identification and estimation of demand for differentiated ...€¦ · vertical example: quality...

Identification and Estimation of Demand forDifferentiated Products

Jean-Francois HoudeCornell University & NBER

September 24, 2016

Demand for Differentiated Products 1 / 58

Starting Point: The Characteristic Approach

Ultimate goal: Measure the elasticity of substitution between goods.Why?

I Measure the degree of market powerI Predict the effect of proposed mergersI Evaluate the value of new goodsI . . .

Natural starting point: Linear inverse demand

pjt = α0 +∑k∈J

αj ,kqkt + εjt

Curse of dimensionality: Even with exogenous variation in q’s, thenumber of parameters to estimate grows with the number of products.Solution: The characteristic approach (Lancaster 1966)

I Each product is defined as a bundle of characteristics: xj = x1, . . . , xkI Consumers assigned values to each characteristics, and choose the

product maximize their utility given their budget.I Demand for product j is function of the distribution of valuations for

each attribute (finite dimension).

Demand for Differentiated Products Introduction 2 / 58

Representative Agent Models

Almost Ideal Demand System (Deaton and Muellbauer 1980)I Assumption: Products are grouped in K categories (i.e. x ’s), and

utility is separable in the sub-utility of the categories.I Multi-stage budgeting: Representative consumer allocates total

expenditures in category and sub-categories. At each stage, only theprices (or price indices) of members of the group matter.

I Typically leads to log-linear category demand systemsI Examples:

F Hausman, Leonard, and Zona (1994): The effect of mergers in themarket for beers (i.e. categories = segments)

F Hausman (1994): The valuation of new goods in the market forready-to-eat cereals

Related approach: Pinkse et al. (2002) and Pinkse and Slade (2004)I Semi-parametric representative-agent demandI The degree of substitutability between products is modeled using

flexible functions of the distance between products.I Examples: Physical distance, alcohol content, calories, etc.

Demand for Differentiated Products Introduction 3 / 58

Discrete Choice Models

Two classes of modelsI Local: Each product has only few close substitutesI Global: Each product is equally substitutable with all others

Global competition: Logit (McFadden 1974, Anderson et al. 1992)I Indirect utility:

uij = δj − pj︸︷︷︸average

+ εij︸︷︷︸T1EV

I Demand (shares):

sj =exp((δj − pj)/σε)∑

j′∈J exp((δj′ − pj′)/σε)

I Elasticity of substitution:

ηjk =

(1

σεsjsk

)pksj

=1

σεpksj

I Two key properties: (i) Independence of Irrelevance Alternatives (IIA),and (ii) monopolistic competition in the limit (i.e. constant markup).

Demand for Differentiated Products Discrete choice models of differentiation 4 / 58

Local competition: Vertical or horizontal differentiation?Definition:

I Horizontal differentiation: Consumers have different valuations ofproducts available (even with equal prices)

I Vertical differentiation: Consumers would choose the same product ifprices were equal.

Horizontal Example: Linear city model (Hotelling 1929)I Preferences for product j = 1, 2:

uij = δ − pj − λ|li − xj |

where li ∼ U(0, 1) is the location (or taste) of consumer i , andxj ∈ (0, 1) is the “address” of products (i.e. characteristic) and x2 > x1.

I Demand:

Type l ∈ [x1, x2]: δ − p1 − λ(l − x1) = δ − p2 − λ(x2 − l)

l =(p2 − p1) + λ(x2 − x1)

2

s1 =

∫ l

0

f (li )dli =λ(x2 − x1)− p1 + p2

2


Local competition: Vertical or horizontal differentiation?

Vertical Example: Quality ladder (Shaked and Sutton 1982,Bresnahan 1987)

I Preferences for product j = 1, 2:

uij = θixj − pj

where li ∼ U(0, λ) is the location (or income) of consumer i , andxj ∈ (0, 1) is the “quality” of products (i.e. characteristic) and x2 > x1.

I Demand:

Type θ: θx1 − p1 = θx2 − p2

s1 =

∫ θ

0

f (θi )dθi = θ/λ =1

λ

p2 − p1

x2 − x1


Local competition: Vertical or horizontal differentiation?

