Discrete panel data Michel Bierlaire Transport and Mobility Laboratory School of Architecture, Civil and Environmental Engineering Ecole Polytechnique F´ ed´ erale de Lausanne M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 1 / 40
Outline Outline Introduction 1 Static model 2 Static model with panel effect 3 Dynamic model 4 Dynamic model with panel effect 5 Application 6 Summary 7 M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 2 / 40
Introduction Introduction Panel data Type of data used so far: cross-sectional. Cross-sectional: observation of individuals at the same point in time. Time series: sequence of observations. Panel data is a combination of comparable time series. M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 3 / 40
Introduction Introduction Panel data Data collected over multiple time periods for the same sample of individuals. Multidimensional Individual Day Price of stock 1 Price of stock 2 Purchase n t x 1 nt x 2 nt i int 1 1 12.3 15.6 1 1 2 12.1 18.6 2 1 3 11.0 25.3 2 1 4 9.2 25.1 0 2 1 12.3 15.6 2 2 2 12.1 18.6 0 2 3 11.0 25.3 0 2 4 9.2 25.1 1 M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 4 / 40
Introduction Introduction Examples of discrete panel data People are interviewed monthly and asked if they are working or unemployed. Firms are tracked yearly to determine if they have been acquired or merged. Consumers are interviewed yearly and asked if they have acquired a new cell phone. Individual’s health records are reviewed annually to determine onset of new health problems. M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 5 / 40
Introduction Model: single time period x ε U i M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 6 / 40
Static model Outline Introduction 1 Static model 2 Static model with panel effect 3 Dynamic model 4 Dynamic model with panel effect 5 Application 6 Summary 7 M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 7 / 40
Static model Static model x t − 1 x t ε t − 1 ε t U t − 1 U t i t − 1 i t M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 8 / 40
Static model Static model Utility U int = V int + ε int , i ∈ C nt . Logit e V int P ( i nt ) = j ∈C nt e V jnt � Estimation: contribution of individual n to the log likelihood T � P ( i n 1 , i n 2 , . . . , i nT ) = P ( i n 1 ) P ( i n 2 ) · · · P ( i nT ) = P ( i nt ) t =1 T � ln P ( i n 1 , i n 2 , . . . , i nT ) = ln P ( i n 1 )+ln P ( i n 2 )+ · · · +ln P ( i nT ) = ln P ( i nt ) t =1 M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 9 / 40
Static model Static model: comments Views observations collected through time as supplementary cross sectional observations. Standard software for cross section discrete choice modeling may be used directly. Simple, but there are two important limitations: Serial correlation unobserved factors persist over time, in particular, all factors related to individual n , ε in ( t − 1) cannot be assumed independent from ε int . Dynamics Choice in one period may depend on choices made in the past. e.g. learning effect, habits. M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 10 / 40
Static model with panel effect Outline Introduction 1 Static model 2 Static model with panel effect 3 Dynamic model 4 Dynamic model with panel effect 5 Application 6 Summary 7 M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 11 / 40
Static model with panel effect Dealing with serial correlation x t − 1 x t ε t − 1 ε t U t − 1 U t i t − 1 i t M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 12 / 40
Static model with panel effect Panel effect Relax the assumption that ε int are independent across t . Assumption about the source of the correlation individual related unobserved factors, persistent over time. The model ε int = α in + ε ′ int It is also known as agent effect, unobserved heterogeneity. M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 13 / 40
Static model with panel effect Panel effect Assuming that ε ′ int are independent across t , we can apply the static model. Two versions of the model: with fixed effect: α in are unknown parameters to be estimated, with random effect: α in are distributed. M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 14 / 40
Static model with panel effect Static model with fixed effect Utility U int = V int + α in + ε ′ int , i ∈ C nt . Logit e V int + α in P ( i nt ) = j ∈C nt e V jnt + α jn � Estimation: contribution of individual n to the log likelihood T � P ( i n 1 , i n 2 , . . . , i nT ) = P ( i n 1 ) P ( i n 2 ) · · · P ( i nT ) = P ( i nt ) t =1 T � ln P ( i n 1 , i n 2 , . . . , i nT ) = ln P ( i n 1 )+ln P ( i n 2 )+ · · · +ln P ( i nT ) = ln P ( i nt ) t =1 M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 15 / 40
