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
Dec 30, 2015
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