Demand Estimation - Endogeneity in Priceseconlizhao.weebly.com/uploads/8/8/8/6/88869258/eio_t1...IV: Product char. of the same rm, product char. of competing rms. IV: Price of product

Demand Estimation - Endogeneity in Prices

Li Zhao, SJTU

Spring, 2017

Li Zhao Demand 1 / 42

Outline

1 2nd Generation

2 BLP (1995)

3 Adding Supply Side

4 Combination of Micro and Macro Data


Discrete Choice Model

Traditional models: Product space models.

Discrete choice models (characteristics space).I 1st generation: Logit / Probit, Nested Logit, Mixed Logit.I 2nd generation: Logit with endogenous variable.I 3rd generation: Random coe�cient + price endogeneity.


Upward-Sloping Demand Curves?

Study of demand for CAT scanners.

Coe�cient on price is positive, implying that people prefer moreexpensive machines!

Possible explanation:I Quality di�erentials across products not adequately controlled for.I In equilibrium of a di�erentiated product market where each product isvalued on the basis of its characteristics, brands with highly-desiredcharacteristics (higher quality) command higher prices.

I Unobserved quality leads to price endogeneity.


A further look of the model

Random utility model takes the form

uij = xiβ −αpj + εij .

εij may contain unobserved product characteristics, ξj .

Firms know ξ when they set prices so prices are correlated with theerror which in turn is buried in a highly non-linear set of equations.

In many situations, the explanatory variables are endogenous, that is,are correlated with the unobserved factors.

I Unobserved attributes of a product can a�ect its price.I Marketing e�orts can be related to prices.I Interrelated choices of decision makers.


Mean Utility of Product j

Let δj denote the mean utility of product j :

uij = xjβ −αpj + ξj + εij ≡ δj + εij .

I Econometrician observes neither ξj or εij , but household i observesboth: these are both �structural errors".

I ξ1, ...ξJ are interpreted as �unobserved quality". All else equal,consumers more willing to pay for brands for which ξj is high.

I It is the source of the endogeneity problem in this demand model.

Main idea of Berry (1994)I Because price is endogenous, we need instruments Z so that

E (Z ′ξ ) = 0.I ξj is part of δj , which a�ects market share. So we can recover ξj fromobserved(actual) market share.


Estimation: Logit with Unobserved Product Char

In Logit, εij ∼ i .i .d . the extreme value distribution.

The market share of product j = 0,1, ...J is

sj(θ) =exp(δj)

∑Jk=0

exp(δk).

First step, we �invert� system of equations s(δ ) to get δj forj = 1,2, ...J.

log(sj)− log(s0) = δj .

Second step, IV estimation. We recover ξj = δj −Xjβ + αpj , andminimize sample moment condition

g =1

J

J

∑j=1

Z ′j ξj .

Note we can estimate it using aggregate data. (Recall we show MLEand GMM of Bernoulli distribution yield the same result).


Instruments

δj = xjβ −αpj + ξj .

Recall we need instruments Z such thatI Z is correlated with pj .I Z is uncorrelated with ξj .

The usual demand cases: cost shifters.I Example: wages. But it does not work in Berry (1994)'s model,because the wage in Michigan is same across all car brands.

IV: Product char. of the same �rm, product char. of competing �rms.

IV: Price of product j in other markets t.

The choice of instruments depends on the context of the industryunder study. No standard solution.


Example

The Impact of Media on Trade, working in progress.

2008 milk contamination scandal.

Markets: importing countries - year.

Products: exporting countries.

δj = quality(2009 milk, China, media exposure)−αpj + ξj .


Summary of Berry (1994)

Endogenous price in utility function

uij = xjβ −αpj + ξj + εij ≡ δj + εij .

Log(share) is a linear function of mean utility δj

log(sj)− log(s0) = δj .

Once a non-linear problem has been transferred to a linear problem.We can use IV to deal with endogeneity.


Outline

1 2nd Generation

2 BLP (1995)




BLP

Berry, S., Levinsohn, J. and Pakes, A., 1995. Automobile prices inmarket equilibrium. Econometrica: Journal of the EconometricSociety, pp.841-890.

