1 Outline.

1. MSL.

2. MSM and Indirect Inference.

3. Example of MSM-Berry(1994) and BLP(1995).

4. Ackerberg’s Importance Sampler.

2 Some Examples.

• In this chapter, we will be concerned with exam-ples of the form:

f(y|x, θ) =Zf(y|x, γ)g(γ|θ)

• We shall discuss some methods for estimatingthese problems.

• Then we shall discuss, in detail, two examples.

• The first is the approach to demand estimationfollowing Berry (1994) and BLP(1995).

• The second is discrete games using the Ackerbergimportance sampler.

• That is, the density f(y|x, γ) is allowed to dependon parameters γ that vary within the population.

• The population distribution of these parametersis g(γ|θ).

• Examples of this are random effects models of

unobserved heterogeneity where an intercept is

allowed to vary stochastically.

• A second is the random coefficient logit.

• In both cases, integration needs to be performedin order to evaluate the likelihood function.

• We can evaluate the above integral using eitherdeterministic methods for intergration or stochas-

tic methods based on draw pseudo random num-

bers.

• In practice, the latter approach seems to be morecommon since the asymptotic theory takes ac-

count of the approximation error in evaluating the

integral.

• To do Monte Carlo integration, we need to draws = 1, ..., S pseudo-random deviaties γ(s) from

the density g(γ).

• Our estimate of the integral is then:

bf(yi|xi, θ) = 1

S

SXs=1

f(yi|xi, γ(s)i , θ)

• Note that we are drawing a seperate set of devi-ates for each yi and xi in our simulator.

• Also note that we have assumed that θ does notenter into g(γ).

• In practice, we may have to carefully parameterizeor problems to make sure that this is true.

• We shall consider an example of this shortly.

• Note that we could form a confidence interval for

the mean in order to evaluate the accuracy of our

integral.

• Also, by doing many monte carlos of size S, wecan generate a distribution of bf(y|x, θ) to see howaccurate our integral is evaluated.

• This simulation is unbiased and consistent as S →∞.

• The MSL (method of simulated likelihood) esti-mator is:

lnLN(θ) =1

N

Xi

ln bf(yi|xi, θ)

• In some, but not all cases, bf(yi|xi, θ) is a smoothand differentiable function of θ.

• Our estimator bθMSL is defined as:

bθMSL = argmax lnLN(θ)

• Prop 21.1 from Gourieroux and Monfort (stated

in the text) demonstrates that if

• (i)the likelihood satisfies the regularity conditionsfor asymptotic normality with limit variance Λ−1(θ0)

Λ(θ0) = −p lim⎡⎣N−1 NX

i=1

∂2 ln f(y|x, θ)∂θ∂θ0

⎤⎦

• (ii) the density is simulated with an unbiased sim-ulator

• (iii)if S,N→∞ witih N−1/2/S → 0, then

• N−1(bθMSL − θ0)→d N(0,Λ−1(θ0))

• There are two approaches one can take to calcu-lating the variance.

• A first is to bootstrap your standard errors.

• A second is based on differentiating ln bf(yi|xi, θ) =1S

PSs=1 ln f(yi|xi, γ

(s)i )

• We shall talk about the mechanics of impliment-ing this estimator in a demand estimation exam-

ple from Berry (1994) which is closesly related to

BLP(1995).

3 MSM and Indirect Inference

• A MSM estimator starts by specifying a moment

equation that depends on the distribution of some

random variable.

m(yi, xi, θ) =Zh(yi, xi, γ, θ)g(γ)

• We proceed analogously to the MSL estimatorand use Monte Carlo to simulate this integral.

• We need to draw s = 1, ..., S pseudo-random de-

viaties γ(s) from the density g(γ|θ).

cm(yi, xi, θ) = SXi=1

h(yi, xi, γ(s)i , θ)

• As in the case of SML, we assume that the para-meters θ do not enter into g(γ).

• This may require a careful parameterization of ourmodel.

• If we could perfectly evaluate our integral, in thejust identified case, our GMM estimator would be:

QN(θ) =

⎡⎣ NXi=1

wim(yi, xi, θ)

⎤⎦0 ⎡⎣ NXi=1

wim(yi, xi, θ)

⎤⎦

• where wi corresponds to our weights.

• In MSM, we plug in the sample analogue-

QN(θ) =

⎡⎣ NXi=1

wicm(yi, xi, θ)⎤⎦0 ⎡⎣ NX

i=1

wicm(yi, xi, θ)⎤⎦

• Under the regularity conditions stated in the text,MSM is consistent and asymtotically normal for

a fixed S.

• Unlike MSL, we do not need to let S→∞ in order

to estimate the model.

• Given that we have asymptotic normality, this jus-tifies the use of the bootstrap to compute our

standard errors.

• A final method discussed in the text is indirect

inference.

