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
Agenda
• Motivation & Managerial issues
• Contribution
• Model & Methodology
• Empirical Results
• Managerial Implications
• Conclusions
• Big picture
Motivation
• 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:
…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…
…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.
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.
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
consumer choice beg inv A beg inv B oos A oos B
1 A 10 5 no no
2 A 9 5 no no
3 A 8 5 no no
4 B 7 5 no no
5 A 7 4 no no
6 O 6 4 no no
7 A 6 4 no no
8 A 5 4 no no
9 A 4 4 no no
10 O 3 4 no no
11 A 3 4 no no
12 A 2 4 no no
13 A 1 4 no no
14 O 0 4 yes no
15 B 0 4 yes no
16 O 0 3 yes no
17 O 0 3 yes no
18 B 0 3 yes no
19 O 0 2 yes no
N=20 O 0 2 yes no
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: 100.
• IB= inventory at the beginning and the end of the period for brand B: 52.
Example:
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: 100.
• IB= inventory at the beginning and the end of the period for brand B: 52.
consumer choice beg inv A beg inv B oos A oos B
1 A 10 5 no no
2 A 9 5 no no
3 A 8 5 no no
4 B 7 5 no no
5 A 7 4 no no
6 O 6 4 no no
7 A 6 4 no no
8 A 5 4 no no
9 A 4 4 no no
10 O 3 4 no no
11 A 3 4 no no
12 A 2 4 no no
13 O 1 4 no no
14 O 1 4 no no
15 B 1 4 no no
16 O 1 3 no no
17 O 1 3 no no
18 B 1 3 no no
19 O 1 2 no no
N=20 A 1 2 no no
Demand Model:
• Multinomial Logit Model with heterogeneous customers.
1
( )1
itm jtm jtm
itm ktm ktm
xijtm
itm Jx
iktmk
a eP y j
a e
consumer
product
period
choice
availability indicator
marketing variables
market
demand shock
Demand Model:
• Multinomial Logit Model with heterogeneous customers.
• Heterogeneity:
~ MVN( , ), 'itm m m mZ
demographics
1
( )1
itm jtm jtm
itm ktm ktm
xijtm
itm Jx
iktmk
a eP y j
a e
consumer
product
period
choice
availability indicator
marketing variables
market
demand shock
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: 1. Use aggregate data to formulate constraints on the
unobserved individual behavior.
2. Define a mechanism to sample availability & choices from their posterior distribution.
Simulating Sequence of Choices
1
1
1
01
ijtm
N
ijtm jtmi
i
ijtm jtm hjtmh
ijtm I
w S
I I w
a
choice indicator
Choices
Inventory
Product Availability
initial inventory
sales
inventory faced by customer i
product availability indicator
Constraints
• Constraints:
consumer choice beg inv A beg inv B 1-aiA 1-aiB
1 A 10 5 no no
2 A 9 5 no no
3 A 8 5 no no
4 B 7 5 no no
5 A 7 4 no no
6 O 6 4 no no
7 A 6 4 no no
8 A 5 4 no no
9 A 4 4 no no
10 O 3 4 no no
11 A 3 4 no no
12 A 2 4 no no
13 A 1 4 no no
14 O 0 4 yes no
15 B 0 4 yes no
16 O 0 3 yes no
17 O 0 3 yes no
18 B 0 3 yes no
19 O 0 2 yes no
N=20 O 0 2 yes no
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: 100.
• IB= inventory at the beginning and the end of the period for brand B: 52.
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: 100.
• IB= inventory at the beginning and the end of the period for brand B: 52.
consumer choice beg inv A beg inv B 1-aiA 1-aiB
1 A 10 5 no no
2 A 9 5 no no
3 A 8 5 no no
4 B 7 5 no no
5 A 7 4 no no
6 O 6 4 no no
7 B 6 4 no no
8 A 6 4 no no
9 A 5 4 no no
10 O 4 4 no no
11 A 4 4 no no
12 A 3 4 no no
13 A 2 4 no no
14 O 1 4 no no
15 A 1 4 no no
16 O 0 3 yes no
17 O 0 3 yes no
18 B 0 3 yes no
19 O 0 2 yes no
N=20 O 0 2 yes no
Out-of-Stocks (OOS)
15
*7
15 15
*7 7
( *)( | *)
( *) ( )
i
i i
iy ii
iy i iy ii i
p ap swap
p a p a
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
Estimation:
Initial Values: Sequence of Choices,
Availability and Demand Parameters
IndividualChoices & Availability
Individual Parameters
Hyper Parameters
Gibbs Sampler:
MCMC Simulation
DemandShocks
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)
m1 2, Z =1; Z ~ ( 1.5,1.5)m m m mZ U
Product x1 x2 x3 x4
1 1 0 1 0.042 1 0 1 -0.203 1 0 1 -0.024 0 1 1 0.165 0 1 1 -0.606 0 1 1 0.617 0 0 1 0.578 0 0 1 -0.509 0 0 1 -0.48
10 0 0 1 -0.12
…Numerical Example
• Two models:
1. Ignoring OOS (Benchmark): all products are available all the time
2. Full model: jointly modeling demand and availability
First Case: OOS=29%
mean of pref. coefficients interaction with z2 heterogeneity var()
Second Case: OOS=1.3%
mean of pref. coefficients interaction with z2 heterogeneity var()
Simulation Study: 50 replications
mean of pref. coefficients interaction with z2 heterogeneity var()
Summary statistics for the posterior mean for each model across 50 replications.
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:
Estimating Lost Sales:
• Lost Sales:
MCMC draws
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)
Summary Statistics
Empirical Results:
Empirical Results:
Estimating Lost Purchases:
Store 1 Store 2
Store 3 Store 4
Store 5 Store 6
Number of OOS products
% L
ost
Sal
es% Lost Sales vs. OOS
incidence
9.5%
30%
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.
Market 5, Day 3 (10 products missing)
2.6%
15%
0.6%
3.5%
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
16.0%
Lost Sales Reduction Profit Change
Herbal Essence (17)
All other products
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)
Lost Sales Reduction
• Market 2, Day 15 (p=-20%): – Only 1 missing product: SKU 15 (Pantene)
Market 2, Day 15 (1 product missing)
4.50%
-33%
3.20%
-8%
-40.00%
-35.00%
-30.00%
-25.00%
-20.00%
-15.00%
-10.00%
-5.00%
0.00%
5.00%
10.00%
Lost Sales Reduction Profit Change
Pantene (13)
Herbal Essence (17)
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
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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