3256:
2922:
1078:
344:
3763:
3251:{\displaystyle U(x_{t},\xi _{t},d,\theta )={\begin{cases}-c(x_{t},\theta )+\xi _{t,{\text{keep}}},&\\-RC-c(0,\theta )+\xi _{t,{\text{replace}}},&\end{cases}}=u(x_{t},d,\theta )+{\begin{cases}\xi _{t,{\text{keep}}},&{\textrm {if}}\;\;d={\text{keep}},\\\xi _{t,{\text{replace}}},&{\textrm {if}}\;\;d={\text{replace}},\end{cases}}}
882:
2157:
1465:
4190:
5622:
2709:
An alternative to full-solution methods is non-solution methods. In this case, the researcher can estimate the structural parameters without having to fully solve the backwards recursion problem for each parameter guess. Non-solution methods are typically faster while requiring more assumptions, but
5635:
The conditional choice probabilities method of Hotz and Miller can be applied in this setting. Hotz, Miller, Sanders, and Smith proposed a computationally simpler version of the method, and tested it on a study of the bus engine replacement problem. The method works by estimating conditional choice
45:
of the agent's decision process. Once these parameters are known, the researcher can then use the estimates to simulate how the agent would behave in a counterfactual state of the world. (For example, how a prospective college student's enrollment decision would change in response to a tuition
84:
5016:
1073:{\displaystyle {\begin{alignedat}{5}U_{nit}\left(x_{nt},\varepsilon _{nit}\right)&&\;=\;&&u_{nit}&&\;+\;&&\varepsilon _{nit}\\&&\;=\;&&X_{nt}\alpha _{i}&&\;+\;&&\varepsilon _{nit}\end{alignedat}}}
2690:(MPEC). Specifically, the likelihood function is maximized subject to the constraints imposed by the model, and expressed in terms of the additional variables that describe the model's structure. This approach requires powerful optimization software such as
3536:
2697:
In the later article Rust and coauthors show that the speed advantage of MPEC compared to NFXP is not significant. Yet, because the computations required by MPEC do not rely on the structure of the model, its implementation is much less labor intensive.
26:, model an agent's choices over discrete options that have future implications. Rather than assuming observed choices are the result of static utility maximization, observed choices in DDC models are assumed to result from an agent's maximization of the
1902:
1220:
1818:
3877:
are respectively transition densities for the observed and unobserved states variables. Time indices in the
Bellman equation are dropped because the model is formulated in the infinite horizon settings, the unknown optimal policy is
5367:
3965:
339:{\displaystyle V\left(x_{n0}\right)=\max _{\left\{d_{nt}\right\}_{t=1}^{T}}\mathbb {E} \left(\sum _{t^{\prime }=t}^{T}\sum _{i=1}^{J}\beta ^{t'-t}\left(d_{nt}=i\right)U_{nit}\left(x_{nt},\varepsilon _{nit}\right)\right),}
2631:
4794:
4467:
2641:
Estimation of dynamic discrete choice models is particularly challenging, due to the fact that the researcher must solve the backwards recursion problem for each guess of the structural parameters.
2753:
and bear the associated replacement cost, or to continue operating the bus at an ever raising cost of operation, which includes insurance and the cost of lost ridership in the case of a breakdown.
5372:
3937:
3875:
5777:
1572:
6074:
757:
3821:
2694:
because of the high dimensionality of the optimization problem. Once it is solved, both the structural parameters that maximize the likelihood, and the solution of the model are found.
533:
3758:{\displaystyle V(x,\xi ,\theta )=\max _{d={\text{keep}},{\text{replace}}}\left\{u(x,d,\theta )+\xi _{d}+\iint V(x',\xi ',\theta )q(d\xi '\mid x',\theta )p(dx'\mid x,d,\theta )\right\}}
3419:
3349:
3384:
3314:
2221:
2152:{\displaystyle v_{nit}(x_{nt})=u_{nit}\left(x_{nt}\right)+\beta \int _{x_{t+1}}\mathbb {E} \max _{j}\left\{v_{njt+1}(x_{nt+1})+\varepsilon _{njt+1}\right\}\,dF(x_{t+1}\mid x_{t})}
887:
3495:
2687:
5349:
5292:
4762:
2499:
2444:
820:
697:
3456:
1212:
872:
The flow utility can be written as an additive sum, consisting of deterministic and stochastic elements. The deterministic component can be written as a linear function of the
5168:
2847:
1460:{\displaystyle V_{nt}(x_{nt})=\mathbb {E} \max _{i}\left\{u_{nit}(x_{nt})+\varepsilon _{nit}+\beta \int _{x_{t+1}}V_{nt+1}(x_{nt+1})\,dF\left(x_{t+1}\mid x_{t}\right)\right\}}
4721:
4613:
4545:
4237:
1684:
1138:
5123:
2182:
1692:
1510:
1490:
5220:
2655:
Aside from estimation methods, there are also solution methods. Different solution methods can be employed due to complexity of the problem. These can be divided into
5627:
This method is faster to compute than non-optimized implementations of the nested fixed point algorithm, and takes about as long as highly optimized implementations.
