A Computationally Practical Simulation Estimation Algorithm for - PDF document

A Computationally Practical Simulation Estimation Algorithm for Dynamic Panel Data Models with Unobserved Endogenous State Variables and Implications of Classification Error for the Dynamics of Female Labor Supply: A Comment on Hyslop (1999) Michael P. Keane University of Technology Sydney Arizona State University Robert M. Sauer University of Southampton

Introduction  Missing endogenous state variables is a widespread problem in panel discrete choice models  Present when unobserved initial conditions and missing choices during sample period  Assess and implement a new SML algorithm that specifically addresses these issues  Main advantage: computationally easy  relies on unconditional simulation of data from a model  conditional simulation of choice probabilities when history not observed is difficult (e.g., GHK, MCMC, EM)

Plan of Presentation  Specify Panel Data Probit Model  Discuss Models of Classification Error (CE)  Describe the SML Algorithm  Show Monte Carlo Results  Application to Female Labor Supply with PSID data  Implications of CE itself for endogeneity of fertility and nonlabor income

Panel Probit Model t − 1 u it   0   1 x it  ∑ d i      it   0 1 if u it ≥ 0 d it  0 otherwise.  it   i   it  it   1  i , t − 1   it x it   2 x i , t − 1   it 2   i  N  0,    it  N  0,   2   it  N  0,  v 2  simulate data from model so algorithm can easily handle wider range of distributions

Classification Error (CE)  Required to form likelihood in our approach  CE probabilistically "matches" simulated choice and reported choice  General model of CE we consider ∗  0 | d it  1  10 t  Pr d it ∗  1 | d it  0  01 t  Pr d it  00 t  1 −  01 t  11 t  1 −  10 t as in Poterba and Summers  1986,1995  and HAS  1998   HAS  1998  develops identification conditions (more later)  Only need tractable expression for  jkt ’s to form likelihood

CE Model 1: Unbiased Classification Error  imposes unconditional prob report an option equals true prob ∗  1   Pr  d it  1  Pr  d it generates tractable linear expressions for  jkt ’s ∗  1 | d it  1  11 t  Pr d it  E   1 − E  Pr  d it  1  ∗  1 | d it  0  01 t  Pr d it   1 − E  Pr  d it  1  because ∗  1    11 t Pr  d it  1    01 t Pr  d it  0  Pr  d it  11 t  E   1 − E  Pr  d it  1  ∗  1   Pr  d it  1   Pr  d it

CE Model 1: Unbiased Classification Error  In  11 t  E   1 − E  Pr  d it  1   E is estimable parameter where  low prob events have prob equal to E of being classified correctly  prob of correct classification increases linearly in E  Pr  d it  1  is easily simulated

CE Model 2: Biased Classification Error ∗  1   Pr  d it  1   Do not impose Pr  d it rather assume: ∗ l it   0   1 d it   2 d it − 1   it 1 if l it ≥ 0 ∗  d it 0 otherwise.  it  logistic  Get tractable expressions for  jkt ’s: ∗ e  0   1   2 d it − 1 ∗  1 | d it  1  11 t  Pr d it  ∗ 1  e  0   1   2 d it − 1 ∗ e  0   2 d it − 1 ∗  1 | d it  0  01 t  Pr d it  ∗ 1  e  0   2 d it − 1  Note that easily incorporates dynamic (persistent) misreporting ∗  d it − 1 can be simulated from l it model if missing

Identification ∗  1   Again, rewrite Pr  d it   11 t Pr  d it  1    01 t Pr  d it  0    1 −  10 t  Pr  d it  1    01 t  1 − Pr  d it  1    01 t   1 −  10 t −  01 t  Pr  d it  1   need non-linear Pr  d it  1  and monotonicity  10 t   01 t  1  otherwise standard identification issues  SD effects from causal effect of lagged X’s  SD effects sensitive to modeling of serial correlation (RE or AR(1), etc.)  in biased CE model, lagged reported choice identified because not perfectly correlated with lagged X’s

The SML Estimation Algorithm ∗   d it N , D i T , x i   x it  t  1 ∗ , x i  i  1 ∗  t  1 Data:  D i T 1 . Draw M times from the  it distribution to T  i  1 M N   it m  t  1 form m  1 T  i  1 N and 2 . Given  x it  t  1 M , construct T  i  1 N   it m  t  1 m  1 T  i  1 M N  d it m  t  1 according to model m  1 3 . Construct conditional probs  m  t  1 M T  jkt m  1 4 . Form a simulator likelihood contribution: ∗ |  , x i  P D i ∗ observed 1  I d it 1 M T m  j , d it ∗  k  m I  d it M ∑  ∑ ∑  jkt 1 m  1 t  1 j  0 k  0

