Conditional Choice Probability Estimators of Single-Agent Dynamic Discrete Choice Models Hotz and Miller (REStud, 1993), Aguirregabiria and Mira (Econometrica, 2002) presented by Anton Laptiev ECON 565, UBC February 7, 2013
Outline Single-Agent Models Hotz & Miller Estimator Aguirregabiria & Mira Estimator
Single-Agent Models � time is discrete t = 0,1,.., T � agents have preferences defined over a sequence of states of the world { s it , a it } � s it a vector of state variables that is known at period t � a it denotes a decision chosen at t with a it ∈ A = {0,1,..., J } � agents’ preferences are represented by a utility function T β j U � � � a i , t + j , s i , t + j , β ∈ (0,1) j = 0 � the optimization problem of an agent � � T β j U � � � max a ∈ A E a i , t + j , s i , t + j | a it , s it j = 0
Single-Agent Models � Bellman equation � � �� � � � V ( s it ) = max U ( a , s it ) + β V s i , t + 1 d F s i , t + 1 | a , s it a ∈ A � choice-specific value function � � � � � v ( a , s it ) ≡ U ( a , s it ) + β V s i , t + 1 d F s i , t + 1 | a , s it � the optimal decision rule α ( s it ) = arg max a ∈ A { v ( a , s it )} � assume that s it is divided into two subvectors s it = ( x it , ǫ it )
Assumptions � Assumption AS � the one-period utility function is additively separable in observable and unobservable components U ( a , x it , ǫ it ) = u ( a , x it ) + ǫ it ( a ) � Assumption IID � the unobserved state variables ǫ it are identically distributed over agents and over time with cdf G ǫ ( ǫ it ) � Assumption CI � conditional on the current values of the decision and ob- servable state variables, next period state variables does nor depend on the current ǫ � � � � F x x i , t + 1 | a it , x it , ǫ it = F x x i , t + 1 | a it , x it
Assumptions � Assumption Logit � the unobserved state variables are independent across al- ternatives and have an extreme value type I distribution � Assumption DS � the support of x it is discrete and finite � by assumptions IID and CI, the solution to DP problem is fully characterized by ex ante value function � V ( x it ) = V ( x it , ǫ it ) d G ǫ ( ǫ it ) � by assumption AS the choice-specific value function can be written as follows � � � � � v ( a , x it ) = u ( a , x it ) + β V x i , t + 1 f x x i , t + 1 | a , x it + ǫ it ( a ) x i , t + 1
Forming the Likelihood � � � � Pr a it , x it | ˜ a i , t − 1 , ˜ x i , t − 1 = Pr ( a it | x it ) f x x it | a i , t − 1 , x i , t − 1 T i T i − 1 � � � � l i ( θ ) = log P ( a it | x it , θ ) + log f x x i , t + 1 | a it , x it , θ f + log Pr ( x i 1 | θ ) t = 1 t = 1 � P ( a | x , θ ) ≡ I { α ( x , ǫ ; θ ) = a }d G ǫ ( ǫ ) �� ∀ a ′ �= a a ′ , x it a ′ � � � � � = v ( a , x it ) + ǫ it ( a ) > v + ǫ it d G ǫ ( ǫ it )
The Hotz and Miller Inversion � suppose that payoff function is linear in parameters u ( a , x it ; θ u ) = z ( a , x it ) ′ θ u � the choice-specific value function v ( a , x t ; θ ) = ˜ z ( a , x t ) θ u + ˜ e ( a , x t ; θ ) T − t β j E ( x t + j , ǫ t + j ) | a t = a , x t � � � � � �� z ( a , x t ; θ ) = z ( a , x t ) + ˜ z α x t + j , ǫ t + j ; θ , x t + j j = 1 � � T − t J β j E x t + j | a t = a , x t � � � � � � = z ( a , x t ) + P a t + j | x t + j ; θ z a t + j , x t + j j = 1 a t + j = 0
The Hotz and Miller Inversion T − t β j E ( x t + j , ǫ t + j ) | a t = a , x t � � � � ��� e ( a , x t ; θ ) = ˜ ǫ α x t + j , ǫ t + j ; θ j = 1 � � T − t J β j E x t + j | a t = a , x t � � � � � � P a t + j | x t + j ; θ e a t + j , x t + j = a t + j = 0 j = 1 ǫ t ( a ) | x t , v ( a , x t ) + ǫ t ( a ) ≥ v ( a ′ , x t ) + ǫ t ( a ′ ) ∀ a ′ �= a � � e ( a , x t ) = E 1 � ǫ t ( a ′ ) − ǫ t ( a ) ≤ v ( a , x t ) − v ( a ′ , x t ) ∀ a ′ �= a � � = ǫ t ( a ) I d G ǫ ( ǫ t ) P ( a | x t ) � ǫ t ( a ′ ) − ǫ t ( a ) ≤ v ( a , x t ) − v ( a ′ , x t ) ∀ a ′ �= a � � P ( a | x t ) = ǫ t ( a ) I d G ǫ ( ǫ t ) = ⇒ e ( a , x t ) = f ( P ( a | x t ), G ǫ , ˜ v ( x t )) P ( a | x t ) = f ( G ǫ , ˜ v ( x t )) � where ˜ v ( x t ) is a vector of value differences
The Hotz and Miller Inversion � H&M prove that the letter mapping is invertible = ⇒ e ( a , x t ) = f ( P ( a | x t ), G ǫ ) � by assumption LOGIT, conditional probabilities look as follows � P ( a , x t ) θ u + ˜ P ( a , x t ) � z ˆ e ˆ exp ˜ P ( a | x t ; θ ) = � � P ( a ′ , x t ) θ u + ˜ P ( a ′ , x t ) � J z ˆ e ˆ ˜ a ′ = 0 exp z ˆ P and ˜ e ˆ P are defined on all a ∈ A � where ˜