5299:
5240:
5191:
5085:
5052:
4782:
4637:
4565:
2867:
2730:
is one of the first dynamic stochastic models of discrete choice estimated using real data, and continues to serve as classical example of the problems of this type.
2356:
2323:
1854:
1605:
853:
571:
2910:
787:
664:
449:
414:
380:
4669:
4501:
3522:
2781:
2290:
2259:
1632:
2890:
3957:
3279:
2805:
2408:
2388:
1894:
1874:
1182:
1162:
718:
634:
614:
594:
489:
469:
72:
5617:{\displaystyle {\begin{aligned}\max &\qquad L(\theta )&\\{\text{subject to}}&\qquad EV(x,d,\theta )=\int \leftp(x'\mid x,d,\theta )\end{aligned}}}
4185:{\displaystyle P(d\mid x,\theta )={\frac {\exp\{u(x,d,\theta )+\beta EV(x,d,\theta )\}}{\sum _{d'\in D(x)}\exp\{u(x,d',\theta )+\beta EV(x,d',\theta )\}}}}
2515:
855:. That is, the agent is uncertain about future transitions in the states, and is also uncertain about future realizations of unobserved factors.
2454:
5011:{\displaystyle L(\theta )=\sum _{i=1}^{N}\sum _{t=1}^{T_{i}}\log(P(d_{it}\mid x_{it},\theta ))+\log(p(x_{it}\mid x_{it-1},d_{it-1},\theta )),}
6037:
4248:
2701:
Despite numerous contenders, the NFXP maximum likelihood estimator remains the leading estimation method for Markov decision models.
3422:
2506:
2450:
5862:
Hotz, V. J.; Miller, R. A.; Sanders, S.; Smith, J. (1994-04-01). "A Simulation
Estimator for Dynamic Models of Discrete Choice".
5653:
2645:
6079:
2713:
The leading non-solution method is conditional choice probabilities, developed by V. Joseph Hotz and Robert A. Miller.
2649:
2410:. Writing the conditional value function in this way is useful in constructing formulas for the choice probabilities.
1515:
729:
5815:
Hotz, V. Joseph; Miller, Robert A. (1993). "Conditional Choice
Probabilities and the Estimation of Dynamic Models".
1634:. The expectation over state transitions is accomplished by taking the integral over this probability distribution.
6069:
6064:
5952:
4568:
496:
5358:
3389:
3319:
2683:
3888:
3826:
2737:
stochastic dynamic problem faced by the decision maker, Harold
Zurcher, superintendent of maintenance at the
2413:
To write down the choice probabilities, the researcher must make an assumption about the distribution of the
864:
It is standard to impose the following simplifying assumptions and notation of the dynamic decision problem:
3771:
3354:
3284:
2228:
2187:
3461:
3351:
represent the component of the utility observed by Harold
Zurcher, but not John Rust. It is assumed that
5917:
5313:
5256:
4726:
2471:
2416:
873:
792:
669:
42:
3428:
2671:
The foremost example of a full-solution method is the nested fixed point (NFXP) algorithm developed by
1187:
5249:
Rust's implementation of the nested fixed point algorithm is highly optimized for this problem, using
5128:
2810:
4682:
4574:
4506:
4198:
1813:{\displaystyle V_{nt}(x_{nt})=\mathbb {E} \max _{i}\left\{v_{nit}(x_{nt})+\varepsilon _{nit}\right\}}
3142:
2978:
1640:
1094:
5295:
4473:
4240:
23:
5090:
2165:
1495:
1473:
6008:
5895:
5887:
5832:
3879:
2742:
2458:
5250:
5196:
6043:
6033:
6000:
5940:
5879:
5797:
5751:
5173:
The joint algorithm for solving the fixed point problem given a particular value of parameter
417:
5225:
5176:
5057:
5024:
4767:
4622:
4550:
2852:
2328:
2295:
1826:
1577:
825:
543:
6025:
5992:
5968:
5932:
5871:
5824:
5789:
5741:
5733:
3528:
2895:
2734:
1085:
762:
639:
424:
389:
355:
75:
4642:
4479:
3500:
2759:
2682:
in 2012 implements another approach (dismissed as intractable by Rust in 1987), which uses
2268:
2237:
1610:
4616:
2913:
35:
5087:
represent data on state variables (odometer readings) and decision (keep or replace) for
5983:(1987). "Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher".
2872:
5956:
5641:
4785:
3942:
3264:
2790:
2691:
2393:
2373:
1879:
1859:
1167:
1147:
703:
619:
599:
579:
574:
474:
454:
383:
57:
31:
6029:
5776:
Iskhakov, Fedor; Lee, Jinhyuk; Rust, John; Schjerning, Bertel; Seo, Kyoungwon (2016).
