1 Demand Estimation - UW Faculty Web Serverfaculty.washington.edu/bajari/iosp07/lecture2.pdf · 1...
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1 Demand Estimation
1. Motivation.
2. Estimation.
3. Consistent Estimation
4. Limitations of Logit.
5. BLP
6. Microdata
7. Semiparametrics
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2 Motivation
• We begin our study of differentiated product mar-kets by describing the method of BLP (1995) for
demand estimation in differentiated product mar-
kets.
• We will also discuss some limitations of this methodand some possible extensions.
• BLP is a method for estimating demand in differ-entiated product markets using aggregate data.
• The method allows for endogenous prices and ran-dom coefficients.
• The method also allows for consistent estimationof the model parameters even if there is imperfect
competition.
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3 A simple example
• To motivate the framework, consider the follow-ing simple example based on Berry (RAND, 1994).
• There are i = 1, ..., I (=∞) agents in t = 1, ..., Tmarkets.
• Each agent makes a choice between j = 1, ..., J
mutually exclusive alternatives.
• xj,t = (xjt,1, ..., xjt,K)0 is a K × 1vector of char-
acteristics for product j.
• Let pj,t denote the price of j at time t.
• ξj,t = ξj+ ξt+∆ξj,t denote an unobserved char-
acteristic/demand shock/measurement error in price.
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• ξj is a permanent component for j, ξt is a com-
mon shock and ∆ξj,t is a product/time specific
shock for j.
• Specify the random utility as:
uijt = x0j,tβ − αpj,t + ξj,t + εij
• Assume that the error term corresponds to the
(conditional) logit model.
• Then the market share for j at time t is:
sjt(x, β, α, ξ) =exp(x0j,tβ − αpj,t + ξj,t)PJ
j0=1 exp(x0j0,tβ − αpj0,t + ξj0,t)
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• Berry assumes that we are working with aggre-gate data and that, at the true parameter values,
sjt(x, β, α, ξ) = Sjt where Sjt denotes the ”true”
market share.
• This differs from the standard logit model in two
ways.
• First, we have unobserved heterogeneity/demandshock, ξj,t.
• Why ξj,t?
1. Observe list of product attributes is incomplete.
This goes back to hedonic regressions.
2. Measurement error in prices. Typically price data
is an average.
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3. Without ξj,t, shares should not vary holding x0j,t pj,t
fixed. This is likely to be violated in some data
sets.
• Second, we are working with aggregate data in-stead of individual choices, as in the standard con-
ditional logit.
• Thus, the data set needs to contain market shares.
• Many of the methods we are going to study arenot valid if market shares are measured with error.
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3.1 Estimation.
• Berry notes that the following transformation canbe made:
log(sjt(x, β, α, ξ)) = et + x0j,tβ − αpj,t + ξj,t
et = − log(JX
j0=1exp(x0j0,tβ − αpj0,t + ξj0,t))
• Next we assume a ”law of large numbers” so thatSjt = sjt(x, β, α, ξ) at the true parameters.
• If we normalize the utility of the outside good tozero, this implies that:
s0t(x, β, α, ξ) =exp(0)PJ
j0=1 exp(x0j0,tβ − αpj0,t + ξj0,t)
log s0t(x, β, α, ξ) = 0− et
• This implies that:
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log(Sjt)− log(Sot) = x0j,tβ − αpj,t + ξj,t
• where Sot is the share of the outside good.
• Berry noted that an obvious way to estimate thismodel is by regression.
• The dependent variable is log(Sjt)− log(Sot)
• The independent variables are [x0j,t, pj,t]
• The error term is ξj,t.
• However, in general we would expect cov(pj,t, ξj,t) 6=0.
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• In the presence of a demand shock, oligopolymodels suggest that firms should raise prices.
• Thus, ols estimates of β and α will be biased.
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3.2 Consistent Estimation.
3.2.1 Fixed Effects
• A first approach to consistent estimation would
be to estimate the following fixed effects model:
log(Sjt)− log(Sot) = x0j,tβ − αpj,t + ξj + ξt +∆ξjt
• Where ξj is a brand fixed effect, ξt is a categorymarket/time shock
• The identifying assumption is E[∆ξjt|x0j,t, pj,t] =0
• This is clearly more appealing thatE[ξjt|x0j,t, pj,t] =0
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• However, there are a couple of limitations.