In both cases the elasticity of substitution is positive only forneighboring products

I Spatial differentiation:

ηjk > 0 iff k is located the left or right of j

I Quality differentiation:

ηjk > 0 iff k is located immediately above or below j

General properties of local competition models:I Bertrand-Nash equilibrium prices are increasing in the degree of

differentiation (i.e. x2 − x1)I Principle of maximum differentiation: If firms endogenously choose

their characteristics, they will choose to locate at x1 = 0 and x2 = 1.I In general, as the number of products goes to infinity (e.g. entry cost→ 0), profit converges to zero (i.e. unlike monopolistic competition)


Hybrid model: Random-coefficients

Most products are both horizontally and vertically differentiated.Example: Cars

I Horizontal attributes: Family vs sports vs light-truck segmentsI Vertical attributes: reliability, horse-power, etc.

The work-horse model is the random-utility model with heterogenousvaluations for characteristics (aka random-coefficients):

maxj∈J

xjβi − αipj + εij

Note: Despite the multiplicative form, this functional form nestssome Hotelling-style models.

Example: uij = δj − pj − λ(zj − li )2 = xjβi − αpj .

Let, Aj(x , p) = (αi , βi , εi )|xjβi − αipj + εij > xkβi − αipk + εik ,∀k 6= j.Aggregate demand:

D(pj , p−j ; x) = M ×∫

(βi ,αi ,εi )∈Aj (x ,p)dF (βi , αi , εi ) = M × sj


Example: Nested-logit with demographic characteristics

Source: Goldberg (1995)

Notation:

I t ∈ (1, . . . ,T ): Decision nodes

I (zi , xj ): consumer/car characteristics

Indirect utility for car j :

ui,j =T∑t=1

xk(t,j)βti + εi,k(t,j), where βt

i = πtzi .

where εik is a Generalized Extreme Value(GEV) random variable, and k(t, j) is theoption of car j in node t.

At each node, the conditional probability ofchoosing an option takes a logit form.

Example: 2 nodes

Pj = Pj|kPk :

Pj|k =

exp(xjβ2/λk )∑j′∈C2|k

exp(xj′β1λk )

Pk = exp(wkβ1+λk Ik )∑l∈C1

exp(wlβ1Il/λl )

Ik = ln∑

j∈C2|kexp(xjβ2/λk)


Example: Mixed-Logit

The nested-logit model allows the (unobserved) taste for discretesegments to be vary across consumers.

The mixed-logit model allows the (unobserved) taste for continuouscharacteristics to differ across consumers.

Normal random-coefficient with demographics characteristics (zi ):

uij =

xjβi + εij If j 6= 0

εi0 Else.

where βi = πzi + ηi , ηi ∼ N(0,Σ), and εij ∼ T1EV(0, 1).

Aggregate market share:

sj =

∫ ∫exp(xj(πzi + ηi ))

1 +∑

j ′∈J exp(xj ′(πzi + ηi ))dF (zi)φ(ηi |Σ)dηi

Why is it useful? This is the most flexible discrete choice model. Canapproximate any random-utility model (McFadden and Train 2000).


Side note: Numerical Integration

Challenge: The mixed-logit model requires to numerically integratethe distribution of consumer heterogeneity.How?

I Monte-Carlo integration:

sj ≈1

S

S∑i=1

exp(xj(πzi + ηi ))

1 +∑

j′∈J exp(xj′(πzi + ηi ))

where zi , ηi are S random draws from F (zi ) (e.g. CPS) and Φ(ηi |Σ).I Refinements:

F Halton sequence (i.e. equally-spaced percentiles)F Importance samplingF Useful reference: Train (2002)

I Quadrature methods : Approximate (low dimensional) integrals using afinite number of points and weights∫

f (x)dx ≈n∑

i=1

f (xi )w(xi )

F References: Judd (1998), Skrainka and Judd (2011)


From McFadden to BLP

Panel of market shares and product characteristics:

st , pt , xtt=1,...,T

where t indexes a market, xt = xjt,1, . . . , xjt,Kj=1,...,nt is a matrix ofobserved characteristics, and pt , st = pjt , sjtj=1,...,nt is a matrix ofendogenous prices and market shares.

Market shares:sjt = Mt × qjt

where Mt is the observed market size (exogenous). Examples:I Cars: U.S. populationI Cereals: Population × Avg. servings per month.I Gasoline: Transportation needs (including car, bus, walk, etc.)