2675:
in 1987. The NFXP algorithm is described in great detail in its documentation manual.
6058:
5778:"Comment on "constrained optimization approaches to estimation of structural models""
2749:
in operation in each time period Harold
Zurcher has to decide whether to replace the
2738:
27:
6020:
Rust, John (1994). "Chapter 51 Structural estimation of markov decision processes".
5899:
5721:
2679:
2262:
721:
5936:
5724:(2012). "Constrained Optimization Approaches to Estimation of Structural Models".
5959:(2009). "Empirical applications of discrete choice dynamic programming models".
2462:
2446:'s. As in static discrete choice models, this distribution can be assumed to be
536:
5697:
5972:
5637:
6047:
6004:
5944:
5883:
5801:
5755:
2626:{\displaystyle P_{nit}={\frac {\exp(v_{nit})}{\sum _{j=1}^{J}\exp(v_{njt})}}}
5980:
4679:
The contraction mapping above can be solved numerically for the fixed point
2672:
2784:
6012:
5891:
5836:
5746:
2644:
The most common methods used to estimate the structural parameters are
2370:, because it is the value function conditional on choosing alternative
5793:
5737:
2750:
5996:
5875:
5828:
2849:
cost of operating the bus which depends on the vector of parameters
4462:{\displaystyle EV(x,d,\theta )=\int \leftp(x'\mid x,d,\theta ).}
2726:
The bus engine replacement model developed in the seminal paper
868:
1. Flow utility is additively separable and linear in parameters
4472:
It can be shown that the latter functional equation defines a
2746:
2502:
2447:
181:
3244:
3096:
2710:
the additional assumptions are in many cases realistic.
2366:
The value function in the previous section is called the
2509:), the formulas for the choice probabilities would be:
5918:"Dynamic discrete choice structural models: A survey"
5771:
5769:
5767:
5765:
5370:
5316:
5259:
5228:
5199:
5179:
5131:
5093:
5060:
5027:
4797:
4770:
4729:
4685:
4645:
4625:
4577:
4553:
4509:
4482:
4251:
4201:
3968:
3945:
3891:
3829:
3774:
3539:
3503:
3464:
3431:
3421:
are independent and identically distributed with the
3392:
3357:
3322:
3287:
3267:
2925:
2898:
2875:
2855:
2813:
2793:
2762:
2688:
mathematical programming with equilibrium constraints
2518:
2474:
2419:
2396:
2376:
2331:
2298:
2271:
2240:
2190:
2168:
1905:
1882:
1862:
1829:
1695:
1643:
1613:
1580:
1518:
1498:
1476:
1223:
1190:
1170:
1150:
1097:
885:
828:
795:
765:
732:
706:
672:
642:
622:
602:
582:
546:
499:
477:
457:
427:
392:
358:
87:
60:
2362:
Conditional value functions and choice probabilities
5698:"Nested fixed point algorithm documentation manual"
5849:
5616:
5343:
5286:
5234:
5214:
5185:
5162:
5117:
5079:
5046:
5010:
4776:
4756:
4715:
4663:
4631:
4607:
4559:
4539:
4495:
4461:
4231:
4184:
3951:
3931:
3869:
3815:
3757:
3516:
3489:
3450:
3413:
3378:
3343:
3308:
3273:
3250:
2904:
2884:
2861:
2841:
2799:
2775:
2625:
2493:
2438:
2402:
2382:
2350:
2317:
2284:
2253:
2215:
2176:
2151:
1888:
1868:
1848:
1812:
1678:
1626:
1599:
1566:
1504:
1484:
1459:
1206:
1176:
1156:
1132:
1072:
847:
814:
781:
751:
712:
691:
658:
628:
608:
588:
565:
527:
483:
463:
443:
408:
374:
338:
66:
6075:Mathematical and quantitative methods (economics)
5351:is recalculated for each guess of the parameters
5375:
3568:
2016:
1737:
1567:{\displaystyle dF\left(x_{t+1}\mid x_{t}\right)}
1265:
1084:2. The optimization problem can be written as a
116:
4503:is bounded, so there will be a unique solution
752:{\displaystyle \mathbb {E} \left(\cdot \right)}
5640:, then backing out the implied differences in
2686:of the likelihood function, a special case of
1686:into deterministic and stochastic components:
6024:. Vol. 4. Elsevier. pp. 3081–3143.
5870:(2). Oxford University Press (OUP): 265–289.
1574:represents the probability distribution over
8:
5916:Aguirregabiria, Victor; Mira, Pedro (2010).