• First, there may be colinearity between ξj and
xj,t if some characteristics for product j are time
invariant.
• Thus, a brand fixed effect does not allow us to
learn about the valuation of individual product
characteristics.
• Also, it presumes that cov(pjt,∆ξjt) = 0
• This assumes that in a given time period, productlevel price variation is exogenous.
• Remark: This type of assumption is commonlymade in marketing.
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3.2.2 BLP Instruments
• A second approach to identification is to find a
set of instruments.
• That is, we need to find a variable zjt such thatE[ξjt|zjt] = 0, cov(zjt, [xjt, pjt])6= 0 (i.e. satis-
fies standard rank conditions for IV).
• One obvious instrument is a supply shifter (e.g.change in costs).
• Problem, there are too few instruments and theymay be weak.
• Weak instruments- standard errors incorrect, biaslarge.
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• BLP and Berry(1994) suggest measures of isola-tion in product space.
• e.g. zjtk =Pj06=j xj0tk
• How much does product j contribute to the (un-weighted) average of characteristic k.
• This instrument is usually available and it tendsto be highly correlated with price.
• Models of oligopoly suggest the more isolated youare in product space, the more likely you are to
have a higher margin.
• Thus, prices will be correlated with zjtk.
• Critiques of this instrument-
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1. Little variation over time.
2. Assumes cov(ξjt, xjtk) = 0.
• This assume that omitted product attributes areuncorrelated wth observed attributed.
• This seems hard to believe since the observed at-tributes are correlated with each other.
• This is a classic problem in demand estimation.
• In hedonic, researchers have long worried aboutthe consistency of:
pjt = x0jtβ + ξjt
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• For example, in a home price regression, the ob-served attributes are likely to be correlated with
the unobserved attributes.
• Ackerberg, however, notes that if cov(zjtk, xjtk) =0 for all k, it is possible to consistently estimate
price elasticities for this model (even if other pa-
rameter estimates are biased).
• This condition is testable!
• Many questions can be answered with price elasticities-e.g. measurement of market power.
• As with the the fixed effects case above, it seemsmore appealing to assume:
E[∆ξjt|zjt] = 0
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• This is possible if we include brand/time fixedeffects.
• Remark: Price endogeneity is being accounted forusing only demand side information.
3.3 Hausmann Instruments
• Hausmann proposes using prices in other marketsas instruments.
• E.g. use prices in Iowa, Wisconsin and the Dako-tas as instruments for price endogeneity in Min-
neapolis.
• The idea behind these instruments is that theypick up common cost shocks.
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• However, if they pick up common demand shocks,they are invalid.
• In general, both the BLP and Hausmann instru-ments have the advantage of at least being avail-
able!
4 Limitations of the Logit
• Some Limitations of the Logit
• While the logit model is computationally conve-nient, it imposes some unpleasant restrictions on
the data.
• It is still widely used since there are few other
computationally convenient estimators.
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1. Implausible substitution patters.
• In the logit model exhibits the independence ofirrelevant alternatives (IIA).
• That is, the ratio of the probability of two choicesdoes not change depending on the set of choices
that are available.
Pr(i chooses j)
Pr(i chooses j0)= constant
for all j and j0 regardless of the set of alternativesthat are available.
• A famous example is the red bus/blue bus prob-lem
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• Suppose that we are studying the mode of trans-portation choice.
• Choice set is take the (red) bus to work or todrive.
• Suppose that these choices are equal in probabil-ity.
• Now suppose that the bus company introduces
blue buses in addition to red buses.
• Suppose that consumers are indifferent about thecolor of their bus and that the probability of the
red bus and blue bus is equal.
• IIA implies that prob(red bus)= prob(blue bus)
=prob(drive) =1/3.
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• Amore ”intuitive” answer would be rob(red bus)=prob(blue bus)= 1/4 and prob(drive)=1/2.
• This example shows that IIA can give wierd sub-stitution patterns.
• This can also show up in terms of price elasticities.
• Suppose that we are modeling consumer demandfor a differentiated product.
• Suppose that the latent utilities are:
ynj = x0njβ − αpj + εnj
• where pj is price.
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• Calculating the own and cross price elasticites.
ηjk =∂ Pr(i chooses j)
∂pk
pkPr(i chooses j)
=n−αpj(1− sj) if j = k
−αpksk
• Since in most cases there are many products sothat the market shares are typically small, (1−sj)is approximately equal to price.