• This method is useful for models that are easyto simulate, but where it is hard to form MSL or

MSM estimators.

• For example, a dynamic stochastic model fromStokey and Lucas might be easy to simulate (as-

suming it is specified parsimoniously).

• However, in some cases it might be hard to formmoments (e.x. an (S,s) model of inventory be-

havior).

• Indirect inference (in the literature, the term EMM,efficient method of moments is used for related

estimators) is also common in certain time series

problems.

• Suppose that our model specifies that yi = f(xi, η, θ)

where xi is a random variable, η is a stochastic

shock and θ is a vector of parameters.

• Suppose that g(η) is the density for our shock.

• In indirect inference, we start with an auxiliarymodel.

• For example, we might specify ad hoc regressionsof the dependent variable yi on the exogenous

variables xi.

• We run this regression on the true data and comeup with regression coefficients bβ.

• Given a vector of parameters, θ and we could sim-ulate our model to generate a sequence of psuedo

random (eyi, exi) i = 1, ..., N.

• We could then run our auxilary model on thepsuedo random (eyi, exi) to come up with a bβ(θ).

• The indirect inference estimator is:

bθ = ³bβ − bβ(θ)´0Ω−1 ³bβ − bβ(θ)´

• Where Ω−1 is a weight matrix.

• Intuitive, indirect inference attempts to match theparameters of the auxiliary model on the real and

simulated data.

• Essentially, it is an extremely convenient way toform moments for a simulation based model.

• Another attract feature is that often the weightmatrix Ω−1 can be formed using the data (yi, xi)and does not require simulating the model.

4 Example of MSM

• 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

• 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

• 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:

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

• 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.

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.

4.1 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/monte carlo study.

• 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.

• 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.

• We shall consider these one at a time.

4.2 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 for each j andkeep them fixed for the whole problem.

• Denote the draws as η(s)i , s = 1, ..., S. Let our

simulated shares are:

bsj(δ, σ) = SXs=1

exp(δj +Pk xj,kη

(s)i,kσk)

1 +Pj0 exp(δj0 +

Pk xj0,kη

(s)i,kσk)

• Sometimes importance sampling is useful in orderto improve the speed/accuracy of the integration.

• Imporance sampling is discussed in the text.

• We can compute confidence intervals using stan-dard methods to see whether the simulated mar-

ket shares are well estimated.

4.3 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(bsj(δ, σ))

• Berry proves that this is a contraction.

• Point: Market shares can be inverted very quicklyin a fairly simple manner!

5 Computing the value of ξ

• The next set is simple. Just let:

ξj = δj − (xjβ − αpj)

where δj is computed using the contraction mapping.

5.1 Computing the value of the objective

function.

• Let Z be the set of instruments.

• The objective function is formulated as in all MSMproblems assuming E (ξ|Z) = 0.

• The econometrician then chooses β, α, and σ inorder to minimize the MSM 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.

• 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.

6 Ackerberg’s Importance Sampler.

• Finally, we consider a method that is useful whenthe model is difficult to compute but it is possible

to ”reparameterize” the model.

• With the reparameterization, the algorithm is par-allel and can be computed more efficiently.

• Ackerberg (2006) describes this method in detail.

• As an example, we take the problem of discrete/normalform games as studied in Bajari, Hong and Ryan

(2006).

• Consider static entry game (see Bresnahan andReiss (1990,1991), Berry (1992), Tamer (2002),

Ciliberto and Tamer (2003), and Manuszak and

Cohen (2004)).

• The economist observes a cross section of mar-kets.

• The players in the game are a finite set of poten-tial entrants.

• In each market, the potential entrants simultane-ously choose whether to enter.

• Let ai = 1 denote entry and ai = 0 denote nonen-try.

• In applications, the function fi takes a form such

as:

fi =nθ1 · x+ δ

Xj 6=i

1naj = 1

oif ai = 1

0 if ai = 0(1)

• The covariates x are variables which influence theprofitability of entering a market.

• These might include the number of consumers inthe market, average income and market specific

cost indicators.

• The term δ measures the influence of j’s choice

on i’s entry decision.

• The εi(a) capture shocks to the profitability of

entry that are commonly observed by all firms in

the market.

• This is a simultaneous system of logit models!

• In the paper, we also discuss network effects andpeer effects as other examples.

7 The Model.

• Simultaneous move game of complete informa-tion (normal form game).

• There are i = 1, ..., N players with a finite set of

actions Ai.

• A =Yi

Ai.

• Utility ui : A→ R, where R is the real line.

• Let πi is a mixed strategy.

• A Nash equilibrium is a set of best responses.

• Following Bresnahan and Reiss (1990,1991), econo-metrically a game is a discrete choice model.