5715:
5713:
5711:
5562:
5487:
4675:Estimation with nested fixed point algorithm
4408:
4333:
4176:
4106:
4065:
4005:
5671:
3281:denotes the decision (keep or replace) and
41:The goal of DDC methods is to estimate the
3226:
3225:
3176:
3175:
2916:. Then the per-period utility is given by
1046:
1042:
1012:
1008:
982:
978:
955:
951:
528:{\displaystyle \beta \in \left(0,1\right)}
78:can be written mathematically as follows:
5745:
5474:
5466:
5459:
5398:
5371:
5369:
5315:
5258:
5227:
5198:
5178:
5154:
5130:
5092:
5065:
5059:
5032:
5026:
4978:
4956:
4940:
4897:
4881:
4854:
4849:
4838:
4828:
4817:
4796:
4769:
4728:
4684:
4644:
4624:
4576:
4552:
4508:
4487:
4481:
4320:
4312:
4305:
4250:
4200:
4074:
3996:
3967:
3944:
3890:
3828:
3773:
3629:
3586:
3578:
3571:
3538:
3508:
3502:
3469:
3463:
3436:
3430:
3414:{\displaystyle \xi _{t,{\text{replace}}}}
3404:
3397:
3391:
3369:
3362:
3356:
3344:{\displaystyle \xi _{t,{\text{replace}}}}
3334:
3327:
3321:
3299:
3292:
3286:
3266:
3233:
3219:
3218:
3206:
3199:
3183:
3169:
3168:
3156:
3149:
3137:
3113:
3082:
3075:
3023:
3016:
2994:
2973:
2949:
2936:
2924:
2897:
2874:
2854:
2824:
2812:
2792:
2767:
2761:
2605:
2586:
2575:
2554:
2538:
2523:
2517:
2479:
2473:
2424:
2418:
2395:
2375:
2336:
2330:
2303:
2297:
2276:
2270:
2245:
2239:
2195:
2189:
2170:
2169:
2167:
2140:
2121:
2107:
2084:
2059:
2034:
2019:
2011:
2010:
1996:
1991:
1968:
1948:
1929:
1910:
1904:
1881:
1861:
1834:
1828:
1793:
1774:
1755:
1740:
1732:
1731:
1716:
1700:
1694:
1664:
1648:
1642:
1618:
1612:
1585:
1579:
1553:
1534:
1517:
1497:
1478:
1477:
1475:
1441:
1422:
1406:
1388:
1366:
1348:
1343:
1321:
1302:
1283:
1268:
1260:
1259:
1244:
1228:
1222:
1195:
1189:
1169:
1149:
1118:
1102:
1096:
1054:
1033:
1020:
990:
963:
931:
915:
894:
886:
884:
833:
827:
800:
794:
770:
764:
734:
733:
731:
705:
677:
671:
647:
641:
621:
601:
581:
551:
545:
498:
476:
456:
432:
426:
397:
391:
363:
357:
311:
295:
274:
250:
224:
214:
203:
193:
180:
175:
162:
161:
153:
142:
129:
119:
99:
86:
59:
3932:{\displaystyle q(d\xi '\mid x',\theta )}
3870:{\displaystyle q(d\xi '\mid x',\theta )}
5664:
3939:, the probability of particular choice
3885:Given the distributional assumption on
3527:Then the optimal decisions satisfy the
3816:{\displaystyle p(dx'\mid x,d,\theta )}
3379:{\displaystyle \xi _{t,{\text{keep}}}}
3309:{\displaystyle \xi _{t,{\text{keep}}}}
2227:3. The optimization problem follows a
5357:. The MPEC method instead solves the
5310:In the nested fixed point algorithm,
1856:is the value to choosing alternative
7:
5683:
5631:Estimation with non-solution methods
2727:
2216:{\displaystyle \varepsilon _{njt+1}}
860:Simplifying assumptions and notation
17:Dynamic discrete choice (DDC) models
4788:function can then be formulated as
3490:{\displaystyle \xi _{t-1,\bullet }}
2733:The model is a simple regenerative
596:receives from choosing alternative
5344:{\displaystyle P(d\mid x,\theta )}
5300:Berndt–Hall–Hall–Hausman algorithm
5287:{\displaystyle P(d\mid x,\theta )}
5193:and maximizing the log-likelihood
4757:{\displaystyle P(d\mid x,\theta )}
4723:that yields choice probabilities
2892:cost of replacing the engine, and
2494:{\displaystyle \varepsilon _{nit}}
2439:{\displaystyle \varepsilon _{nit}}
815:{\displaystyle \varepsilon _{nit}}
692:{\displaystyle \varepsilon _{nit}}
14:
3451:{\displaystyle \xi _{t,\bullet }}
3423:Type I extreme value distribution
2507:Type I extreme value distribution
2501:is multinomial logit (i.e. drawn
1207:{\displaystyle \varepsilon _{nt}}
5163:{\displaystyle t=1,\dots ,T_{i}}
2842:{\displaystyle c(x_{t},\theta )}
2739:Madison Metropolitan Bus Company
2678:A recent work by Che-Lin Su and
636:, and depends on both the state
5405:
5380:
5302:, for likelihood maximization.