Demand for Differentiated Products Aggregate demand for differentiated products 12 / 58

From McFadden to BLPIndirect utility function (Nevo 2001):

uijt =

∑Kk=1 βi,kxjt,k − αipjt + ξjt + εij If j 6= 0

εi0 Else.

βi,k = βk + ziπk + ηi,k

αi = α + ziπp + ηi,p

zi ∼ F (·) (known) and ηi ∼ N(0,Σ) (unknown)

Contribution: Berry (1994) and Berry et al. (1995) introduced amethod to account for unobserved product characteristics (ξjt).Why is it important?

I Structural residual: Allow the model to explain differences in marketshares that are not accounted for by xjt and Ft(βi , αi ). The alternativeis to rely on measurement error (e.g Bresnahan 1987).

I Simultaneity: Recast the identification of the model into a “standard”IV problem. Previous literature relied on exogenous characteristics (e.gGoldberg 1995), or on deterministic models (e.g. Bresnahan 1987).

I Failing to account for unobserved product heterogeneity leads to biasedestimates own and cross-elasticities.


Inverse Demand RepresentationChallenge: The error of the model enters non-linearly into theaggregate demand (outcome). Does not look like standard IVmethods should work...

sjt = σj(δt , xt , pt ; Σ) =

∫ ∫exp(δjt + µij)

1 +∑nt

j′=1 exp(δj′t + µi,j′)dF (zi )φ(ηi ; Σ)dηi

where δjt = xjt β − αpjt + ξjt is the average value of (j , t), andµij = xjtηi ,x − ηi ,ppjt is the idiosyncratic value of observedcharacteristics.Solution: The error term is additive and separable in the inversedemand representation

ξjt = σ−1jt (st , xt , pt ; Σ)− xjt β + αpjt , iff θ = θ0

where θ = (Σ, β, α).Important: Berry et al. (1995) and Berry et al. (2013) show thatthere exists a unique solution under fairly general assumptions.This leads to a standard GMM estimator with IV vector Zjt :

minθ

(ZT ξ(θ))TW−1(ZT ξ(θ))


Side note: Nested-fixed point algorithm

To estimate θ you will need two routines: (i) a non-linearoptimization algorithm to minimize the GMM objective function, and(ii) a non-linear equation solver to invert the demand system.

I You should use standard (i.e. canned) routines for (i), but it is typicallyfaster to code your own non-linear solver for (ii).

Sketched of the algorithm:0. Guess Σ1

1. Invert demand: δjt(Σ1) = σ−1j (st , pt , xt |Σ1)

2. Estimate “linear parameters” using IVs Zjt : δjt(Σ1) = xjt β − αpjt + ξjt3. Evaluate moment conditions: m(Σ1) =

∑j,t ξjt(Σ1)Zjt

4. Check convergence of the GMM problem. Return to [1].

Berry et al. (1995) show that the following algorithm is a contractionmapping that can be used in step [1]:

δk+1jt = δkjt + ln sjt − lnσj(δ

k , pt , xt ; Σ)

Convergence is very slow with this algorithm. It is recommended tocombine this with a Newton algorithm.


What in the data identifies the model?

Like any IV problem, we need to have enough instruments to estimate(β, α,Σ). But what is a relevant/valid instrument?

To define the identification problem, we initially considerenvironments without prices.

Assumption: Conditional independence of unobserved attributes andthe menu of product characteristics (Berry et al. 1995)

E (ξjt |xt) = 0 (CI)

This suggests that θ is identified by variation in the choice-set ofconsumers.

I Most identification sections describe the “ideal” experiment in which anew product enters, and steal market shares from existing productswith similar attributes.

This “red-bus/blue-bus” logic is incomplete...

Demand for Differentiated Products The Identification Problem 16 / 58

The Identification Problem

The presence of an unobservable characteristic implies that when xtvaries, so do unobservables ξt = (ξ1t , . . . , ξnt t).

For any hypothesis θ we have a unique quality assignment function(Berry, Gandhi and Haile, 2013)

st = σ (xt , ξt ; θ) ⇐⇒ξt = σ−1 (st , xt ; θ)

Since any θ can explain the data equally well, the model is notidentified by the quality of the fit.

I This is what distinguish a BLP-model from a McFadden-model.

Identification problem: What is the correct quality assignmentmodel?


