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