ccp estimation of dynamic discrete choice models with...
TRANSCRIPT
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).
Back to Harold Zurcher