Identification of the Random Coefficient ModelBerry and Haile (2014)

Partition the characteristics vector, xt =x

(1)jt , x

(2)jt

, such that

uijt = β1x(1)jt + ξjt +

∑kx

(2)jt,kηik + εijt

Express the average quality of products in units of x(1)jt to obtain a

system of simultaneous equations with nt endogenous variables:

x(1)jt =

1

β1σ−1j

(st , x

(2)t ; θ

)+

1

β1ξjt .

Apply CI restriction to obtain the reduced-form of the model:

x(1)jt = E

[1

β1σ−1j

(st , x

(2)t ; θ

)|xt]

+ 0 = gj(xt ; θ)

If xt (nt · K variables) is complete forst , x

(2)t

(nt · K variables)

then F (η) is non-parametrically identified.I Completeness is the non-parametric analogue of the rank condition for

parametric models.Demand for Differentiated Products The Identification Problem 18 / 58

Special Case: Nested-Logit

Indirect utility:

ujt = xjtβx +∑k

ηik1(xjt,k = 1) + ξjt + εijt

Note: Nested-logit is equivalent to a model with random-coefficientson product segments dummies.

Linear quality assignment regression (Berry, 1994):

σ−1j (st , xt ; θ

0) = ln(sjt/s0t)− λ ln(sjt/sg(j),t)− xjtβ = ξjt

↔ ln(sjt/s0t) = xjtβx + λ ln(sjt/sg(j),t) + ξjt

Relevant and Valid IVs are the characteristics of competitors withinthe same nest (e.g. Verboven (1996), Bresnahan et al. (1997))


Simultaneous Equation for the Non-Linear Case

To define relevant instruments, we can linearize the system aroundthe true θ0 (Jorgenson and Laffont (1974) and Amemiya (1974)):

σ−1jt (st , xt ; θ) =

∑k

(Σk − Σ0k)∂σ−1

j (st , xt ; θ0)

∂θk+ ξjt + Error

= Jjt(st , xt ; θ0)b + ujt

This leads to a linear IV regression: Gauss-Newton Regression.

Important: ||bIV || = 0 defines any GMM estimator of θ.

Therefore, the most relevant IV for θk involves calculating the bestpredictor Jjt given xt (Amemiya (1977), Chamberlain (1987)):

hj ,k(xt ; θ0) = E

[Jjt,k(st , xt ; θ

0)|xt]


Side Note: Gauss-Newon Regression

The linear representation can be used to construct a Gauss-Newtonalgorithm to estimate θ.

σ−1jt (st , xt ; θ) =

∑k(θk − θ0

k)∂σ−1

j (st , xt ; θ0)

∂θk+ ξjt + Error

= Jjt(st , xt ; θ0)b + ujt

Iteration k ≥ 1:1 Invert demand: σ−1

jt (st , xt |θk−1) and Jjt(st , xt |θk−1)

2 Estimate bk by 2SLS.3 If |bk | < ε stop. Else, update θk = θk−1 + bk , and repeat steps (1)-(3).

With strong instruments, this procedure typically requires less than 5iterations.

I Weak IVs lead to severe numerical problems (e.g. Knittle andMetaxoglou (2014), Dube et al. (2012))

I Why? The central-limit theorem does no hold, and the quadraticapproximation is bad.


Optimal IV

The “optimal” IV for Σ is (Amemyia (1977), Chamberlain (1987)):

Dj ,k

(x t ; Σ0

)= E

∂σ−1j

(st , x

(2)t ; Σ0

)∂Σk

∣∣∣∣x t

How? Approximate the optimal IV by estimating a non-parametricfirst-stage regression (Newey, 1990)

Curse of Dimensionality: Dj ,k (x t) is a j-specific function of theentire menu of characteristics.

I First order: Dj,k(x t ; Σ0) ≈∑nt

j=1 x jtaj

I Impossible: BLP (1995)

Bottom line: If we can’t estimate the first-stage, how can weconstruct powerful instruments?

Demand for Differentiated Products Choice of Instruments 22 / 58

What does the characteristic structure imply for thereduced-form of the model?

Market-structure facing product j (dropping t):

(w j ,w−j) ≡((δj , x

(2)j

),(δ−j , x

(2)−j

))Properties of the linear-in-characteristics model:

I Symmetry:σj (w j ,w−j) = σk (w j ,w−j) ∀k 6= j

I Anonymity:σ (w j ,w−j) = σ

(w j ,wρ(−j)

)∀ρ

I Translation invariant: for any c ∈ RK

σ (w j + (0, c) ,w−j + (0, c)) = σ (w j ,w−j)


Re-Express the Demand System

Express the “state” of the market in differences relative to j and treatthe outside option just like any other product.

