Discrete Choice Models Dynamic Programming Models Discrete-Choice Dynamic-Programming Models References
Estimating Discrete Choice Dynamic Programming Models
Katsumi Shimotsu1 Ken Yamada2
1Department of Economics, Hitotsubashi University
2School of Economics, Singapore Management University
Japanese Economic Association Spring Meeting Tutorial Session (May, 2011) Discrete Choice Models Dynamic Programming Models Discrete-Choice Dynamic-Programming Models References
Introduction to Part II
• The second part focuses on econometric implementation of discrete choice dynamic programming models. • A simple machine replacement model is estimated using the nested fixed point algorithm (Rust, 1987). • For more applications, see Aguirregabiria and Mira (2010) and Keane, Todd, and Wolpin (2010).
• For audience with different backgrounds, I will briefly review 1. Discrete choice models and 2. Numerical methods for (i) maximum likelihood estimation and (ii) dynamic programming. Discrete Choice Models Dynamic Programming Models Discrete-Choice Dynamic-Programming Models References
Outline
Discrete Choice Models The Random Utility Model Maximum Likelihood Estimation
Dynamic Programming Models Machine Replacement Numerical Dynamic Programming
Discrete-Choice Dynamic-Programming Models The Nested Fixed Point Algorithm The Nested Pseudo Likelihood Algorithm
References Discrete Choice Models Dynamic Programming Models Discrete-Choice Dynamic-Programming Models References
Outline
Discrete Choice Models The Random Utility Model Maximum Likelihood Estimation
Dynamic Programming Models Machine Replacement Numerical Dynamic Programming
Discrete-Choice Dynamic-Programming Models The Nested Fixed Point Algorithm The Nested Pseudo Likelihood Algorithm
References Discrete Choice Models Dynamic Programming Models Discrete-Choice Dynamic-Programming Models References
The Random Utility Model • Consider a static problem in which the agent chooses among J alternatives, such as transportation mode and brand choice. • Assume that the choice-specific utility can be expressed as
Vij = uij + εij i = 1,...,N, j = 0...,J,
where uij is a deterministic component while εij is a stochastic component.
• A simple example is uij = xij θ, where xij is a vector of observed characteristics such as price and income.
• Let A = {1,...,J} denote the choice set. The choice probability is Z ( ) P (a) = 1 a = argmax (uij + εij ) f (εi )dεi , ε j∈A 0 where f (εi ) is a joint density of εi = (εi1,...,εiJ ) . Discrete Choice Models Dynamic Programming Models Discrete-Choice Dynamic-Programming Models References
The Conditional Logit Model
• Assume that each εj is independently, identically distributed −ε −ε extreme value. The density is f (εj ) = e j exp(−e j ). The choice probability follows the logit model (McFadden, 1981).
exp(uij ) exp(uij − ui1) pij = P (a = j) = J = J ∑j=1 exp(uij ) 1 + ∑j=2 exp(uij − ui1) • Note that this is a system of J − 1 equations. For J = 2,
exp(ui2 − ui1) pi2 = = Λ(ui2 − ui1). 1 + exp(ui2 − ui1) There exists a mapping between the utility differences and choice probabilities. −1 pi2 ui2 − ui1 = Λ (pi2) or ui2 − ui1 = ln 1 − pi2 See Hotz and Miller (1993) for the details on invertibility. Discrete Choice Models Dynamic Programming Models Discrete-Choice Dynamic-Programming Models References
Outline
Discrete Choice Models The Random Utility Model Maximum Likelihood Estimation
Dynamic Programming Models Machine Replacement Numerical Dynamic Programming
Discrete-Choice Dynamic-Programming Models The Nested Fixed Point Algorithm The Nested Pseudo Likelihood Algorithm
References Discrete Choice Models Dynamic Programming Models Discrete-Choice Dynamic-Programming Models References
MLE
• The choice probability is
exp(uij ) P (a = j) = J ∑j=1 exp(uij )
• Suppose Vij = xij θ + εij . The log-likelihood function is
N N J l (θ) = ∑ li (θ) = ∑ ∑ 1(ai = j)lnP (ai = j|xij ). i=1 i=1 j=1
• If the model is correctly specified,
2 −1! d ∂ l θb → N θ,E − . ∂θ∂θ 0 Discrete Choice Models Dynamic Programming Models Discrete-Choice Dynamic-Programming Models References
Numerical Maximization (Train, 2009) Discrete Choice Models Dynamic Programming Models Discrete-Choice Dynamic-Programming Models References
Numerical Maximization (Train, 2009) Discrete Choice Models Dynamic Programming Models Discrete-Choice Dynamic-Programming Models References
Numerical Methods for MLE
• The Newton-Raphson procedure uses the following formula.