• This implies that the lower the price the lower theelasticity.
• This implies that markups should be higher incheap products.
• This is clearly not appropriate in many industries.
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• A second limitation is that cross price elasticitiesare determined by αpksk.
• Suppose that Lucky Charms and Grape Nuts aresimilarly priced and have a similar market share.
• An implication of this formula is that both ofthese will have the same cross price elasticity with
CoCo Puffs.
• This is clearly a priori implausible, yet it is anassumption that we have imposed through the
functional form.
3. Treatment of Heterogeneity.
• In the logit model, consumers are only heteroge-nous because of εij.
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• εij can be thought of as adding additional product
characteristics into the model for each j and an iid
random preference shock for that characteristic.
• Caplin and Nalebuff argue that this generates toomuch ”taste for variety”.
• Applied studies of welfare, such as Petrin (JPE2002, Quantifying the Benefits of New Products:
The Case of the Minivan), argue that to much of
the utility comes from implausbly large draws of
the εij.
• Leads to pathological implications (e.g. markupsin Bertrand may not converge to zero as market
becomes thick).
• See Anderson, DePalma and Thisse.
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5 BLP-Random Coefficients Logit.
• In Berry (1994) and BLP (1995), consumer pref-erences can be written as:
u(xj, ξj, pj, vi; θd)
where:
• xj = (xj,1, ..., xj,K) is a vector of K character-
istics of product j that are observed by both the
economist and the consumer.
• ξj is a characteristic of product j observed by the
consumer but not by the economist.
• pj is the price of good j
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• vi vector of taste parameters for consumer i
• θd vector of demand parameters.
• One commonly used specification is the logit modelwith random (normal) coefficients:
uij = xjβi − αpj + ξj + εij
• The K random coefficients are:
βi,k = βk + σkηi,k
ηi,k ∼ N(0, 1), iid
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• Consumer i will purchase good j if and only if it isutility maximizing, just as in the previous lecture.
• Question: How do we interpret the parameters ofthis model?
• It is useful to decompose utility into two parts, thefirst is a “mean” level of utility and the second is
a heteroskedastic error terms that captures the
effect of random tastes parameters:
υij =
⎡⎣Xk
xjkσkηi,k
⎤⎦+ εij
δj = xjβ − αpj + ξj
• We can now write utility of person i for product
j as:
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uij = δj + υij
• Next, we will write the market shares for aggre-gate demand in a particularly convenient fashion.
First define the set of “error terms” that make
product j utility maximizing given the J dimen-
sional vector δ = (δj)
Aj(δ) =nυi = (vij)|δj + vij ≥ δj0 + vij0 for all j
0 6= jo
• The market share of product j can then be writtenas (assuming a law of large numbers):
sj(δ(x, p, ξ), x, θ) =ZAj(δ)
f(υ)dυ
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• In this case, the parameter θ is β, α and σ.
• Given θ and the demand for product j actually
observed in the data, esj it must be the case that:
esj = sj(δ(x, p, ξ), x, θ)
• Given θ, this can be expressed as a system of J
equations in J unknowns (the ξj).
• To estimate, we find a set of instruments for theξj.
• We must find a set of instruments correlated withthe endogenous variable pj, but uncorrelated with
the residual ξj.
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Commonly used instruments:
1. The product characteristics.
2. Prices of products in other markets (interpret ξjas a demand shifter).
3. Measures of isolation in product space (Pj06=j xj0,k)
4. Cost shifters.
• Question: Are these really valid instruments?
• Typically we think of product characteristics as achoice variable.
• Suppose that a firm chose product characteristics
optimally.
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• Then the unobserved characteristics (to the econo-metrician) of a product would be independent of
the observed characteritics only under strong sep-
arability assumptions about cost and demand.
• The model written down probably violates theseparability assumptions on demand.
• A number of empirical case studies have been
done. They find that BLP style estimators typi-
cally find more elastic demand curves.
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6 Firm Behavior.
• In the model above, we abstracted from the be-
havior of the firm.
• Suppose that firms engage in Bertrand price com-petition.
• Let firm f produce some set of products Pf .
• Then to profit maximization problem for firm f is
to choose prices pj for j ∈ Pf that maximize ex-
pected profit holding the prices of the other firms
fixed:
πf =Xj∈Pf
(pj −mcj)Msj(x, p, ξ, θ)
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• Suppose that we know the function sj, then the
first order conditions for all of the products are a
system of J equations in J unknowns where the
unknowns are the latent cost parameters mcj.