4716:{\displaystyle EV(x,d,\theta )}
4608:{\displaystyle EV(x,d,\theta )}
4540:{\displaystyle EV(x,d,\theta )}
4232:{\displaystyle EV(x,d,\theta )}
2265:. That is, attainment of state
1470:where the expectation operator
5864:The Review of Economic Studies
5850:Aguirregabiria & Mira 2010
5654:Inverse reinforcement learning
5607:
5578:
5559:
5531:
5516:
5493:
5430:
5412:
5390:
5384:
5338:
5320:
5281:
5263:
5209:
5203:
5002:
4999:
4933:
4927:
4915:
4912:
4874:
4868:
4807:
4801:
4751:
4733:
4710:
4692:
4658:
4646:
4602:
4584:
4534:
4516:
4453:
4424:
4405:
4377:
4362:
4339:
4276:
4258:
4226:
4208:
4173:
4150:
4135:
4112:
4095:
4089:
4062:
4044:
4029:
4011:
3990:
3972:
3926:
3895:
3864:
3833:
3810:
3778:
3747:
3715:
3709:
3678:
3672:
3644:
3619:
3601:
3561:
3543:
3131:
3106:
3065:
3053:
3006:
2987:
2967:
2929:
2836:
2817:
2617:
2598:
2566:
2547:
2146:
2114:
2074:
2052:
1938:
1922:
1783:
1767:
1725:
1709:
1679:{\displaystyle V_{nt}(x_{nt})}
1673:
1657:
1403:
1381:
1311:
1295:
1253:
1237:
1144:value function for individual
1133:{\displaystyle V_{nt}(x_{nt})}
1127:
1111:
1:
6030:10.1016/s1573-4412(05)80020-0
5937:10.1016/j.jeconom.2009.09.007
5251:Newton–Kantorovich iterations
2646:maximum likelihood estimation
30:of utility, generalizing the
5244:nested fixed point algorithm
5118:{\displaystyle i=1,\dots ,N}
4239:is a unique solution to the
3882:, i.e. independent of time.
2787:reading (mileage) at period
2722:Bus engine replacement model
2177:{\displaystyle \mathbb {E} }
1637:It is possible to decompose
1505:{\displaystyle \varepsilon }
1485:{\displaystyle \mathbb {E} }
5961:Review of Economic Dynamics
2650:method of simulated moments
50:Mathematical representation
6096:
5817:Review of Economic Studies
5215:{\displaystyle L(\theta )}
5125:individual buses, each in
2368:conditional value function
2292:depends only on the state
2162:where now the expectation
5973:10.1016/j.red.2008.07.001
5931:(1). Elsevier BV: 38–67.
4569:implicit function theorem
2455:generalized extreme value
21:discrete choice models of
6022:Handbook of Econometrics
5359:constrained optimization
2684:constrained optimization
5925:Journal of Econometrics
5672:Keane & Wolpin 2009
5242:was named by John Rust
5235:{\displaystyle \theta }
5186:{\displaystyle \theta }
5080:{\displaystyle d_{i,t}}
5047:{\displaystyle x_{i,t}}
4777:{\displaystyle \theta }
4764:for any given value of
4632:{\displaystyle \theta }
4560:{\displaystyle \theta }
2862:{\displaystyle \theta }
2351:{\displaystyle x_{t-2}}
2318:{\displaystyle x_{t-1}}
2229:Markov decision process
1849:{\displaystyle v_{nit}}
1600:{\displaystyle x_{t+1}}
848:{\displaystyle U_{nit}}
759:is taken over both the
666:and unobserved factors
566:{\displaystyle U_{nit}}
471:'s decision from among
5618:
5345:
5288:
5236:
5216:
5187:
5164:
5119:
5081:
5048:
5012:
4861:
4833:
4778:
4758:
4717:
4665:
4633:
4609:
4561:
4541:
4497:
4463:
4233:
4186:
3953:
3933:
3871:
3817:
3759:
3518:
3491:
3452:
3415:
3380:
3345:
3310:
3275:
3252:
2906:
2905:{\displaystyle \beta }
2886:
2863:
2843:
2801:
2777:
2627:
2591:
2495:
2440:
2404:
2384:
2352:
2319:
2286:
2255:
2217:
2178:
2153:
1890:
1870:
1850:
1814:
1680:
1628:
1601:
1568:
1506:
1486:
1461:
1208:
1178:
1158:
1134:
1074:
849:
816:
783:
782:{\displaystyle x_{nt}}
753:
714:
693:
660:
659:{\displaystyle x_{nt}}
630:
610:
590:
567:
529:
485:
465:
445:
444:{\displaystyle d_{nt}}
410:
409:{\displaystyle x_{n0}}
376:
375:{\displaystyle x_{nt}}