Nevo, A., 2000. A Practitioner's Guide to Estimation ofRandom-Coe�cients Logit Models of Demand. Journal of economics

& management strategy, 9(4), pp.513-548.

BLP is an extension to the Logit model, that allows for unobservedproduct characteristics and consumer heterogeneity in tastes forcharacteristics.

The single most important contribution of BLP is showing how to dothe inversion in a random-coe�cient Logit model, that allows the errorto be popped out, and thus allowing endogeniety problems to beaddressed.

The next most important contribution is showing that all themachinery can produce results that make a lot of sense.


Why Consumer Heterogeneity

If a consumer chooses product j , it means that she values thecharacteristics of product j , therefore, when pj increases, she will tendto choose a product with similar characteristics.

In Logit, the IIA property suggests the consumer whose �rst choice is jis no di�erent from the rest.

A better modeling approach is to allow for correlation in εij .I Why not assume that they are correlated across products and estimatethe variance-covariance matrix of εij?

I An alternative approach is to impose structure on the form ofcorrelation in idiosyncratic shocks: the idea is to specify the correlationas a linear function of the product characteristics.


Random Coe�cients Logit

Without consumer heterogeneity, a Logit model takes the form

uij = Xjβ −αpj + ξj + εij .

In a random coe�cient Logit, consumer i 's utility for product j is

uij = Xjβi −αipj + ξj + εij

where (αi

βi

)∼(

α

β

)+ ΠDi + vi .

(αi ,βi ) di�er across consumers. They are called �random coe�cients�.

(α, β ), Π and Σ are additional parameters to be estimated.


Subsitution Patterns of RC Logit

uij = Xjβi −αipj + ξj + εij(αi

βi

)∼(

α

β

)+ ΠDi + vi .

We can decomposition uij as the sum of mean utility δj = Xj β − αpj + ξj

and individual tastes.

uij = Xj β − αpj + ξj +Xj(βi − β )− (αi − α)pj + εij .

= δj + [ −pj Xj ](ΠDi + vi ) + εij

= δj(Xj ,pj ,ξj ; α, β ) + µij(Xj ,pj ,Di ,vi ;Π,Σ) + εij .

The interaction between (observed and unobserved) consumercharacteristics and product characteristics kills IIA problem.

Aggregate substitution patterns are now far more reasonable.


Estimation - Main Idea


As in Logit, we want to deal with endogeneity in prices. The IVssatisfy E (Z ′ξ ) = 0,

I (Inversion step) We use observed market share sj to recover mean

utility δj and hence ξj .

I (IV step) We then minimize g = 1

J ∑Jj=1

Z ′j ξj .

With random coe�cients, the inversion step is complicated.


Estimation - Inner Loop

Work out the aggregate shares conditional on (δ , α, β ,Σ)

For a given (αi ,βi ), the choice probabilities for consumer i take MNLform

Pij =exp(Xjβi −αipj + ξj)

1+ ∑k exp(Xkβi −αipk + ξk).

In the whole population, the predicted aggregate market share ofproduct j is

sj =∫

exp(Xjβi −αipj + ξj)

1+ ∑k exp(Xkβi −αipk + ξk)dF (αi ,βi )

=∫

[δj(Xj ,pj ′ ,ξj ; α, β ) + µij(Xj ,pj ,Di ,vi ;Π,Σ)]

1+ ∑k [δk(Xk ,pk ,ξk ; α, β ) + µik(Xk ,pk ,Di ,vi ;Π,Σ)]dF (vi ,Di )

≈1

S ∑s

[δj + µ(s)ij ]

1+ ∑k [δk + µ(s)ik ]≡ sj(δ ,Π,Σ)


Estimation - Inner LoopRecover ξ the from the shares.

Recall that δj = Xj β − αpj + ξj , if we know δj , we can recover ξj .Thenwe can construct the moments for the estimator.