• Note that if we recover the marginal cost parame-ters by assuming Bertrand price competition and
that the first order conditions hold, we could do
policy experiments.
• For instance, some have used this approproach tosimulate the effects of a merger.
• BLP (1995) proceeds in a similar fashion to Berry,except that it models the supply side as well by
assuming that firms are Bertrand price competi-
tors.
• We then need to find instruments for a set ofunobserved supply shifters.
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• BLP propose the use of product characteristics.
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7 Computation.
• In this section, I shall outline some of the keysteps needed to actually compute Berry (1994).
• A key step in many programming projects is to
do a fake data experiment.
• Simulate the model using fixed parameter values.
• Pretend you don’t know the parameter values andestimate.
• This tests the code and sometimes shows you lim-itations of the models.
• One of the best ways to really learn the econo-metrics in a paper is to do a fake data experiment.
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• We shall consider as an example the random co-
efficinet logit model.
There are basically 4 things we need to do in order to
compute the value of the objective function in order
to do GMM.
1. For a given value of σ and δ, compute the vector
of market shares.
2. For a given value of σ, find the vector δ that
equates the observed market shares and those pre-
dicted by the model using the contraction map-
ping.
3. Given δ and β, α compute the value of ξ
4. Search for the value of ξ that mimizes the objec-
tive function.
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• We shall consider these one at a time.
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7.1 Computing Market Shares.
• In the random coefficient logit model, we can
compute the market shares, given δ as follows:
sj(δ, σ) =Z exp(δj +
Pk xj,kηi,kσk)
1 +Pj0 exp(δj0 +
Pk xj0,kηi,kσk)
df(ηi)
• In practice, the integral above is computed usingsimulation.
• Make a set of S simulation draws and keep themfixed for the whole problem.
• Sometimes importance sampling is useful in orderto improve the speed/accuracy of the integration.
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• See Judd for an overview of numerical integration.
• We can compute confidence intervals using stan-dard methods to see whether the simulated mar-
ket shares are well estimated.
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7.2 The contraction mapping.
• Next, we wish to find the δ that matches the
observed market shares given σ.
• In Berry and BLP they demonstrate that the fol-lowing is a contraction:
δ(n+1)j = δ
(n)j + ln(esj)− ln(sj(δ, σ))
• Therefore, given that we can compute marketshares, we can use the formula above to find the
value of δ by making an initial guess at δ and then
evaluting the equation above until convergence is
(approximately) achieved.
• A mapping T that maps S → S is a contraction
with modulus β if for all x, y d(T ◦ x, T ◦ y) ≤βd(x, y).
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• A contraction mapping has a unique fixed point.
• Let vo be an initial guess about the fixed pointv. Let Tn(vo) denote applying the mapping n
times, as in the previous equation.
• This converges to the fixed point at an exponen-tial rate.
• Point: Market shares can be inverted very quicklyin a fairly simple manner!
• Contraction mappings are used all the time in eco-nomics, particularly in modern Macro.
• See Stokey and Lucas, chapters 4 and 5 for proofs.
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8 Computing the value of ξ
• The next set is simple. Just let:
ξj = δj − (xjβ − αpj)
where δj is computed using the contraction mapping.
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8.1 Computing the value of the objective
function.
• Let Z be the set of instruments.
• The objective function is formulated as in all GMMproblems assuming E (ξ|Z) = 0.
• The econometrician then chooses β, α, and σ inorder to minimize the objective function.
• Standard mathematical programs (MATLAB, GAUSS,IMSL,NAG) contain software for optimization prob-
lems.
• One standard way to proceed is to do a roughglobal search first and then use a derivative based
method second once you have a very rough sense
of the overall shape of the objective function.
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• Multiple starting points commonly used in orderto search for multiple local solutions to minimiza-
tion problem..
• See Judd for an overview of numerical minimiza-tion.
• Doing a ”fake data experiment” is a good way tolearn how well the estimator works.
• Fix true parameters, simulate the model. Then
see if your computations allow you to get back
the correct answer.
9 Individual Level Data
• These models are discussed in some detail in Cameronand Travedi, Chapter 15.