I Characteristic differences:

d (2)j,k = x (2)

k − x (2)j

I New normalization:

τj =exp(δj)

1 +∑

j′ exp(δj′),∀j = 0, . . . , n.

I Product k attributes: ωj,k =(τk ,d

(2)jt,k

)Demand for product j is a fully exchangeable function of ωj :

σ (w j ,w−j) = D(ωj)

where ωj = ωj ,0, . . . , ωj ,j−1, ωj ,j+1, . . . , ωj ,n.


Main Theory Result

Define the exogenous state of the market facing product j :

d j ,k = xk − x j

d j = (d j ,0, . . . ,d j ,j−1,d j ,j+1, . . . ,d j ,n)

Theorem

If the distribution of ξjj=1,...,n is exchangeable, then the reduced formbecomes

Dj ,k

(x ; Σ0

)= E

[∂σ−1

j

(s, x (2); Σ0

)∂Σk

∣∣∣∣∣x]

= hk (d j)

where hk is a symmetric function of the state vector.

Implication: hk is a vector symmetric function (see Briand 2009)


Why is it useful?1 Curse of dimensionality: In any fixed order approximation the

number of basis functions is independent of the number of products.

I A vector generalization of Pakes (1994), Pakes and McGuire (1994).

2 Examples: Basis functions

I First-order (i.e. market):

h(dj) ≈∑k 6=j

akdjt,k = a×

∑k 6=j

djt,k

(≡∑j′ 6=j

x jt

)I Second-order (i.e. distance):∑

j′ 6=j

(d kjt,j′

)2

, ∀k = 1, . . . ,K

I Histogram basis (i.e. nested-logit or hotelling):∑j′ 6=j

1(|dkjt,j′ | < κk)d jt,j′ , ∀k = 1, . . . ,K


Closing the loop

Let Aj (x t) be an L vector of basis functions summarizing theempirical distribution of characteristic differences: d jt,kk=0,...,nt .

Differentiation IV: These functions are moments describing therelative isolation of each product in characteristic space.

Donald, Imbens, and Newey (2003): Using basis functions directlyas IVs, is asymptotically equivalent to approximating the optimal IV.

I Recommended practice is to use low-order basis functions (Donald,Imbens, and Newey 2008).

Alternatively use Lasso methods with higher order polynomials (e.g.Belloni, Chernozhukov, and Hansen (2012))


Side Note: Demographic Panel

In many settings, product characteristics are fixed across markets, butthe distribution of consumer types vary (e.g. Nevo 2001).

Focus on a single characteristics xj

Assumption: Consumer heterogeneity ηit follows distribution of ademographic can be decomposed as

ηit = µt + σtνi

where (µt , σt) are known, and Pr(νi < x) = F (x) is common across t.

Preferences

uijt = δjt + θηitxj + εijt

= δjt + θνi xjt + εijt

where xjt = σtxj , and δjt = xj(β + θµt) + ξjt = xjβt + ξjt .

Differentiation IVs can be constructed as before: dxjt,k = xkt − xjt .


Illustration: A Motivating Example

One dimension random coefficient model:

uijt = β0 + β1x(1)jt + β2x

(2)jt + ξjt + σxvix

(2)jt + εijt .

Unobserved heterogeneity:I vi ∼ N(0, 1)I Numerical integration: 10 points Gauss-Hermite quadrature.

Data:I Panel structure: 100 markets × 15 productsI Characteristics: (ξjt , xjt) ∼ N(0, I).I Dimension: |xjt | = 2

Demand for Differentiated Products Illustration: Exogenous Characteristics 29 / 58

Identification in a PictureMarket IV: Sum of rivals’ characteristics

(A) Residual quality at σx = σ0x

-4-2

02

4True Residual Quality

(σ0)

-15 -10 -5 0 5 10(residual) Sum of rivals' characteristics

Note: Regression R2=0.

(B) Residual quality at σx = 0

-4-2

02

4IIA

Residual Quality

(σ=0

)

-15 -10 -5 0 5 10(residual) Sum of rivals' characteristics


Moment restriction: Independence of ξjt and the sum of rivalscharacteristics cannot distinguish (A) from (B)

Weak identification: The moment restrictions are “almost” satisfied atσx 6= σ0

x (Stock and Wright, 2000).