• Except actions of others are right hand size vari-ables.

ui(a) = fi(x, a; θ1) + εi(a). (2)

• Mean utility, fi(x, a; θ1)

• a, the vector of actions, covariates x, and a pa-

rameters θ.

• εi(a) preference shocks.

• εi(a) ∼ g(ε|θ2) iid.

• Standard random utility model, except utility de-

pends on actions of others.

• E(u) set of Nash equilibrium given a vector of

utilities u.

• λ(π;E(u), β) is probability of equilibrium, π ∈E(u) given parameters β.

• λ(π;E(u), β) corresponds to a finite vector of

probabilities.

• In an application, might let λ depend on

1. Satisfies a particular refinement concept (e.g. trem-

bling hand perfection).

2. The equilibrium is in pure strategies.

3. Maximizes joint payoffs (efficiency).

4. Maximizes profit of incumbent firms (as in airlines

examples).

• In practice, we could create dummy variables forwhether a given equilibrium, π ∈ E(u) satisfies

1-4 above.

• Let x(π, u) be this vector of dummies.

• A straightforward way to model λ is:

λ(π;E(u), β) =exp(β · x(π, u))P

π0∈E(u) exp(β · x(π0, u))(3)

• Computing the set E(u), all of the equilibrium to

a normal form game, is a well understood prob-

lem.

• McKelvy and McLennan (1996) survey the avail-able algorithms in detail.

• Software package Gambit.

• Also not hard to program directly.

8 Estimation.

• P (a|x, θ, β) is probability of a given x, θ and β

P (a|x, θ, β) =Z ⎧⎪⎨⎪⎩X

π∈E(u(x,θ,ε))λ(π;u(x, θ1, ε), β)

³YNi=1π(ai)

´⎫⎪⎬⎪⎭ g(ε|θ2)dε

• Computation of the above integral is facilitated bythe importance sampling procedure of Ackerberg.

• Make a change of variables to integrate over la-tent utility ui instead of over εi.

• With this change, we won’t need to recomputethe equilibria to the game during estimation.

• Estimation not feasible without this insight.

• Often g(ε|θ2) is a simple parametric distribution(e.g. normal, extreme value, etc...)

• For instance, suppose it is normal and let φ(·|μ, σ)denote the normal density.

• Then, the density h(u|θ, x) for the vNM utilities

u is:

h(u|θ, x) =Yi

Ya∈A

φ(εi(a); fi(θ, x, θ) + μ, σ)

where for all i and all a, εi(a) = fi(x, a; θ1)− ui(a)

• Evaluating h(u|θ, x) is cheap.

• Draw s = 1, ..., S vectors of vNM utilities, u(s) =

(u(s)1 , ..., u

(s)N ) from an importance density q(u).

• We can then simulate P (a|x, θ, β) as follows:

bP (a|x, θ, β) =PSs=1

⎧⎪⎨⎪⎩X

π∈E(u)λ(π;E(u(s)), β)

³YNi=1π(ai)

´⎫⎪⎬⎪⎭ h(u(s)|θ,x)q(u(s))

• Precompute E(u(s)) for a large number of ran-

domly drawn games s = 1, ..., S.

• Evaluating bP (a|x, θ, β) at new parameters DOESNOT REQUIRE RECOMPUTINGE(u(s)) for new

s = 1, ..., S!

• Evaluating simulation estimator of bP (a|x, θ, β) ofP (a|x, θ, β) only requires “reweighting” of theequilibrium by new λ and h(u(s)|θ,x)

q(u(s)).

• This is a cheap computation.

• Normally, the computational expense of structuralestimation comes from recomputing the equilib-

rium many times.

• Also, note that this approach is naturally parallel.

• This saves on the computational time by ordersof magnitude.

• Given bP (a|x, θ, β) we can simulate the likelihoodfunction or simulate the moments.

• The asymptotics are standard.

9 Application.

• Study entry decisions by highway paving contrac-tors in CA- 1999-2000.

• Focus on decisions by 4 largest contractors in oursample.

• 414 projects, 271 contractors, $369.2 million dol-lars awarded.

• Simultaneous move-antitrust laws forbid commu-nication.

• Static-bid for the right to complete a single job.

• fi(a, x)−expected profits less markup.

• Use nonparametric procedure to estimate markupsGuerre, Perrigne and Vuong (Emet, 2000).

• Literature is well developed and seems to give rea-sonable answers (see Bajari and Hortacsu (JPE,

2005)).

• Median margin is 2.7% similar to reported mar-

gins of publicly traded companies in our sample.

• Firms compete spatially.

• Transportation costs significant- closest firm has

costs advantage and market power.

• Only part of fi to be estimated is entry costs(around 1-2% which is reasonable according to

industry books, e.g. Park and Chapin).

• Goal- which equilibrium is most likely?

• Pure Strategy, efficient, dominated, Nash prod-uct?