Lecture 6 - George Washington Universityhome.gwu.edu/~bdwilliams/Site/econ379_files/lecture6.pdf ·...

Overview Maximum Likelihood Simulation methods Moment Inequalities

Lecture 6

Economics 379George Washington University

Instructor: Prof. Ben Williams

Economics 379 George Washington University

Lecture 6


Methods

Suppose Y = (Y1, . . . ,Yn) and X = (X1, . . . ,XN) and(Y,X) ∼ f (y | x, θ).

The (conditional) log-likelihood function isLN(θ) = log(f (Y | X, θ)).θ̂MLE = arg maxθ∈Θ LN(θ)Sometimes θ is identified based on moments:E(m(Y | X, θ)) = 0.Sometimes computing f (y | x, θ) requires anapproximation.Sometimes computing f (y | x, θ) requires simulation.Sometimes we rely only on moment inequalities:E(m(Y | X, θ)) ≤ 0.


Lecture 6


Applications

standard MLE – static model of female labor force participationdit = 1(v∗it ≥ 0) where

v∗it = v∗(yit ,w(zit , ηit ),nit , κit , εit )

= (cit (1))− (cit (0) + αit )

budget constraint: cit (d) = yit + d(w(zit , ηit )− πnit )utility of “leisure”: αit = κitβ + εit


Lecture 6


Applications

standard MLE – static model of female labor force participationno dynamics (agents are myopic or no effect of dit onfuture utility stream)can estimate with cross section:

Pr(dit = 1 | yit , zit ,nit , κit ) = Fε−η(zitγ − nitπ − κitβ)

if wages, wit are observed more can be identified – recallRoy model


Lecture 6


Applications

approximate MLE – Rust (1987)dynamic discrete choice programming problem:

maxi1,i2,...

E

( ∞∑t=1

βt−1u(xt , εt , it ; θ)

)

stationary solution of infinite-horizon dynamic optimization→ Bellman equationleads to likelihood LN(θ) =

∑ni=1 li(θ) where

li(θ) =∑T

t=1 log(Pr(it | xt ; θ))


Lecture 6


Applications

) approximate MLE – Rust (1987)but the probabilities are

Pr(it = 1 | xt ; θ) =exp(ū(it , xt , θ) + βE(V (x ′, �′) | xt , it )∑

i=0,1 exp(ū(i , xt , θ) + βE(V (x ′, �′) | xt , i))

where

V (x , �) = maxi=0,1{u(x , �, i ; θ) + βE(V (x ′, �′) | xt = x , �t = �, i)}

is the value function


Lecture 6


Applications

approximate MLE – Rust (1987)to compute the likelihood we need to be able to computethe value function for given parameter valuesnested fixed point algorithm – use value function iterationwithin the optimization algorithm


Lecture 6


Applications

simulated MLE or MSM – Berry, Levinsohn, Pakes (1995)data on J differentiated products; shares, sj = Pr(yij = 1),and product characteristics, (xj ,pj)ξj – unobserved product characteristicsmodel predicts that sj = s∗j (ξ1, . . . , ξJ , θ)Berry (1994) – sj = s̃j(δ1, . . . , δJ) where δj = Xjβ − pjα + ξjBLP – sj =

∫sj(δ1 + X1(βi − β) + p1(αi − α), . . . , δJ +

XJ(βi − β) + pJ(αi − α))dG(αi , βi)


Lecture 6


Applications

simulated MLE or MSM – Berry, Levinsohn, Pakes (1995)GMM based on E(ξjzj) = 0 where zj are, for example, costshifters.in simple logit case of Berry (1994), δj = log(sj)− log(s1)BLP – compute by combination of simulation and nestedfixed point algorithm


Lecture 6


Applications

indirect inferenceGuvenen and Smith (2010)Del Boca and Flinn (2012)Morten (2013)


Lecture 6


Applications

moment inequalities – Ciliberto and Tamer (2009)entry game – suppose two firms are considering whetheror not to enter a marketbest response functions: y = 1(αX + δy ′ + ε > 0)exhibits multiple equilbria


Lecture 6


Applications

moment inequalities – Ciliberto and Tamer (2009)data: Pr(y1, y2 | X1,X2)model implies these are related to the probability that(ε1, ε2) lies in some region R(X1,X2).in some regions where multiple equilbria occur we cannotassign probabilitiesinstead we know that the probability is between 0 and 1


Lecture 6


Applications

moment inequalities – Ciliberto and Tamer (2009)thus, for example, when δ1, δ2 < 0