340:
219:
198:
68:
5619:
5346:
5289:
5237:
5217:
5188:
5165:
5120:
5082:
5049:
5013:
4834:
4813:
4779:
4759:
4718:
4666:
4664:{\displaystyle (x,d)}
4634:
4610:
4562:
4542:
4498:
4496:{\displaystyle x_{t}}
4464:
4234:
4187:
3954:
3934:
3872:
3818:
3760:
3519:
3517:{\displaystyle x_{t}}
3492:
3453:
3416:
3381:
3346:
3311:
3276:
3253:
2907:
2887:
2864:
2844:
2802:
2778:
2776:{\displaystyle x_{t}}
2667:Full-solution methods
2657:full-solution methods
2628:
2571:
2496:
2441:
2405:
2385:
2353:
2320:
2287:
2285:{\displaystyle x_{t}}
2256:
2254:{\displaystyle x_{t}}
2218:
2179:
2154:
1891:
1871:
1851:
1815:
1681:
1629:
1627:{\displaystyle x_{t}}
1602:
1569:
1507:
1487:
1462:
1209:
1179:
1159:
1135:
1075:
874:structural parameters
850:
817:
784:
754:
715:
694:
661:
631:
611:
591:
568:
530:
491:discrete alternatives
486:
466:
446:
411:
377:
341:
199:
171:
69:
43:structural parameters
5636:probabilities using
5368:
5314:
5306:Estimation with MPEC
5296:quasi-Newton methods
5257:
5226:
5197:
5177:
5129:
5091:
5058:
5025:
4795:
4768:
4727:
4683:
4643:
4623:
4575:
4551:
4507:
4480:
4249:
4199:
3966:
3943:
3889:
3827:
3772:
3537:
3501:
3462:
3429:
3390:
3355:
3320:
3285:
3265:
2923:
2896:
2873:
2853:
2811:
2791:
2760:
2705:Non-solution methods
2661:non-solution methods
2516:
2472:
2451:Type I extreme value
2417:
2394:
2374:
2358:or any prior state.
2329:
2296:
2269:
2238:
2188:
2166:
1903:
1880:
1860:
1827:
1693:
1641:
1611:
1578:
1516:
1496:
1474:
1221:
1188:
1168:
1148:
1095:
883:
826:
793:
763:
730:
704:
670:
640:
620:
600:
580:
544:
497:
475:
455:
425:
390:
356:
85:
76:maximization problem
58:
6080:Dynamic programming
5696:Rust, John (2008).
4476:if the state space
4474:contraction mapping
4241:functional equation
3458:are independent of
2468:For the case where
158:
24:dynamic programming
5957:Wolpin, Kenneth I.
5614:
5612:
5480:
5341:
5284:
5232:
5212:
5183:
5160:
5115:
5077:
5044:
5008:
4774:
4754:
4713:
4661:
4629:
4605:
4567:, and further the
4557:
4537:
4493:
4459:
4326:
4229:
4182:
4099:
3949:
3929:
3867:
3813:
3755:
3592:
3514:
3487:
3448:
3411:
3376:
3341:
3306:
3271:
3248:
3243:
3095:
2902:
2885:{\displaystyle RC}
2882:
2859:
2839:
2797:
2773:
2743:Madison, Wisconsin
2623:
2491:
2459:multinomial probit
2436:
2400:
2380:
2348:
2315:
2282:
2251:
2213:
2184:is taken over the
2174:
2149:
2024:
1896:and is written as
1886:
1866:
1846:
1810:
1745:
1676:
1624:
1597:
1564:
1502:
1482:
1457:
1273:
1204:
1174:
1154:
1130:
1070:
1068:
845:
812:
779:
749:
710:
689:
656:
626:
606:
586:
563:
525:
481:
461:
441:
406:
372:
336:
160:
120:
64:
38:models are based.
6039:978-0-444-88766-5
5953:Keane, Michael P.
5794:10.3982/ECTA12605
5477:
5469:
5455:
5401:
4323:
4315:
4301:
4180:
4070:
3952:{\displaystyle d}
3589:
3581:
3567:
3407:
3372:
3337:
3302:
3274:{\displaystyle d}
3236:
3222:
3209:
3186:
3172:
3159:
3085:
3026:
2800:{\displaystyle t}
2621:
2403:{\displaystyle t}
2383:{\displaystyle i}
2015:
1889:{\displaystyle t}
1869:{\displaystyle i}
1736:
1264:
1177:{\displaystyle t}
1157:{\displaystyle n}
713:{\displaystyle T}
629:{\displaystyle t}
609:{\displaystyle i}
589:{\displaystyle n}
484:{\displaystyle J}
464:{\displaystyle n}
418:initial condition
115:
67:{\displaystyle n}
6087:
6070:Economics models
6065:Choice modelling
6051:
6016:
5976:
5948:
5922:
5904:
5903:
5859:
5853:
5847:
5841:
5840:
5812:
5806:
5805:
5773:
5760:
5759:
5749:
5738:10.3982/ECTA7925
5732:(5): 2213–2230.