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• In the notation of Cameron and Trivedi, j =
1, ..., J indexes choices and i = 1, ..., I indexes
households.
• That is:
Uij = x0ijβi + εijβi ∼ N(β,Σβ)
• In the above, εij comes from the Weibull distrib-
ution as before.
• Each household i is allowed to have a unique setof marginal utilities which come from a normal
distribution with unknown mean and variance.
• In this model, the probability that houshold i choosesproduct j, pij is therefore:
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pij =exp(x0njβi)
1 +JX
j0=1exp(x0nj0βi)
• The probability of choice j, pj is therefore:
pj(β,Σβ) =Z exp(x0njβi)
1 +JX
j0=1exp(x0nj0βi)
φ(βi|β,Σβ)dβi
• where φ(βi|β,Σβ) is the normal density.
• We could in principal estimate the model usingMLE since our model generates a likelihood for
the choice probabilities.
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• If x0nj has a large dimension (e.g. there are manycharacteristics), then evaluation of the above in-
tegral is difficult.
• Therefore, we need to estimate these models us-ing simulation.
• We will study the theory of simulation in detailnext week, however, we will sketch how to form
a simulated likelihood function.
• Suppose that the βi can be written as:
βi,k = βk + ηiσk k = 1, ...,K
ηi standard normal
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• In this specification, we are assuming that therandom coefficients are independently distributed
across k with a normal distribution of mean βkand standard deviation σk.
• In the simplest simulation based estimator, wecould make s = 1, ..., S monte carlo draws η
(s)i
of the random coefficients for each household i.
• A monte carlo estimator of bpj(β,Σβ) is then:
bpj(β,Σβ) =1
S
X exp(x0njβk +Pηiσkxnjk)
1 +JX
j0=1exp(x0nj0βk +
Pηiσkxnj0k)
• The ”simulated” likelihood function would thenbe:
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ln bL(β,Σβ) =NXi=1
JXj=1
ynj log³bpj(β,Σβ)
´
• If we let the number of simulations become infi-nite (at an appropriate rate) as the sample size
N → ∞, this will yield a consistent estimator ofour model parameters.
• It is also possible to derive the asymptotic vari-ance matrix in a reasonably straightforward way.
• There are some limitations, however.
• A first limitation is that this estimator is in generalbiased.
• An alternative, unbiased estimator is based on aNLLS approach:
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NXn=1
xn,r³yn,j − bpj(β,Σβ)
´= 0
r = 1, ..,K and j = 1, ..., J
• Unfortunately, this estimator is not efficient ingeneral and may not even be smooth without us-
ing some fairly sophisticated numerical approaches.
• A second limitation is the variance of our esti-
mates may be high if the distribution of random
coefficients is flexibly specified.
• Hence, tightly parameterized models are required.
• Computational burden increases considerably innumber of choices.
• An alternative approach is to use Gibbs sampling.
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10 A semiparametric alternative
• Briefly discuss a computationally simple, but flex-ible estimator due to Bajari, Fox and Ryan (2006).
• Let (β(r)1 , ..., β(r)K ) for r = 1, ...,∞ be a sequence
of real vectors that is dense in the domain of βi.
• Assume that the random preference shock comes
from the Weibull distribution as before.
• We will chose a large, but finite number of pointsof support r = 1, ..., R for the distribution of ran-
dom coefficients.
• Let p(r) denote the probability that βi = (β(r)1 , ..., β
(r)K ).
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• Let P (j) denote the probability that the choice jis made. Then
P (j|xij) =RXr=1
p(r)
⎛⎜⎜⎜⎜⎜⎜⎜⎝exp(β(r)xij)JX
j0=1exp(β(r)xij0)
⎞⎟⎟⎟⎟⎟⎟⎟⎠
• Note that in the above we let regressors vary byboth j and i.
• Let yij = 1 if consumer i chooses j and zero
otherwise.
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• Straightforward algebra implies that:
yij =IX
i=1
RXr=1
p(r)
⎛⎜⎜⎜⎜⎜⎜⎜⎝exp(β(r)xij)JX
j0=1exp(β(r)xij0)
⎞⎟⎟⎟⎟⎟⎟⎟⎠+ ejtm
(1)
for j = 1, ..., J, i = 1, ..., I (2)
where ejtm = yij − P (j|xij)
• Since ejtm is pure forecast error due to random
sampling, it is orthogonal to all of our regressors
and functions of our regressors.