Example: Renewal Actions � logit errors �� � V ( x t ) = log exp [ v ( x t , a )] + γ a ∈ A � ��� a ∈ A exp [ v ( x t , a )] �� v ( x t , a ′ ) � V ( x t ) = log exp + γ exp [ v ( x t , a ′ )] �� �� v ( x t , a ) − v ( x t , a ′ ) + v ( x t , a ′ ) + γ � = log exp a ∈ A p ( a ′ | x t ) + v ( x t , a ′ ) + γ � � = − log � conditional value function �� v ( x t + 1 , a ′ ) − log p ( a ′ | x t + 1 � �� v ( x t , a ) = u ( x t , a ) + β f x ( x t + 1 | x t , a )d x t + 1 + βγ
Example: Renewal Actions � let a = R indicate the renewal action so that � � x t + 1 | x t , a ′ � � f x ( x t + 1 | x t , a ) f x ( x t + 2 | x t + 1 , R )d x t + 1 = f x f x ( x t + 2 | x t + 1 , R )d x t + 1 ∀ { a , a ′ }, x t + 2 � substitute into the conditional value function �� � �� v ( x t , a ) = u ( x t , a ) + β v ( x t + 1 , R ) − log p ( R | x t + 1 f x ( x t + 1 | x t , a )d x t + 1 + βγ �� � �� = u ( x t , a ) + β u ( x t + 1 , R ) − log p ( R | x t + 1 f x ( x t + 1 | x t , a )d x t + 1 �� + βγ + β 2 V ( x t + 2 ) f x ( x t + 2 | x t + 1 , R ) f x ( x t + 1 | x t , a )d x t + 2 d x t + 1 � the last term is constant across all choices made at time t
Estimation � first stage � P ( a | x t ;) and f x � � x i , t + 1 | a it , x it , θ f are estimated nonpara- metrically directly from the data P and ˜ P by backward induction (finite T ) or us- z ˆ e ˆ � estimate ˜ ing an iterative procedure (requires stationarity) � second stage � estimate θ by GMM using moment conditions of the form N T i � � I { a it = a } − ˆ � � H ( x it ) P ( a | x it ; θ ) = 0 t = 1 i = 1
Advantages of H&M estimator � computational simplicity relative to full solution methods z ˆ P and ˜ e ˆ P are computed only once and remain fixed � ˜ during the search for θ � conditional logit assumption along with the assump- tion of linearity result in the unique solution for the system of GMM equations � pseudo maximum likelihood version of H&M es- timator is asymptotically equivalent to two-step NFP estimator
Limitations of H&M approach � computational gain relies on (strong) assump- tions regarding the form of utility function and the structure of unobservable state variables � in finite samples produces larger bias than the two step NFP estimator (Aguirregabiria & Mira, 2002) � subject to the "curse of dimensionality" � cannot accommodate unobserved heterogene- ity � consistent estimates of conditional choice probabili- ties cannot be recovered in the first stage
Aguirregabiria & Mira estimator � start from the pseudo likelihood version of H&M estima- tor � � z ˆ P ( a it , x it ) θ u + ˜ e ˆ P ( a it , x it ) exp ˜ N T i P , ˆ θ u , ˆ � � � � Q θ f log = � � � J P ( a ′ , x it ) θ u + ˜ P ( a ′ , x it ) z ˆ e ˆ ˜ a ′ = 0 exp i = 1 t = 1 � obtain estimates ˆ θ u � compute new estimates of the choice probabilities � � z ˆ P ( a , x ) θ u + ˜ e ˆ P ( a , x ) exp ˜ P 1 = ˆ ˆ P 1 ( a | x ) = � � � J P ( a ′ , x ) θ u + ˜ P ( a ′ , x ) z ˆ e ˆ ˜ a ′ = 0 exp
Aguirregabiria & Mira estimator P and ˜ z ˆ e ˆ � given ˆ P P 1 one can compute new values ˜ as well as a new pseudo likelihood function Q ( · ) and maximize it to get a new value of ˆ θ u � iterating in this way we can generate a sequence of estimators of structural parameters and CCPs � ˆ θ u , K , ˆ � P K : K = 1,2,... : ∀ K ≥ 1 ˆ θ u , ˆ P K − 1 , ˆ � � θ u , K = arg max θ u ∈ Θ Q θ f � � z ˆ e ˆ P K − 1 ( a , x ) ˆ P K − 1 ( a , x ) exp ˜ θ u , K + ˜ ˆ P K ( a | x ) = � � � J z ˆ P K − 1 ( a ′ , x ) ˆ e ˆ P K − 1 ( a ′ , x ) a ′ = 0 exp ˜ θ u , K + ˜
Advantages of the Aguirregabiria & Mira estimator � Aguirregabiria & Mira (2002) present Monte Carlo evidence that iterating in this procedure produces significant reductions in finite sample bias � as K → ∞ the recursive procedure converges to the two-step NFP estimator even if the initial CCP estimator was inconsistent � A&M show that CCP mapping used in the iter- ative estimation is a contraction mapping and therefore can be used to compute the solution of the DP problem in the space of CCPs
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