Bayesian Structural Estimation of Retail Demand Under PartiallyObserved OutofStocks

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Bayesian Structural Estimation of Retail Demand
Under Partially-Observed Out-of-Stocks

• Eric T. Bradlow U. of Pennsylvania (Wharton)
• Andrés Musalem Duke U. (Fuqua)
• Marcelo Olivares Columbia U. (CBS)
• Christian Terwiesch U. of Pennsylvania (Wharton)
• Daniel Corsten IE Business School

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Agenda

• Motivation
• Big Picture
• Contribution
• Model Methodology
• Empirical Results
• Managerial Implications
• Extensions
• Conclusions

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Motivation iPhone
How would the analyst with Apple store data know
whether when you went to buy the product the
store was OOS?
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Big Picture

• Many situations in which we dont 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
• Environment Out-of-stocks
• Other applications Shopping paths

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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 via improved inventory
  methods?
• What is the impact of these policies on the
  retailers profits?
• Can OOSs lead to misleading demand estimates?
  (assortment planning, inventory decisions)

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Motivation

• Dealing with OOSs
• Operations Management
• Tools for assortment and inventory management
  (e.g., Mahajan and van Ryzin 2001) given a choice
  model.
• Economics
• Conlon and Mortimer (2007) ECM algorithm, E-step
  becomes harder to derive/implement as the number
  of simultaneous out-of-stocks increases.
• Marketing
• Most applications of demand estimation in the
  marketing literature ignore out-of-stocks (OOS)
  or treat it as an outcome to be modeled
  exogenously.

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Contribution Whats 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
  OOSs.

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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
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Example

• Available information
• N total number of customers20.
• 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.

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Example

• Available information
• N total number of customers20.
• 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.

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Demand Model

• Multinomial Logit Model with heterogeneous
  customers.

marketing variables
demand shock
availability indicator
product
choice
market
consumer
period
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Demand Model

• Multinomial Logit Model with heterogeneous
  customers.
• Heterogeneity

marketing variables
demand shock
availability indicator
product
choice
market
consumer
period
demographics
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Estimation

• If availability and individual choices were
  observed (aijtm) standard methods
• Solution data augmentation conditional on
  aggregate data (following Chen 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.

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Simulating Sequence of Choices

• Constraints

choice indicator
sales
Choices
initial inventory
inventory faced by customer i
Constraints
Inventory
product availability indicator
Product Availability
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Out-of-Stocks (OOS)

• Available information
• N total number of customers20.
• NA number of customers buying A 10.
• NB 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.

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Out-of-Stocks (OOS)

• Available information
• N total number of customers20.
• NA number of customers buying A 10.
• NB 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.

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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
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Estimation
Initial Values Sequence of Choices, Availability
and Demand Parameters
Gibbs Sampler
Individual Choices Availability
Hyper Parameters
Demand Shocks
MCMC Simulation
Individual Parameters
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Simulation Study

• Choice Set J10 products no-purchase.
• Markets M12 markets
• Utility function
• Covariates
• X1-X3 dummy variables (2 brands, purchase
  option)
• X4 continuous variableN(2,1)
• Preferences in each market N( ,?)
• ?diag( 0, 0, 0.5, 2)
• ?jtmN(0,0.5)

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Simulation Study

• Two models
• Ignoring OOS all products are available all the
  time
• Full model jointly modeling demand and
  availability

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First Case OOS35
mean of pref. coefficients
interaction with z2
heterogeneity
var(?)
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Second Case OOS1.4
mean of pref. coefficients
interaction with z2
heterogeneity
var(?)
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Results

• Fraction of consumers experiencing an OOS for
  product 1

estimated
estimated
R 0.70
true
true
est() 1-0.5(S1tmNtm)/Ntm
est() simulated from posterior
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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

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Estimating Lost Sales

• Lost Sales

MCMC draws
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Real Data Set

• M6 stores from a major retailer in Spain
• J24 SKUs (shampoo)
• T15 days
• Sales and price data for each SKU in each day and
  periodic inventory data
• Demographics (income)

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Summary Statistics
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Empirical Results
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Estimating Lost Purchases
Market 1
Market 2
Market 3
Market 4
Market 5
Market 6
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Lost Sales vs. OOS incidence
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Dynamic Pricing Sales Improvement
Missing products in Day 5 Market 5 4
(Timotei), 9 (Other), 10-13 (Pantene), 14
(Other), 18-19 (HS), 23 (Cabello Sano)
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Dynamic Pricing Profit Improvement
Item implied by Lost Revenue ? Lost Profit ? Most
Frequent OOS
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Extensions / Next Steps

• Behavioral issues (e.g., complexity, variety)
• Backorder effects
• Purchase quantity model
• Price Endogeneity
• Sampling k components instead of 2
• Infer OOS without (periodic) inventory data
• In-Store Shopping Behavior

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Behavioral Issues

• Choice Complexity
• Current model Ui0tm?i0tm
• Instead Ui0tm f( ?) ?i0tm

outside good
Proxy for Complexity
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Backorder Effects

• Backorder
• Current model Ui0tm?i0tm
• Instead Ui0tm g(
  ?)?i0tm

outside good
Previous OOS
Previous no purchase
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Quantity Decisions

• Sampling Choices and Quantities

choices

• For simplicity no variety seeking.
• What are the feasible values of the choices and
  quantities of consumers 2 and 4 in period 1?
• (BB, A) (A, BB)
• Update product inventory for customers 2, 3 and
  4.

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Price Endogeneity

• Very limited price variation for each SKU within
  market.
• Price endogeneity could arise from price
  differences across markets.
• Bayesian instrumental variables approach (e.g.,
  Yang, Chen and Allenby 2003 Conley et al. 2008)
• pjtm ?? zjtm ??jtm

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Sampling Choices in groups of k components.

• Example k3

• What values of y21, y31 and y41 are consistent
  with the sales data?
• (A,A,B) (A,B,A) (B,A,A)
• Assign (A,A,B) with the following probability
• Prob((A,A,B))

choices
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Sampling Choices in groups of k components.

• Example k3

• What values of y21, y31 and y41 are consistent
  with the sales data?
• (A,A,B) (A,B,A) (B,A,A)
• Assign (A,A,B) with the following probability
• Prob((A,A,B))

choices
Note number of terms in the denominator may
increase at k! rate (e.g., ABC, ACA, BAC, BCA,
CAB, CBA).
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In-Store Shopping Behavior

• Using RFID technology it is possible to track the
  location of shopping carts in a grocery store
  every 5 seconds (disaggregate data).
• Alternatively record the number of shopping
  carts that pass through a measuring point
  (aggregate data).
• Infer the trajectory of shopping carts using only
  these aggregate measurements.

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In-Store Shopping Behavior
qD 10
qC 7
qCD 7
C
D
qBC 4
qAD 2
qBD 1
qAC 3
A
B
qB 5
qA 10
qAB 5
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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 OOSs
• Several extensions are possible

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Questions?