BLP point out that iterating on the system

δkj (Π,Σ) = δ

k−1j (Π,Σ) + ln[s0j ]− ln[sj(δ

k−1,Π,Σ]

has a unique solution.

Once the sequence converges, we get δ and thus ξj :

ξj(Π,Σ,s0) = δj(Π,Σ,s0)−Xj β − αpj .

Construct the Moments

We interact ξ with the instruments to exploit population momentrestriction E (Z ′ξ ) = 0:

1

J

J

∑j=1

Z ′(δj(Π,Σ,s0)−Xj β − αpj) = 0

.Li Zhao Demand 18 / 42

Estimation Procedure

In the outer loop, we iterate over di�erent values of the parameters(α, β ,Π,Σ). Let θ be the current values of the parameters.

In the inner loop, for the given parameter values θ , we wish toevaluate the objective function G (·). In order to do this we must

I 1) Starting from δ start ,get sj (δ start , Π, Σ) by simulation;I 2) Find �xed point δ k

j = δk−1j + lns0j − ln sj (δ k−1, Π, Σ). Run a loop to

�nd the �xed point δj .

I 3) Get ξj = δ kj −Xj β + αpj for outer loop.

I 4) Calculate sample moments g = 1

J ∑Jj=1

Z ′ξj = 0. Construct GMMcriterion function.


Example: BLP (1995) - Data

All models marketed, 1971 - 1990. 2217 model/years.

The least expensive car in the sample is the 1990 Yugo at $3393(1983 dollars) while the top-of-the-line Porsche 911 Turbo Cabrioletcosts $68,597.

Distinguish which �rms produce which models.I Buick, Oldsmobile, Cadillac, Chevrolet, and Pontiac are all part of one�rm, General Motors.

Variables: ratio of horsepower to weight (HPWT), a dummy forwhether air conditioning is standard, miles per dollar (MP$), size, anda constant.


Outline

1 2nd Generation

2 BLP (1995)




Why Adding Supply Side

In some cases we will want to fully specify a supply relationship andestimate it jointly with the demand-side equations (for example, seeBLP).

This �ts into the model easily by adding moment conditions to theGMM objective function.

When supply side is added to the model, we canI Gain e�ciency, because pricing decision depends on elasticities.I Estimate marginal costs and markups.I Conduct counterfactual analysis.

The bene�ts comes at the cost of requiring more structural (andassumptions).


How to Model Supply Side (1)

We assume Bertrand-Nash competition: oligopoly �rms set prices tomaximize pro�ts.

Suppose there are F �rms, each of which produces some subset of Ffof the product.

Assume that there are no (dis-)economies of scope, so that productioncosts are simply additive across di�erent products for a multi-product�rm.

The pro�ts of �rm f are

Πf = ∑j∈Ff

(pj −mcj) ·M · sj(p)−FixedCost.


How to Model Supply Side (2)

Πf = ∑j∈Ff


Take FOC, equilibrium prices are characterized by

sj(p) + ∑r∈Ff

(pr −mcr )∂ sr (p)

∂pj= 0.

In matrix form,s(p)−Λ · (p−mc) = 0,

where Λjr =− ∂ sr∂pj·1((j,r) produced by the same �rm).

In Logit, ∂ sr∂pj

=−αsr (1− sr ) if j = r , and ∂ sr∂pj

=−αsr sk otherwise.

In RC Logit, Λ is a integration function of (δ , Π, Σ) over Di and vi .As before, we can simulate Λ .


Moments from Supply Side

Parameterize marginal cost by

ln(mc) = λw + ω.

ω include some of x , but with some overlaps. In BLP, mileage perdollar is in x , but mileage per gallon is in ω .

E (ω|ZsupplyIV ) = 0. Recall we have

s(p)−Λ · (p−mc) = 0,

therefore

ω = ln(mc)−λw

= ln(p−Λ−1 · s(p))−λw .


Estimation Procedure - With Supply Side

In the outer loop, we iterate over di�erent values of the parameters(α, β , λ ,Π,Σ). Let θ be the current values of the parameters.

