CCP Estimation of Dynamic Discrete ChoiceModels With Unobserved Heterogeneity

Yitian (Sky) LIANG

Department of MarketingSauder School of Business

March 7, 2013

Roadmap

I Summary of the paper (5 mins)I Motivating example: bus engine replacement model (Rust,

1987) (10 mins)I Estimator and algorithm (10 mins)I Application result in the motivating example (5 mins)

Summary

I Motivation: unobserved heterogeneity (unobserved correlatedstate variables)

I Can’t have consistent first-stage estimates of CCPI Violation of CI

I Develop a modified EM algorithm to estimate the structuralparameters and the distribution of unobserved state variables

I Develop the concept of “finite dependence” (will not covered)I identification?I facilitate estimation?

Motivating Example (Setup): Our Friend - Harold Zurcher

I Infinite horizon (later in the application, they set it to be finitehorizon)

I Choice space {d1t , d2t}, i.e. replace the engine v.s keep it.I State space {xt , s, εt}, i.e. accumulated mileage since the last

replacement, brand of the bus and transitory shocks (notobserved by the econometrician)

I Controlled transition rule:I xt+1 = xt + 1 if d2t = 1.I xt+1 = 0 if d1t = 1.

I Per-period payoff:u (d1t , xt , s) = d1t · ε1t + (1− d1t) · (θ0 + θ1xt + θ2s + ε2t).

Harold Zurcher Cont.

I Hotz and Miller (1993): difference between conditional valuefunction can be represented by flow payoff and CCP, i.e.

v2 (x , s)−v (x1, s) = θ0 +θ1x +θ2s +β log [p1 (0, s)]−β log [p1 (x + 1, s)] .

I Then we have: p1 (x , s) = 11+exp[v2(x ,s)−v(x1,s)] .

I Let πs be the probability a bus is brand s.

Harold Zurcher Cont. (Suppose know p̂)

I MLE,{θ̂, π̂

}= argmaxθ,π

∑n log [

∑s πsΠt l (dnt | xnt , s, p̂1, θ)].

I EM AlgorithmI Expectation step:

I q̂ns = Pr{

sn = s | dn, xn; θ̂, π̂, p̂1

}=

π̂sΠt l(dnt | xnt , s, p̂1, θ)∑s′ π̂s′Πt l(dnt | xnt , s′, p̂1, θ)

I π̂s = 1N

∑Nn=1 q̂ns .

I Maximization step:θ̂ = argmaxθ

∑n log [

∑s π̂sΠt l (dnt | xnt , s, p̂1, θ)].

Harold Zurcher Cont. (Update p̂)

I Two ways to update CCP: model-based v.s non-model-basedI Non model based update of CCP

p1 (x , s) = Pr {d1nt = 1 | sn = s, xnt = x}

=E [d1ntqns | xnt = x ]

E [qns | xnt = x ]

I Sample analogue:

p̂1 (x , s) =

∑n∑

t d1nt q̂ns I (xnt = x)∑n∑

t q̂ns I (xnt = x)

I Model based update:p(m+1)1 (xnt , s) = l

(dnt | xnt , s, p(m)

1 , θ(m)).

General Model

I Larger choice space, non-stationarity (i.e. finite horizon)I Unobserved heterogeneity changes over time: need to estimate

its transition π (st+1|st).I Initial value problem: need to estimate π (s1|x1).I Sketch of the algorithm

I Expectation step: sequential updateqns → π (s1|x1) , π (st+1|st)→ pjt (x , s).

I Maximization step: maximize the conditional likelihood w.r.tstructural parameters.

General Model - Likelihood

L (dn, xn | xn1; θ, π, p) =∑s1

∑s2

· · ·∑sT

[π (s1|xn1)L1 (dn1, xn2| xn1, s1; θ, π, p)

×(

ΠTt=2

)π (st |st−1)Lt (dnt , xn,t+1| xnt , st ; θ, π, p)

].

where

Lt (dnt , xn,t+1| xnt , st ; θ, π, p)

= ΠJj=1 [ljt (xnt , snt , θ, π, p) fjt (xn,t+1|xnt , snt , θ)]djnt .

The Algorithm - Expectation Step

Update q(m)nst :

q(m+1)nst =

L(m)n (snt = s)

L(m)n

,

where

Lnt (snt = s)

=∑s1

· · ·∑st−1

∑st+1

· · ·∑sT

π (s1|xn1)Ln1 (s1)(Πt−1

t′=2π (st′ |st′−1)Lnt′ (st′))

×π (st |st−1)Lnt (s)π (st+1|s)Ln,t+1 (st+1)(ΠT

t′=t+2π (st′ |st′−1)Lnt′ (st′)).

The Algorithm - Expectation Step Cont.

Update π(m) (s|x):

π(m+1) (s|x) =

∑Nn=1 q(m+1)

ns1 I (xn1 = x)∑Nn=1 I (xn1 = x)

.

Update π(m+1) (s ′|s):

π(m+1)(s ′|s)

=

∑Nn=1

∑Tt=2 q(m+1)

ns′t|s q(m+1)ns,t−1∑N

n=1∑T

t=2 q(m+1)ns,t−1

,

where the definition of q(m+1)ns′t|s is on page 1847.

The Algorithm - Expecation Step Cont. & MaximizationStep

Update p(m+1)jt (x , s):

p(m+1)jt (x , s) =

∑Nn=1 dnjtq

(m+1)nst I (xnt = x)∑N

n=1 q(m+1)nst I (xnt = x)

.

Maximization step:

θ(m+1) = argmaxθ∑n

∑t

∑s

∑j

q(m+1)nst logLt

(dnt , xn,t+1|xnt , snt = s; θ, π(m+1), p(m+1)

).

Alternative Algorithm - Two Stage Estimator

I Stage 1: recover θ1, π (s1|x1), π (s ′|s), pjt (xt , st) by using theEM algorithm.

I Stage 2: recover θ2.I Key idea: non-parametric representation of the likelihood (free

of structural parameters):

Lt (dnt , xn,t+1|xnt , snt ; θ1, π, p)

= ΠJj=1 [ljt (xnt , snt , θ, π, p) fjt (xn,t+1|xnt , snt , θ1)]djnt

= ΠJj=1 [pjt (xnt , snt) fjt (xn,t+1|xnt , snt , θ1)]djnt .

Alternative Algorithm - Two Stage Estimator Cont.

I Stage 1 expectation step: update q and πI Stage 1 maximization step: maximize the conditional

likelihood w.r.t p and θ1I Stage 2: given stage 1 estimates, can apply any CCP based

method to recover θ2, i.e. Hotz and Miller (1993), BBL(2007).

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