5722:Judd, Kenneth L.
5717:
5706:
5705:
5693:
5687:
5681:
5675:
5669:
5623:
5621:
5620:
5615:
5613:
5588:
5574:
5570:
5569:
5565:
5552:
5541:
5509:
5479:
5478:
5475:
5470:
5467:
5402:
5399:
5394:
5356:
5350:
5348:
5347:
5342:
5293:
5291:
5290:
5285:
5241:
5239:
5238:
5233:
5222:with respect to
5221:
5219:
5218:
5213:
5192:
5190:
5189:
5184:
5169:
5167:
5166:
5161:
5159:
5158:
5124:
5122:
5121:
5116:
5086:
5084:
5083:
5078:
5076:
5075:
5053:
5051:
5050:
5045:
5043:
5042:
5017:
5015:
5014:
5009:
4992:
4991:
4970:
4969:
4948:
4947:
4905:
4904:
4889:
4888:
4860:
4859:
4858:
4848:
4832:
4827:
4783:
4781:
4780:
4775:
4763:
4761:
4760:
4755:
4722:
4720:
4719:
4714:
4670:
4668:
4667:
4662:
4638:
4636:
4635:
4630:
4614:
4612:
4611:
4606:
4566:
4564:
4563:
4558:
4546:
4544:
4543:
4538:
4502:
4500:
4499:
4494:
4492:
4491:
4468:
4466:
4465:
4460:
4434:
4420:
4416:
4415:
4411:
4398:
4387:
4355:
4325:
4324:
4321:
4316:
4313:
4238:
4236:
4235:
4230:
4191:
4189:
4188:
4183:
4181:
4179:
4166:
4128:
4098:
4082:
4068:
3997:
3958:
3956:
3955:
3950:
3938:
3936:
3935:
3930:
3919:
3908:
3876:
3874:
3873:
3868:
3857:
3846:
3822:
3820:
3819:
3814:
3791:
3764:
3762:
3761:
3756:
3754:
3750:
3728:
3702:
3691:
3665:
3654:
3634:
3633:
3591:
3590:
3587:
3582:
3579:
3529:Bellman equation
3523:
3521:
3520:
3515:
3513:
3512:
3496:
3494:
3493:
3488:
3486:
3485:
3457:
3455:
3454:
3449:
3447:
3446:
3420:
3418:
3417:
3412:
3410:
3409:
3408:
3405:
3385:
3383:
3382:
3377:
3375:
3374:
3373:
3370:
3350:
3348:
3347:
3342:
3340:
3339:
3338:
3335:
3315:
3313:
3312:
3307:
3305:
3304:
3303:
3300:
3280:
3278:
3277:
3272:
3257:
3255:
3254:
3249:
3247:
3246:
3237:
3234:
3224:
3223:
3220:
3212:
3211:
3210:
3207:
3187:
3184:
3174:
3173:
3170:
3162:
3161:
3160:
3157:
3118:
3117:
3099:
3098:
3093:
3088:
3087:
3086:
3083:
3034:
3029:
3028:
3027:
3024:
2999:
2998:
2954:
2953:
2941:
2940:
2911:
2909:
2908:
2903:
2891:
2889:
2888:
2883:
2868:
2866:
2865:
2860:
2848:
2846:
2845:
2840:
2829:
2828:
2806:
2804:
2803:
2798:
2782:
2780:
2779:
2774:
2772:
2771:
2735:optimal stopping
2632:
2630:
2629:
2624:
2622:
2620:
2616:
2615:
2590:
2585:
2569:
2565:
2564:
2539:
2534:
2533:
2500:
2498:
2497:
2492:
2490:
2489:
2445:
2443:
2442:
2437:
2435:
2434:
2409:
2407:
2406:
2401:
2389:
2387:
2386:
2381:
2357:
2355:
2354:
2349:
2347:
2346:
2324:
2322:
2321:
2316:
2314:
2313:
2291:
2289:
2288:
2283:
2281:
2280:
2260:
2258:
2257:
2252:
2250:
2249:
2222:
2220:
2219:
2214:
2212:
2211:
2183:
2181:
2180:
2175:
2173:
2158:
2156:
2155:
2150:
2145:
2144:
2132:
2131:
2106:
2102:
2101:
2100:
2073:
2072:
2051:
2050:
2023:
2014:
2009:
2008:
2007:
2006:
1980:
1976:
1975:
1959:
1958:
1937:
1936:
1921:
1920:
1895:
1893:
1892:
1887:
1875:
1873:
1872:
1867:
1855:
1853:
1852:
1847:
1845:
1844:
1819:
1817:
1816:
1811:
1809:
1805:
1804:
1803:
1782:
1781:
1766:
1765:
1744:
1735:
1724:
1723:
1708:
1707:
1685:
1683:
1682:
1677:
1672:
1671:
1656:
1655:
1633:
1631:
1630:
1625:
1623:
1622:
1606:
1604:
1603:
