74 6O64 7A64 8A54 9A44 10O34no 11A34no 12A24no 13A14no 14O04yesno 15B04yesno 16O03yesno 17O03yesno 18B03yesno 19O02yesno N=20O02yesno Available information: N= total number of customers=20. S A = number of customers buying A = 10. S B = number of customers buying B =3. I A = inventory at the beginning and the end of the period for brand A: 10  0. I B = inventory at the beginning and the end of the period for brand B: 5  2. Out-of-Stocks (OOS)

16. Available information: N= total number of customers=20. S A = number of customers buying A = 10. S B = number of customers buying B =3. I A = inventory at the beginning and the end of the period for brand A: 10  0. I B = inventory at the beginning and the end of the period for brand B: 5  2. consumerchoicebeg inv Abeg inv B 1-a iA 1-a iB 1A105no 2A95 3A85 4B75 5A74 6O64 7B64 8A64 9A54 10O44no 11A44no 12A34no 13A24no 14O14no 15A14no 16O03yesno 17O03yesno 18B03yesno 19O02yesno N=20O02yesno Out-of-Stocks (OOS)

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 Individual Choices & Availability Individual Parameters Hyper Parameters Gibbs Sampler: MCMC Simulation Demand Shocks

19. Numerical Example: Choice Set: J=10 products + no-purchase. Markets: M=12 markets Utility function: –Covariates: X 1 -X 3 : dummy variables (2 brands, purchase option) X 4 : 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: 1.Ignoring OOS (Benchmark): all products are available all the time 2.Full model: jointly modeling demand and availability

21. First Case: OOS=29% mean of pref. coefficientsinteraction with z 2 heterogeneity var(  )

22. Second Case: OOS=1.3% mean of pref. coefficientsinteraction with z 2 heterogeneity var(  )

23. Simulation Study: 50 replications mean of pref. coefficientsinteraction with z 2 heterogeneity var(  ) Summary statistics for the posterior mean for each model across 50 replications.

24. Estimating Lost Sales: Let A*: Set of all products Let A i : 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:

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

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

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