CCP Estimation of Dynamic Discrete Choice Models With Unobserved - - PowerPoint PPT Presentation

CCP Estimation of Dynamic Discrete Choice Models With Unobserved Heterogeneity

Yitian (Sky) LIANG

Department of Marketing Sauder School of Business

March 7, 2013

Roadmap

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

1987) (10 mins)

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

Summary

◮ Motivation: unobserved heterogeneity (unobserved correlated

state variables)

◮ Can’t have consistent first-stage estimates of CCP ◮ Violation of CI

◮ Develop a modified EM algorithm to estimate the structural

parameters and the distribution of unobserved state variables

◮ Develop the concept of “finite dependence” (will not covered)

◮ identification? ◮ facilitate estimation?

Motivating Example (Setup): Our Friend - Harold Zurcher

◮ Infinite horizon (later in the application, they set it to be finite

horizon)

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

replacement, brand of the bus and transitory shocks (not

• bserved by the econometrician)

◮ Controlled transition rule:

◮ xt+1 = xt + 1 if d2t = 1. ◮ xt+1 = 0 if d1t = 1.

◮ Per-period payoff:

u (d1t, xt, s) = d1t · ǫ1t + (1 − d1t) · (θ0 + θ1xt + θ2s + ǫ2t).

Harold Zurcher Cont.

◮ Hotz and Miller (1993): difference between conditional value

function 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)] .

◮ Then we have: p1 (x, s) = 1 1+exp[v2(x,s)−v(x1,s)]. ◮ Let πs be the probability a bus is brand s.

Harold Zurcher Cont. (Suppose know ˆ p)

◮ MLE,

• ˆ

θ, ˆ π

• = argmaxθ,π
• n log [

s πsΠtl (dnt | xnt, s, ˆ

p1, θ)].

◮ EM Algorithm

◮ Expectation step: ◮ ˆ

qns = Pr

• sn = s | dn, xn; ˆ

θ, ˆ π, ˆ p1

• =

ˆ πsΠtl(dnt | xnt, s, ˆ p1, θ)

• s′ ˆ

πs′ Πtl(dnt | xnt, s′, ˆ p1, θ) ◮ ˆ

πs = 1

N

N

n=1 ˆ

qns.

◮ Maximization step:

ˆ θ = argmaxθ

• n log [

s ˆ

πsΠtl (dnt | xnt, s, ˆ p1, θ)].

Harold Zurcher Cont. (Update ˆ p)

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

p1 (x, s) = Pr {d1nt = 1 | sn = s, xnt = x} = E [d1ntqns | xnt = x] E [qns | xnt = x]

◮ Sample analogue:

ˆ p1 (x, s) =

• n
• t d1ntˆ

qnsI (xnt = x)

• n
• t ˆ

qnsI (xnt = x)

◮ Model based update:

p(m+1)

1

(xnt, s) = l

• dnt | xnt, s, p(m)

1

, θ(m) .

General Model

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

its transition π (st+1|st).

◮ Initial value problem: need to estimate π (s1|x1). ◮ Sketch of the algorithm

◮ Expectation step: sequential update

qns → π (s1|x1) , π (st+1|st) → pjt (x, s).

◮ Maximization step: maximize the conditional likelihood w.r.t

structural 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) ×

• ΠT

t=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) = ΠJ

j=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) = N

n=1 q(m+1) ns1

I (xn1 = x) N

n=1 I (xn1 = x)

. Update π(m+1) (s′|s): π(m+1) s′|s

• =

N

n=1

T

t=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. & Maximization Step

Update p(m+1)

jt

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

jt

(x, s) = N

n=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

log Lt

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

.

Alternative Algorithm - Two Stage Estimator

◮ Stage 1: recover θ1, π (s1|x1), π (s′|s), pjt (xt, st) by using the

EM algorithm.

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

• f structural parameters):

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

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

= ΠJ

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

Alternative Algorithm - Two Stage Estimator Cont.

◮ Stage 1 expectation step: update q and π ◮ Stage 1 maximization step: maximize the conditional

likelihood w.r.t p and θ1

◮ 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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