Pr((y1, y2) = (1,0) | X1,X2) = Pr((ε1, ε2) ∈ R1(X , θ))+ E(Pr((y1, y2) = (1,0) | X1,X2, ε1, ε2) | (ε1, ε2) ∈ R2(X , θ))× Pr((ε1, ε2) ∈ R2(X , θ))

which implies that

Pr((ε1, ε2) ∈ R1(X , θ)) ≤ Pr((y1, y2) = (1,0) | X1,X2)≤ Pr((ε1, ε2) ∈ R1(X , θ)) + Pr((ε1, ε2) ∈ R2(X , θ))


Lecture 6


Maximum Likelihood

an example of a general class of estimators calledextremum estimatorsobjective function: QN(θ) = QN(Y,X, θ)score function: sN(θ) = ∂∂θQN(θ)


Lecture 6


Maximum Likelihood

assumptions for consistency (Newey and McFadden)QN(θ) converges uniformly in probability to Q(θ)Q(θ) is uniquely maximized at θ0Q(θ) is continuousΘ is compact

then θ̂ →p θ0


Lecture 6


Maximum Likelihood

assumptions for asymptotic normality (Newey and McFadden)θ0 is an interior point of ΘsN(θ) is differentiable with derivative HN(θ) inneighborhood of θ0√

NsN(θ0)→d N(0,Σ)HN(θ) converges uniformly in probability to H(θ), which iscontinuous and nonsingular at θ0

then√

N(θ̂ − θ0)→d N(0,H−1(θ0)ΣH−1(θ0)′)


Lecture 6


Maximum Likelihood

a few implications for MLEMLE may be inconsistent if density is misspecified or if thesupport of y depends on θquasi-(or pseudo-) MLE: if density is misspecified, θ̂ isconsistent for pseudo-true value but information matrixequality failscomputation:

global versus local solutionsdifferentiability“empirical identification”formula for f (y | x, θ) known?


Lecture 6


Maximum Likelihood

Rust (1987)the log-likelihood function is

N∑i=1

T∑t=1

(iit log(Pr(it = 1 | xit , θ))+(1−iit ) log(1−Pr(it = 1 | xit , θ)))

calculation of θ̂ requires computing the value function,V (x , �; θ) for different values of θ


Lecture 6


Maximum Likelihood

Rust (1987)compute the expected value fn:EV (x , i) = E(V (x ′, �′) | xt = x , it = i)initial guess: EV 0(x , i))update: EV 1(x , i) =∑

y log{∑

j=0,1 exp(ū(y , j ; θ) + βEV0(y , j))

}p(y | x , i)

requires estimation of p(y | x , i) in a first stagecontinue iterating until convergencevalue function iteration required every time thelog-likelihood is computed


Lecture 6


Simulation methods

Often the objective function is difficult to compute becauseit involves high-dimensional integration.Quadrature and other integral approximations lead toapproximation error that typically cannot be quantified.High-dimensional problems suffer from the “curse ofdimensionality”.Monte Carlo integration

is less effected by the dimension of the problemleads to sampling error which can be easily quantifiedprovides framework for improving accuracy of the algorithm


Lecture 6


Simulation methods

early examples:Multinomial Probit:LN(β) =

∑ni=1∑J

j=1 1(yi = j) ln(Pr(yi = j | Xi)) where

Pr(yi = j | Xi) = Pr(X ′ijβ + εij ≥ maxl 6=jX ′ilβ + εil)

and εi = (εi1, . . . , εiJ) ∼ N(0,Σ)


Lecture 6


Simulation methods

early examples:Random coefficients logit: same form for likelihood with

Pr(yi = j | Xi) =∫ exp(X ′ijβi)∑J

l=1 exp(X′ilβi)

f (βi)dβi

and βi ∼ N(β,Σβ).


Lecture 6


Simulation methods

Maximum Simulated LikelihoodSuppose generally thatf (yi | Xi ; θ) =

∫g(yi | Xi , ; u, θ)ψ(u)du.

simulate ui1, . . . ,uiS ∼i.i.d . ψ(·) for each i and replace`i(θ) = log(f (yi | Xi ; θ)) with

ˆ̀i(θ) = log

(S−1

S∑s=1

g(yi | Xi , ; uis, θ)

)

then, θ̂MSL = arg max L̂(θ) where L̂(θ) =∑N

i=1ˆ̀i


Lecture 6


Simulation methods

Maximum Simulated LikelihoodOnly consistent and asymptotically normal if

√N/S → 0.

Take S as a multiple of the sample size if feasible.do not draw new simulation sample in each iteration of theoptimization routine!Sometimes this simulation is quite naive and can beimproved upon by importance sampling and othervariance-reduction techniques. (See 12.7 in CT; see pp.866-867 in BLP (1995) for an example.)Bayesian methods provide an alternative.