• An attractive feature of this model is that it is lin-ear in the parameters p(r) and we do not require
nonlinear maximization to find the estimator.
• If we let R be sufficiently large, this can approx-
imate any discrete choice model to an arbitrary
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degree of precision due to a result by McFadden
and Trian (2000).
• A first (naive) estimator for this model would beto minimize:
bp = argminp
1
I
IXi=1
⎛⎜⎜⎜⎜⎜⎜⎜⎝yij −RXr=1
p(r)
⎛⎜⎜⎜⎜⎜⎜⎜⎝exp(β(r)xij)JX
j0=1exp(β(r)xij0)
⎞⎟⎟⎟⎟⎟⎟⎟⎠
⎞⎟⎟⎟⎟⎟⎟⎟⎠
2
• Note that this is just regression!
• We would then naively interpret our estimates asthe probabilities p(r).
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• In Monte Carlo studies, this performed poorly.Many of the coefficients p(r) were negative forinstance.
Instead, we propose using the following estimator:
bp = argminp
1
I
IXi=1
⎛⎜⎜⎜⎜⎜⎜⎜⎝yij −RXr=1
p(r)
⎛⎜⎜⎜⎜⎜⎜⎜⎝exp(β(r)xij)JX
j0=1exp(β(r)xij0)
⎞⎟⎟⎟⎟⎟⎟⎟⎠
⎞⎟⎟⎟⎟⎟⎟⎟⎠
2
s.t. p(r) ≥ 0 andXrp(r) = 1
• This is an inequality constrained regression, asconsidered by Judge and Takayam (1966), Geweke(1986) and Wolak (1987).
• This is a straightforward quadratic programmingproblem.
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• The algorithm for finding the solution is impli-
mented in standard software packages including
Matlab.
• The standard errors are simple for this model andcorrepond to simple modifications of OLS formu-
las.
11 Identification.
• An important question to ask is whether our ran-dom coefficient discrete choice models are identi-
fied.
• That is, can the primitives (i.e. random utilities)
be uniquely recovered from the data.
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• The answer to this question in general is no.
• To see why, suppose that there was only a singleconsumer with a deterministic utility.
• This is a special case of our more general, randomutility framework.
• Utility functions cannot be identified from choice
behavior.
• We can always make monotonic transformations.
• Therefore, in general, distributions over utility func-tions cannot be identified.
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11.1 Quasi Linear Preferences.
• One case where we can identify our model is thecase of quasi linear preferences.
• Consider a simple example where there are i =
1, ..., 3 consumers choose between two goods j =
1 or 2 and an outside option ( j = 0).
• If utility is quasi linear, WLOG we can write theutility function for consumer i as:
ui(j, c) = βi,11{j = 1}+ βi,21{j = 2}+ c
• For tour simple example, suppose that³β1,1, β1,2
´=
(1, 2) and³β2,1, β2,2
´= (5, 6).
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• If p2 is sufficiently high, the demand for good 1will be two if p1 is less than 1, one unit if p1 is
less than 5 and zero if p1 exceeds 5.
• One consumer has a marginal utility for good 1equal to 1 and another person has marginal utility
of 5.
• In a similar fashion, the economist can learn thatthe marginal utilities for good 2 are equal to 2
and 6.
• At this point, cannot determined whether³β1,1, β1,2
´=
(1, 2) or³β1,1, β1,2
´= (1, 6).
• However, note that when p1 = 1 and p2 = 2, theconsumer is exactly indifferent between consum-
ing good 1 and good 2.
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• Therefore, the demand changes discontinuouslyat this point.
• In fact, demand changes discontinuously alongthe plane where βi,1 − p1 = βi,2 − p2 and p1 ≤1, p2 ≤ 2.
• Therefore, we can conclude that the preferencesof consumers in this market can be represented
by³β1,1, β1,2
´= (1, 2) and
³β2,1, β2,2
´= (5, 6).
• More generally, using this type of logic, we candemonstrate that the distribution of random co-
efficients for the model below is identified:
ui(j, c) =JX
j0=1βi,11{j = j0}+ c (3)
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• It is also possible to prove that if (i) there is paneldata on individual decisions and (ii)individual pref-
erences remain fixed, then the distribution of pref-
erences is identified (up to monotonic transforma-
tions of the utility function.
• See Bajari, Fox and Ryan for a proof.