Important Things to Note  any observed choice history has non-zero prob conditional on any simulated history  building likelihood off of unconditional simulations  state space updated according to simulated outcomes  completely circumvents problem of partially observed choice history  consistency and asymptotic normality N →  as N →  (as in M require that Pakes and Pollard  1989  and Lee  1992  )  but still need to check small sample properties of estimator

Some More Important Things to Note  if missing x it ’s, can simulate them and modify likelihood to include density of x it ’s  if initial conditions problem can simulate model from t  0,..., T with d i 0  x i 0  0  or can imbed Heckman’s solution - specify marginal distribution for d i   , then simulate from t    ,..., T  or can imbed Wooldridge’s solution - random effect function of d i   and covariates, simulate from t     1,..., T  for standard errors, or to use gradient based optimization, have smooth version of the algorithm

Importance Sampling (Smooth) Version of Algorithm  construct  d it m   0  t  1 and  U it m   0  t  1 T T and hold fixed as vary   calculate importance sampling weights as vary  m   0  ,..., U iT m   0  |  , x i g U i 1 W m     m   0  ,..., U iT m   0  |  0 , x i g U i 1 T m   0  |  , x i    g  U i     a    1 t  1 t − 1 m   0  −  0 −  1 x it − ∑ a  U it m   0    d i    0  likelihood contribution becomes M T ∗ |  , x i M ∑ W m      1 f m  x it  I x it observed P D i t   m  1  ∗ observed 1  I d it 1 m  j , d it ∗  k  m I  d it ∑ ∑  jkt j  0 k  0

Repeated Sampling Experiments  RE  AR(1) Polya Model with Exponential Decay: t − 1 u it   0   1 x it  ∑ d i      it   0     e −   t −  − 1   it   i   it  it   1  i , t − 1   it 2  1 −  1  it  N  0,  1 −   2  x it   2 x i , t − 1   it ,  it  N  0,  v 2   Sample Size: N  500, T  10  Replications: R  50  Simulated histories per person: M  1000  Also do experiments on Markov model

Table 3 Repeated Sampling Experiments Random E ff ects Polya Model Unbiased Classi fi cation Error (Missing X’s, No Initial Conditions Problem) Mean b Median b Std ( c Parameter True Value β β β ) RMSE t-Stat 20% Missing Choices and X’s ( t = 1 , ..., 10) β 0 -.1000 -.1051 -.1023 .0436 .0439 -.83 β 1 1.0000 1.0167 1.0191 .0611 .0634 1.92 ρ 1.0000 1.0479 1.0446 .0444 .0653 7.63 α .5000 .4977 .5031 .0656 .0657 -.24 φ 2 .2500 .2520 .2505 .0176 .0177 .80 σ ν .5000 .5015 .5016 .0057 .0059 1.86 σ μ .8000 .8056 .8017 .0287 .0292 1.38 E .7500 .7428 .7430 .0172 .0187 -2.95 40% Missing Choices and X’s ( t = 1 , ..., 10) β 0 -.1000 -.1087 -.1099 .0539 .0546 -1.15 β 1 1.0000 1.0141 1.0233 .0678 .0692 1.48 ρ 1.0000 1.0458 1.0374 .0636 .0784 5.10 α .5000 .4953 .4949 .0600 .0602 .56 φ 2 .2500 .2521 .2546 .0253 .0254 .59 σ ν .5000 .5012 .5012 .0069 .0070 1.21 σ μ .8000 .8046 .8063 .0347 .0350 .94 E .7500 .7474 .7416 .0245 .0246 -.74 60% Missing Choices and X’s ( t = 1 , ..., 10) β 0 -.1000 -.0997 -.1116 .0542 .0543 .05 β 1 1.0000 1.034 1.0258 .0894 .0924 1.85 ρ 1.0000 1.0401 1.0512 .0682 .0791 4.15 α .5000 .4957 .4973 .0721 .0722 -.42 φ 2 .2500 .2507 .2498 .0372 .0373 .13 σ ν .5000 .5011 .5017 .0089 .0090 .88 σ μ .8000 .8096 .8044 .0421 .0432 1.61 E .7500 .7493 .7440 .0288 .0288 -.16 Note: The number of replications in each experiment is 50 and the number of individuals in the sample is 500 . Std ( c β ) and RMSE refer to the sample standard deviation and the root mean square µ ¶ √ Mean � β − β error, respectively, of the estimated parameters. The t-statistics are calculated as 50 . Std ( � β ) The model is the same as in Table 1 . 1