Lecture 6


Simulation methods

Method of Simulated MomentsSuppose we want to estimate θ based on the momentcondition: E(wim(yi , xi , θ0)) = 0where m(yi , xi , θ) =

∫h(yi , xi ,u, θ)ψ(u)du requires

simulationthen, draw uis, s = 1, . . . ,S for each i and computem̂(yi , xi , θ) = S−1

∑Ss=1 h(yi , xi ,uis, θ)


Lecture 6


Simulation methods

Method of Simulated Momentsif E(m̂(yi , xi , θ) | yi , xi) = m(yi , xi , θ) and usual GMMconditions are satisfiedthen minimizing

∑Ni=1 wim̂(yi , xi , θ) provides consistent,

asymptotically normal estimatorasymptotically equivalent if S →∞ but otherwise there isefficiency lossthe asymptotic variance for fixed S is inflated by a factor of1 + S−1 though this can often be improved, e.g., byimportance sampling


Lecture 6


Simulation methods

Method of Simulated Momentsvariance estimation requires either simulation or abootstrap procedureGourieroux and Monfort (1991) provide more generalconditions under which S →∞ is not necessary.Pakes and Pollard (1989) provide some examples


Lecture 6


Simulation methods

Example. BLP (1995)utility of option j : uij = X ′j βi + pjαi + ξj + εij and ui0 = εi0with εij independent type 1 extreme value distributed.αi = α + σαvi0 and βik = β + σβvik are random coefficientswhere (vi0, . . . , viK ) ∼ N(0, I)write uij = δj + µij + εij + εij where δj = X ′j β + pjα + ξjhere I am suppressing the additional notation related todata on individual income.


Lecture 6


Simulation methods

Example. BLP (1995)For each individual i the probability of choosing product j isgiven by

exp(δj + σαvi0pj +∑K

k=1 σβvikXjk )

1 +∑J

l=1 exp(δl + σαvi0pl +∑K

k=1 σβvikXlk )

the aggregate shares are∫exp(δj + σαvi0pj +

∑Kk=1 σβvikXjk )

1 +∑J

l=1 exp(δl + σαvi0pl +∑K

k=1 σβvikXlk )dF (v)


Lecture 6


Simulation methods

Example. BLP (1995)let sj denote observed sharesdenote the individual level shares implied by model –s∗j (X ,p, δ, vi ; θ2) where θ2 is subset of parameters θdenote the aggregate shares implied by model –s∗j (X ,p, δ; θ2)moment conditions of the form E(ξjzj) = 0would like to minimize GJ(θ)′GJ(θ) where GJ(θ) is thevector of

∑Jj=1 Hj(z)ξj where z = {zj} and zj = (Xj ,Wj)


Lecture 6


Simulation methods

Example. BLP (1995)fix simulation draws vs for s = 1, . . . ,Sfor given θ compute the objective function

start with initial guess δ0 = (δ01 , . . . , δ0J )

estimate s∗j (X ,p, δ0; θ2) using

ŝ∗j (X ,p, δ0; θ2) =

1S

∑Ss=1 s

∗j (X ,p, δ, vis; θ2)

update δ1 = δ0 + sj − ŝ∗j (X ,p, δ0; θ2)iterate this until ||δm − δm−1|| < �take ξ̂j = δm − X ′j β − pjαform ĜJ(θ)′ĜJ(θ) using

∑Jj=1 Hj (z)ξ̂j

iterate this procedure over candidate values of θ untiloptimum obtained.


Lecture 6


Simulation methods

Indirect Inference:auxiliary model, e.g., a likelihoodLN(θ) =

∑Ni=1 log(f (yi | Xi , θ))

auxiliary estimate – θ̂ = arg maxθ LN(θ)economic model, e.g., yi = G(Xi ,ui ;β) for i = 1, . . . ,N andui ∼i.i.d . Fusimulate {ymi (β)} and obtain θ̃(β) by maximizing∑M

m=1∑N

i=1 log(f (ymi (β) | Xi , θ))


Lecture 6


Simulation methods

Indirect Inference:β̂ = arg minβ D(θ̂, θ̃(β))D is a metric function; Smith (2008) suggest Wald, LR, LMmetricsconsistent and asymptotically normal for M fixed, N →∞variance inflated by (1 + M−1)huge efficiency questionvery easy to implement


Lecture 6


Simulation methods

Indirect Inference:asymptotic results depend on the exact formulation but thefollowing conditions are usually required:the economic model is correctly specified andwell-behavedthe auxiliary likelihood function is well-behaved in the limit,despite the fact that it is misspecifiedbinding function –