Identification in a PictureDifferentiation IV: Number of competitors located within 1 std-dev of x

(2)jt

(A) Residual quality at σx = σ0x

-4-2

02

4True Residual Quality

(σ0)

-10 -5 0 5 10(residual) Number of local competitors


(B) Residual quality at σx = 0

-4-2

02

4IIA

Residual Quality

(σ=0

)

-10 -5 0 5 10(residual) Number of local competitors

Note: Regression R2=.11.

Moment restriction: Independence of ξjt and the number of closecompetitors rules out (B).


Monte-Carlo Experiment

Random coefficient model:

uijt = δjt +K∑

k=1

vikx(2)jt,k + εijt , vi ∼ N(0, σ2

x I ).

Data:I Panel structure: 100 markets × 15 productsI Characteristics: (ξjt , x jt) ∼ N(0, I).I Dimension: |x jt | = K + 1I Monte-Carlo replications = 1,000

Instruments (K + 1):I Market IVs (first-order): Aj(x t) =

∑ntl 6=j xlt,k ,∀k = 1, . . . ,K

I Diff. IVs (2nd-order): Aj(x t) =∑nt

j′=1

(dkjt,j′

)2,∀k = 1, . . . ,K


Result 1: Weak IV ProblemSpecification: One dimension of heterogeneity

01

23

4Ke

rnel

den

sity

0.0

5.1

Frac

tion

0 2 4 6 8 10Random coefficient parameter

Market IV Diff. IV


Result 2: Weak IVs with Multiple Dimensions

VARIABLES Dimensions of consumer heterogeneity1 2 3 4

σ1 2.012 1.931 1.919 2.055(1.3) (1.423) (1.383) (1.467)

σ2 1.978 1.934 2.055(1.302) (1.313) (1.412)

σ3 1.992 2.088(1.353) (1.564)

σ4 2.106(1.494)

Nb. IVs 2 3 4 5

Parenthesis: RMSE. Starting at true parameter = 2.


Result 3: Differentiation IVs with Multiple Dimensions

VARIABLES Dimensions of consumer heterogeneity1 2 3 4 6

σ1 2.002 1.998 1.994 1.994 1.991(.107) (.109) (.105) (.111) (.115)

σ2 1.995 1.996 1.992 1.993(.108) (.105) (.111) (.115)

σ3 1.995 1.994 1.991(.109) (.113) (.116)

σ4 1.995 1.990(.11) (.115)

σ5 1.997(.118)

σ6 1.993(.118)

Nb. IVs 2 3 4 5 7

Parenthesis: RMSE. Starting at true parameter = 2.


Result 4: Weak IVs and Numerical ProblemsSimulation: 1,000 samples and 10 random starting values each

Differentiation IV Market IVEst. RMSE Est. RMSE

σ1 1.994 0.111 2.107 1.536σ2 1.992 0.111 2.053 1.477σ3 1.994 0.113 2.088 1.618σ4 1.995 0.110 2.131 1.640

Algorithm Simplex SimplexNb. IVs 5 5CPU time (sec) 0.718 33Local optima (fraction) 0.002 0.53Global min/Local min 46.6 53


Experiment 2: Correlated Random Coefficients

Consumer heterogeneity:

ηi ∼ N(0,Σ)

Note: 4 dimensions ⇒ 10 non-linear parameters (choleski)

Panel structure:100 markets × 50 products

Differentiation IVs: Second-order polynomials (with interactions):

nt∑j ′=1

(dkjt,j ′

)2and

nt∑j ′=1

(dkjt,j ′ × d l

jt,j ′

)for all characteristics k and l .


Simulation Results: Unrestricted Covariances

Σ.1 Σ.2 Σ.3 Σ.4

Estimated parameters:Σ1. 4.066

(.214)Σ2. -2.080 4.268

(.132) (.242)Σ3. 1.993 0.697 4.502

(.143) (.154) (.212)Σ4. -1.147 -0.405 -0.458 3.050

(.123) (.124) (.135) (.204)True parameters:

Σ1. 4.064Σ2. -2.083 4.270Σ3. 1.995 0.689 4.505Σ4. -1.152 -0.398 -0.462 3.071

CPU time (sec) 3.151

Parenthesis: RMSE. Starting values = true. Algorithm: Gauss-Newton.