Static model with panel effect Static model with fixed effect Comments α in capture permanent taste heterogeneity. For each n , one α in must be normalized to 0. The α ’s are estimated consistently only if T → ∞ . This has an effect on the other parameters that will be inconsistently estimated. In practice, T is usually too short, the number of α parameters is usually too high, for the model to be consistently estimated and practical. M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 16 / 40
Static model with panel effect Static model with random effect Denote α n the vector gathering all parameters α in . Assumption: α n is distributed with density f ( α n ). For instance: α n ∼ N (0 , Σ) . We have a mixture of static models. Given α n , the model is static, as ε ′ int are assumed independent across t . M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 17 / 40
Static model with panel effect Static model with random effect Utility U int = V int + α in + ε ′ int , i ∈ C nt . Conditional choice probability e V int + α in P ( i nt | α n ) = j ∈C nt e V jnt + α jn � M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 18 / 40
Static model with panel effect Static model with random effect Contribution of individual n to the log likelihood, given α n T � P ( i n 1 , i n 2 , . . . , i nT | α n ) = P ( i nt | α n ) . t =1 Unconditional choice probability T � � P ( i n 1 , i n 2 , . . . , i nT ) = P ( i nt | α ) f ( α ) d α. α t =1 M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 19 / 40
Static model with panel effect Static model with random effect Estimation Mixture model. Requires simulation for large choice sets. Generate draws α 1 , . . . , α R from f ( α ). Approximate T R T � P ( i nt | α ) f ( α ) d α ≈ 1 � � � P ( i nt | α r ) P ( i n 1 , i n 2 , . . . , i nT ) = R α t =1 r =1 t =1 The product of probabilities can generate very small numbers. � T R T R � � � � � P ( i nt | α r ) = ln P ( i nt | α r ) exp . r =1 t =1 r =1 t =1 M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 20 / 40
Static model with panel effect Static model with random effect Comments Parameters to be estimated: β ’s and σ ’s Maximum likelihood estimation leads to consistent and efficient estimators. Ignoring the correlation (i.e. assuming that α n is not present) leads to consistent but not efficient estimators (not the true likelihood function). Accounting for serial correlation generates the true likelihood function and, therefore, the estimates are consistent and efficient. M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 21 / 40
Dynamic model Outline Introduction 1 Static model 2 Static model with panel effect 3 Dynamic model 4 Dynamic model with panel effect 5 Application 6 Summary 7 M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 22 / 40
Dynamic model Dynamics Choice in one period may depend on choices made in the past e.g. learning effect, habits. Simplifying assumption: the utility of an alternative at time t is influenced by the choice made at time t − 1 only. It leads to a dynamic Markov model. M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 23 / 40
Dynamic model Dynamic Markov model x t − 1 x t ε t − 1 ε t U t − 1 U t i t − 1 i t M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 24 / 40
Dynamic model Dynamic Markov model The model U int = V int + γ y in ( t − 1) + ε int , i ∈ C nt . � 1 if alternative i was chosen by n at time t − 1 y in ( t − 1) = 0 otherwise . Captures serial dependence on past realized state Example - utility of bus today depends on whether consumer took bus yesterday (habit). Fails if utility of bus today depends on permanent individual taste for bus (tastes) and whether consumer took bus yesterday. No serial correlation. Estimation: same as for the static model except that observation t = 0 is lost M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 25 / 40
Dynamic model with panel effect Outline Introduction 1 Static model 2 Static model with panel effect 3 Dynamic model 4 Dynamic model with panel effect 5 Application 6 Summary 7 M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 26 / 40
Dynamic model with panel effect Dynamic Markov model with serial correlation x t − 1 x t ε t − 1 ε t U t − 1 U t i t − 1 i t M. Bierlaire (TRANSP-OR ENAC EPFL) Discrete panel data 27 / 40
Recommend
More recommend