LN(θ)→p L(θ;β,Fu) when the data is generated by theeconomic model with parameters β and distribution Fudefine θ(β) = arg maxθ L(θ;β,Fu)θ0 = θ(β0) is the pseudo-true valueθ(β) is the binding function


Lecture 6


Simulation methods

Indirect Inference:under regularity conditions θ̂ →p θ0 and θ̃(β)→p θ(β)thus, the identification condition is: is β0 the unique β suchthat θ0 = θ(β)?requires dim(θ) ≥ dim(β)simulation avoids needing to know the binding functionvery easy to implementhuge efficiency question


Lecture 6


Simulation methods

example. multinomial probitthe likelihood is difficult to compute, as discussed aboveone approach is MSL, though variance-reduction isnecessary in the simulatorindirect inference where the auxiliary model is a linearprobability model offers an alternativeturns out that the II objective function is not smoothKeane and Smith (2003) smooth out the objective functionand find better resultsless efficient than MSL


Lecture 6


Simulation methods

example. Guvenen and Smith (2010)joint model of income processes and life-cycleconsumption/savings/retirement behaviorthe goal is to use economic choices to decomposevariance of earnings into predictable and unpredictablecomponents


Lecture 6


Simulation methods

example. Guvenen and Smith (2010)the economic model specifies (i) an exogenous incomeprocess with a life-cycle component, heterogeneity in levelsand growth, and persistence in stochastic components(ii) dynamic optimization problem that incorporates learningabout individual-specific heterogeneity components(iii) age-dependent borrowing constraints and detailedinformation about social security benefits in retirementand (iv) measurement error in income and consumptionand missing observations


Lecture 6


Simulation methods

example. Guvenen and Smith (2010)the auxiliary model specifies (i) regression of level ofconsumption on past and future levels of consumption andpast and future growth and levels of income, separately byage group(ii) regression of level of income on past and future growthand levels of income, separately by age groupand (iii) estimate of median wealth-income ratio


Lecture 6


Simulation methods

example. Guvenen and Smith (2010)with the exception of how the wealth-income ratio isincorporated they use a likelihood ratio metric for theobjective functionthey combine the different auxiliary regressions into alikelihood-type model (see discussion in paper)identification of discount factor and the coefficient ofrelative risk aversionstandard errorsrobustness


Lecture 6


Moment inequalities

sometimes an economic model leads to an incompleteeconometric modelthe indeterminate part of the model can typically beboundedthis leads to moment inequalitiesE(wim(yi , xi , θ0) ≤ 0the model will not be identified based on these inequalitiesinstead do inference on the identified set :I0 = {θ ∈ Θ : E(wim(yi , xi , θ) ≤ 0}


Lecture 6


Moment inequalities

if I0 = {θ ∈ Θ : E(hL(yi , xi)) ≤ θ ≤ E(hU(yi , xi))} estimationand inference are straightforwardÎ = [EN(hL(yi , xi)),EN(hU(yi , xi))]and standard methods provide confidence intervals for theendpoints (suggested, e.g., by Pakes, Porter, Ho, and Ishii2011)


Lecture 6


Moment inequalities

often the identified set is more complicatedin addition this is not the typical goal of inferenceeither construct a confidence set for θ0 or I0


Lecture 6


Moment inequalities

objective function that counts how many inequalities arenot satisfied, QN(θ).a consistent estimator for I0 is {θ ∈ Θ : QN(θ) < rn} wherern is a prespecified tuning parameter.a confidence region for θ0 is based on subsampling criticalvalues of n(Qn(θ)− inft∈ΘQn(t)); see Chernozhuov, Hong,and Tamer (2007) or Ciliberto and Tamer (2009)other approaches: Romano and Shaikh (2008), Romanoand Shaikh (2010), Andrews and Barwick (2012), Romano,Shaikh, and Wolf (2013)


Lecture 6


Moment inequalities

example. Ciliberto and Tamer (2009)the setting is a static game with K players that is playedout in each of M marketsthe payoff function for player i in market m isπim = X ′imβi +

∑j 6=i δ

ji yjm + εim

there are multiple equilibria for some values of X , ε – as aresult, the likelihood cannot be completely specified


Lecture 6


Moment inequalities

example. Ciliberto and Tamer (2009)moment inequalities: H j1(θ,X) ≤ Pr(ym = y

j | X) ≤ H j2(θ,X)confidence regions constructed as abovethe functions H js(θ,X) for s = 1,2 are hard to compute –simulation is used.exclusion restriction


Lecture 6

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