Endogenous Prices

Berry (1994) discuss the simultaneity of prices and unobservedattributes (ξjt). Conditional on Σ, this is a “simpler” linear-IVproblem:

δjt(Σ) = xjt β − αpjt + ξjt

Note: Since the inverse demand is linear, we can easily includefixed-effects in the model to control for time invariant unobservables.The literature has proposed several alternatives:

I BLP-IVs: Characteristics of products owned by the same firm.Example:

wjt =∑j′∈Fj

xjt

I Hausman-IVs: Price of same and competing brands in neighboringgeographic markets. Why? If the demand shock ξjt is independentacross markets, but production plants deliver to several markets, priceswill be spatially correlated (i.e. cost shocks). E.g.: Nevo (2001).

I Cost IVs: Input prices, shipping distance, merger (e.g. Miller andWeinberg (2016))

Demand for Differentiated Products Illustration: Endogenous Characteristics 40 / 58

Supply Restrictions: Pricing game and cost function

Equilibrium pricing restrictions can be very informative about demandparameters.

Assume the following constant marginal cost expression:

ln(mcjt) = wjtγ + ωjt

The multi-product Bertrand-Nash equilibrium price vector ischaracterized by the following FOC for all j .

sjt +∑k∈Fjt

(pkt −mckt)∂σk(δt , pt , xt ; Σ)

∂pjt= 0

where Fjt is the set of products owned by the same firm as j ’s.


Supply Restrictions: Pricing game and cost function

Matrix representation:

st + (∆t ∗ Ωt)(pt −mct) = 0

where ∆jk,t = ∂σk (δt ,pt ,xt ;Σ)∂pjt

, and Ωjk,t = 1 if (jk) are owned by the

same company (zero otherwise).

Given Σ, the marginal cost vector can be recovered from the data:

mct(Σ, γ) = wtγ + ωt = pt + (Ωt ∗∆t)−1 σ(δt , pt , xt |Σ)

We can therefore form a joint vector of moment conditions:

E

[ξjt(θ)zjtωjt(θ, γ)zjt

]= 0

The demand parameters are therefore identified by the variationcoming from the IVs, and the cross-equation restrictions implied bythe pricing model.


Illustration: Vertical Differentiation

Demand: Mixed-logit with vertical differentiationI Indirect utility

uijt = δjt − αipjt + εijt

where αi = σpy−1i , and log(yi ) ∼ N(µy , σ

2y ) (known).

Supply: Bertrand-Nash with multi-product firms

I Constant marginal cost: mcjt = γ0 + γxxjt + ωjt


How to incorporate endogenous prices?

Price instruments: w t = wjtj=1,...,nt

Reduced-form of the model:

Dj(x t ,pt ,w t ;σ0p) = E

(∂σ−1

j (st , x t ,pt |σ0p)

∂σp

∣∣∣x t ,w t

)6= h(d x

j ,dpj )

Solution: Use insights from Berry et al. (1999) and Reynaert &Verboven (2013) to take expectation for price inside:

Dj(x t ,pt ,w t ;σ0p) ≈ Dj(x t , pt ,w t ;σ

0p) = h

(d xj , d

pj

)I.e., dp

jt,k = pkt − pjt is replaced by dpjt,k = E (pkt |wkt)− E (pjt |w jt).


BLP (1995) Instruments

BLP’s original basis function:

zjt =

wjt ,∑j ′∈Fft

wj ′,t ,∑j ′ /∈Fft

wj ′,t

where wjt = (1, xjt , ωjt), and Fft is the set of products controlled byfirm f .

This corresponds to the first moment of characteristic differences

z ′jt =

wjt ,∑j ′∈Fft

dwjt,k ,

∑j ′ /∈Fft

dwj ′,t

where dw

jt,k =(

1, dxjt,k , d

ωjt,k

), and span(Z ) = span(Z ′).


Differentiation IVs

Characteristic differences: Let djt,k = pkt − pjt denote theexogenous price differences between j and k , where:

pjt = π0 + π1xjt + π2ωjt .

