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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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