In the inner loop, for the given parameter values θ , we wish toevaluate the objective function G (·). In order to do this we must

I 1) Starting from δ start ,get sj (δ start , Π, Σ) by simulation;I 2) Find �xed point δ k

j = δk−1j + lns0j − ln sj (δ k−1, Π, Σ). Run a loop to

�nd the �xed point δj .

I 3-1) Get ξj = δ kj −Xj β + αpj for outer loop.

I 3-2) Get Λ(δ , Π, Σ), s(δ , Π, Σ), and hence ω = ln(p− Λ−1(·s(p))− λw .I 4) Calculate sample moments g1 = 1

J ∑Jj=1

Z ′demand ξj and

g2 = 1

J ∑Jj=1

Z ′supply ωj = 0. Construct GMM criterion function.


Counterfactual Analysis when Supply Side is Modeled

Example, the impact of removing one product, say A.

Demand changes, because choice sets changes.I Customers move to their second choices.

Know the demand function, �rms set their optimal prices.

We can evaluate prices, consumer surplus and other policy relevantquestions after removing product A.

Similarly, we can evaluate the impact of introducing a new product.


Nevo (2001) - Introduction

Nevo, A., 2001. Measuring market power in the ready - to - eat cerealindustry. Econometrica, 69(2), pp.307-342.

The RTE markets has very high margins.

This paper empirically separate price-cost margins into three sources:I A �rm is ability to di�erentiate its brands from those of its competition.I If two brands are perceived as imperfect substitutes, a �rm producingboth would charge a higher price than two separate manufacturers.

I Main players in the industry could engage in price collusion.


Nevo (2001) - Data and Model

Πf = ∑j∈Ff


sj(p) + ∑r∈Ff

(pr −mcr )∂ sr (p)

∂pj= 0.

In matrix form,s(p)−Λ · (p−mc) = 0,

In BLP, we assume Bertrand - Nash, and recover cost parameters.

In Nevo (2001), the author observes MC, so it could test di�erentconduct (single �rm, multi-product �rm, collude).Data

I 25 brands with the highest national market shares.I First quarter of 1988 to the last quarter of 1992.I The author collects data on market shares and prices in each market(city-quarter pair), brand characteristics, advertising, and distributionof demographics.


Outline

1 2nd Generation

2 BLP (1995)




Data Structure

In BLP framework, we typically observeI M markets, each has Jm products.I Market-level (aggregate) data on prices, product characteristics, etc.I Market-level demographic information such as income distribution.

It is possible to combine micro and macro data to better understandconsumer demands.

BLP (2004): Survey of recent purchasers of automobiles, containinginformation of consumer's �second choice�.

Petrin (2002): Linking demographics of purchasers of new vehicles tothe vehicles they purchase.


How to Combine Multiple Data Sets?

Ultimately, demand estimation takes the form

g1 = E (Z ′demandξ ) = 0.

For example, in BLP, each product-market pair is one �sample� of themoment condition.

The residual of mean utilities is independent of the instruments.

We can construct moment conditions for the micro-level data.I For example, if we observe the probability of purchase by consumers ofdi�erent demographic groups in micro-level data. We can �force� ourdemand model to predict the same probability.

We can construct more moment conditions for macro-level data aswell.

I For example, the co-variance of the product characteristics and theobserved demographics.


Petrin (2002) - Introduction

Petrin, A., 2002. Quantifying the bene�ts of new products: The caseof the minivan. Journal of political Economy, 110(4), pp.705-729.

Objective of the study:I The goal is to infer the changes in producer surplus, measuring theextent of �rst-mover advantage and the pro�t cannibalization that tookplace both initially by the innovator and later by the imitators.

I Conclusion: Introduction generated large welfare gains for consumersand surplus for the innovator at the expense of the other producers.

Methodological contribution: augmenting the market-level data withinformation that relates the average demographics of consumers to thecharacteristics of the products they purchase.

I For example, family size conditional on the purchase of a minivan.


Petrin (2002) - Background

In the early 1970s Ford proposed the �Mini/Max�, but the ideareceived little support.