Two basis functions:1 Second moment: Euclidian distance

zjt =

wjt ,∑j′∈Fft

d2j′t,j ,

∑j′ /∈Fft

d2j′t,j

2 Histogram: Characteristics of local competitors

zjt =

wjt ,∑

j ′ ∈ Fft ,∣∣dj′t,j ∣∣ < sd(d)

wj′t ,∑

j ′ /∈ Fft ,∣∣dj′t,j ∣∣ < sd(d)

wj′t


Weak IV problem is very severe without cost-shiftersPanel structure: J = 50 and T = 10

0.0

2.0

4.0

6.0

8.1

Kern

el d

ensi

ty

0.0

5.1

Frac

tion

0 20 40 60 80Random coefficient estimate (price)

BLP (1995) NBH MomentNote: True value = 20


Simulation Results

IV: Sum of charact. IV: Local competitorsw/o cost w/ cost w/o cost w/ cost

βpAverage 0.46 1.08 1.00 1.02RMSE 2.19 1.32 0.22 0.18

σpAverage 13.24 17.47 19.10 19.68RMSE 10.84 7.95 3.93 1.51


The weak IV problem does not go away in large samples...

0.0

5.1

.15

.2Ke

rnel

den

sity

0 20 40 60Random coefficient estimate (price)

BLP (1995) Local competition

Sample size: Solid = 500, Long dash = 1,000, Dash = 2,500.


Natural Experiments

Hotelling example:

uim = ξjm − λ(ηi − xjm)2 + εijm

and E (ξjt |xjt) 6= 0 (i.e. endogenous locations).

Natural experiment: Exogenous entry of a new product (x ′ = 5)

Three-way panel:I Product j , market m, and time (t = 0, 1).I Treatment variable:

Djm = 1 (|xjm − 5| < Cutoff)

“Ideal” first-stage: Difference-in-difference regression

∂σ−1j (st , xt , pt |θ0)

∂θ= µjm + τt + γDjm × 1(t = 1) + ejmt


Natural Experiment: Hotelling ExampleDGP: δjmt = ξjm + ∆ξjmt , where E (ξjm|xm) 6= 0

“Diff-in-Diff” specification:

z jmt = Product Dummyjm, 1(t = 1), 1(|xjm − 5| < 1)1(t = 1)0

24

68

kdensity

theta

-2.5 -2 -1.5 -1 -.5x

Diff. IV + FE Diff. IV BLP (1999)


Evaluating the Relevance of Instruments

Ex-post: First-stage testI First-stage of the non-linear GMM problem at θgmm:

∂σ−1j (st , xt , pt |θgmm)

∂θ= πxxjt + πzzjt + ejt

I Standard tests for weak instruments in linear models can be used totest the relevance of zjt (e.g. IVREG2 in STATA).

Ex-ante: IIA testI A strong instrument for Σ is able to reject the wrong model (Stock

and Wright, 2000)I Under H0 : Σ = 0, the inverse demand equation is independent of x−j :

σ−1jt (st , xt ; Σ = 0) = ln sjt/s0t = xjtβ + zjtγ + ξjt

I Standard test statistics for H0 : γ = 0, can be used to test nullhypothesis of IIA preferences

I With endogenous prices, this test is equivalent to the J-test at Σ = 0.

Demand for Differentiated Products Testing for Weak IVs 52 / 58

Micro-moments in demand estimation

Motivation:I Heterogeneity parameters are often imprecisely estimated with

aggregate data solelyI Propose to add “micro-moment” conditions from consumer surveysI Originally proposed by Petrin (2002)

Additional moment: Difference between predicted and observedconditional market share of “large” cars for large/small families.

Why is it not redundant? We’re inverting the aggregate marketshares, not the conditional market shares.

Demand for Differentiated Products Micro-moments 53 / 58

Empirical Specification

Indirect utility combines demographics and normalrandom-coefficients:

uij = αi ln(yi − pj) + xjβ +∑k

γkνikxik + ξj + εij

Where,I Price sensitivity αi varies across income groupsI Family size categories:

γic = γc ln(Familiy sizei )νi

where c indicates minivan, station wagons, full-size vans, and SUVs.


Moment Conditions

IV moments: Combined demand and supply restrictions

1

n

∑j ,t

ξjtZjt = 0

1

n

∑j ,t

ωjtZjt = 0

Micro-moments:I Share of new vehicles purchased by income group (3)I Average family size per car typeI Average age conditional on buying each car type.

All three sets of moments are “stacked” to form vector Gn(θ,Z ).


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