Introduced in 1984 by the �nancially troubled Chrysler Corporation,the Dodge Caravan (its minivan) was an immediate success.

General Motors (GM) and Ford quickly responded, but they wereunprepared for the Caravan's success.

Over the years, minivans cannibalized station wagon sales.


Petrin (2002) - Model and Data

uij = Xjβi −αi ln(yi −pj) + ξj + εij .

βi depend on demographics, such as income, family size, age ofhousehold head, etc.

Xj contains the type of vehicle, such as minivans, station wagons,full-size passenger vans, and sport-utility vehicles.

The supply side follows Bertrand-Nash fashion.

Macro data: 2,407 nameplates marketed in the United States from theyears 1981 to 1993.

Micro data: A rotating panel that records U.S. household purchasingpatterns.


Petrin (2002) - Moments

Micro moments:I Average probability of new vehicle purchase conditional on income level.I Average family size of purchasers for each of these four family vehiclegroups.

I The probability the head of the household is between certain age groupfor each of these four family vehicle groups.

BLP moments:I Demand supply moments.I Supply supply moments.


Petrin (2002) - Welfare Analysis

Counterfactual:I In the counterfactual environment, there are no minivans. and othervehicle prices solve Bertrand-Nash.

I Compensating variation is the dollar amount a consumer would need tobe just indi�erent between the two scenarios.

Changes in consumer welfare are caused byI Changes in prices. Non-minivan consumers bene�t from the pricecompetition.

I Changes in choice sets. Large families had a strong taste over minivan.

Changes in pro�ts.


BLP (2004) - Introduction

Berry, S., Levinsohn, J. and Pakes, A., 2004. Di�erentiated productsdemand systems from a combination of micro and macro data: Thenew car market. Journal of political Economy, 112(1), pp.68-105.

Objective:I Uncover basic parameters of demand and supply of vehicles.I Make predictions of two out-of-sample changes: the decision of GeneralMotors to shut down its historic Oldsmobile division and theintroduction of luxury sport utility vehicles (SUVs).

Methodological contributions:I Show how rich sources of information on consumer choice can help toidentify demand parameters.

I How to use �second-choice� data on automotive purchases to obtaingood estimates of substitution patterns in the automobile industry.

I �Second choice� data is helpful in estimating the model parametersthat govern the predicted pattern of substitution across products.


BLP (2004) - Model and Data


Micro Data: A propriety survey conducted on behalf of the GeneralMotors Corporation (GM).

The CAMIP questionnaire asks about a limited number of householdattributes, including income, age of the household head, family size,and place of residence (urban, rural, etc.).

CAMIP asks the �second choices��the purchase that they would havemade if their preferred product were not available.

The sampled vehicles consist of almost all vehicles sold in the UnitedStates in 1993, not just GM products.


BLP (2004) - Moments

The covariances of the observed �rst-choice product characteristicswith the observed consumer attributes. (Need Bayes Rule).

E (Di |yi = j).

The covariances between the �rst-choice product characteristics andthe second choice product characteristics.

E (Second Choice Char|yi = j}

The market shares of the J products.


BLP (2004) - Counterfactual Analysis

Evaluating the potential demand for �high-end� SUVs. (Actuallyintroduced in late 1990s).

I The ξ of the new Toyota SUV was set equal to the mean y of allToyota cars marketed in that year

I The price of that vehicle was obtained from a regression of price onto alarge set of vehicle characteristics and company dummies.

Evaluating the impact of shutting down Oldmobile division of GM.(GM announced its intention to close town the division in 2000).


Demand Estimation - Summary

NEIOI Structural estimation, computational intensive, sophisticatedeconometric tools.

Demand estimationI Elasticities generated by di�erent demand models.I IV, MLE, GMM.I Endogeneity and instruments.I Adding supply side.I Combine micro and macro information.I Counterfactual analysis.


Documents

Demand Estimation - Endogeneity in Priceseconlizhao.weebly.com/uploads/8/8/8/6/88869258/eio_t1...IV: Product char. of the same rm, product char. of competing rms. IV: Price of product