1598:
1596:
1595:
1573:
1571:
1570:
1565:
1563:
1559:
1558:
1557:
1545:
1544:
1511:
1509:
1508:
1503:
1491:
1489:
1488:
1483:
1481:
1466:
1464:
1463:
1458:
1456:
1452:
1451:
1447:
1446:
1445:
1433:
1432:
1402:
1401:
1380:
1379:
1361:
1360:
1359:
1358:
1332:
1331:
1310:
1309:
1294:
1293:
1272:
1263:
1252:
1251:
1236:
1235:
1213:
1211:
1210:
1205:
1203:
1202:
1183:
1181:
1180:
1175:
1163:
1161:
1160:
1155:
1139:
1137:
1136:
1131:
1126:
1125:
1110:
1109:
1086:Bellman equation
1079:
1077:
1076:
1071:
1069:
1065:
1064:
1048:
1040:
1038:
1037:
1028:
1027:
1014:
1006:
1005:
1001:
1000:
984:
976:
974:
973:
957:
949:
947:
943:
942:
941:
923:
922:
905:
904:
854:
852:
851:
846:
844:
843:
821:
819:
818:
813:
811:
810:
788:
786:
785:
780:
778:
777:
758:
756:
755:
750:
748:
737:
726:The expectation
719:
717:
716:
711:
698:
696:
695:
690:
688:
687:
665:
663:
662:
657:
655:
654:
635:
633:
632:
627:
615:
613:
612:
607:
595:
593:
592:
587:
572:
570:
569:
564:
562:
561:
534:
532:
531:
526:
524:
520:
490:
488:
487:
482:
470:
468:
467:
462:
450:
448:
447:
442:
440:
439:
415:
413:
412:
407:
405:
404:
381:
379:
378:
373:
371:
370:
345:
343:
342:
337:
332:
328:
327:
323:
322:
321:
303:
302:
285:
284:
269:
265:
258:
257:
240:
239:
232:
218:
213:
197:
192:
185:
184:
165:
159:
157:
152:
141:
137:
136:
111:
107:
106:
73:
71:
70:
65:
19:, also known as
6095:
6094:
6090:
6089:
6088:
6086:
6085:
6084:
6055:
6054:
6040:
6019:
5997:10.2307/1911259
5991:(5): 999–1033.
5979:
5951:
5920:
5915:
5912:
5910:Further reading
5907:
5876:10.2307/2297981
5861:
5860:
5856:
5848:
5844:
5829:10.2307/2298122
5814:
5813:
5809:
5775:
5774:
5763:
5719:
5718:
5709:
5695:
5694:
5690:
5682:
5678:
5670:
5666:
5662:
5650:
5642:value functions
5633:
5611:
5610:
5581:
5545:
5534:
5502:
5454:
5450:
5443:
5439:
5403:
5395:
5393:
5378:
5366:
5365:
5352:
5312:
5311:
5308:
5255:
5254:
5224:
5223:
5195:
5194:
5175:
5174:
5150:
5127:
5126:
5089:
5088:
5061:
5056:
5055:
5028:
5023:
5022:
4974:
4952:
4936:
4893:
4877:
4850:
4793:
4792:
4766:
4765:
4725:
4724:
4681:
4680:
4677:
4641:
4640:
4621:
4620:
4617:smooth function
4573:
4572:
4549:
4548:
4505:
4504:
4483:
4478:
4477:
4427:
4391:
4380:
4348:
4300:
4296:
4289:
4285:
4247:
4246:
4197:
4196:
4159:
4121:
4075:
4069:
3998:
3964:
3963:
3941:
3940:
3912:
3901:
3887:
3886:
3850:
3839:
3825:
3824:
3784:
3770:
3769:
3721:
3695:
3684:
3658:
3647:
3625:
3597:
3593:
3535:
3534:
3504:
3499:
3498:
3497:conditional on
3465:
3460:
3459:
3432:
3427:
3426:
3393:
3388:
3387:
3358:
3353:
3352:
3323:
3318:
3317:
3288:
3283:
3282:
3263:
3262:
3242:
3241:
3216:
3195:
3192:
3191:
3166:
3145:
3138:
3109:
3094:
3092:
3071:
3035:
3033:
3012:
2990:
2974:
2945:
2932:
2921:
2920:
2914:discount factor
2894:
2893:
2871:
2870:
2851:
2850:
2820:
2809:
2808:
2789:
2788:
2763:
2758:
2757:
2724:
2719:
2707:
2669:
2639:
2601:
2570:
2550:
2540:
2519:
2514:
2513:
2475:
2470:
2469:
2420:
2415:
2414:
2392:
2391:
2372:
2371:
2364:
2332:
2327:
2326:
2299:
2294:
2293:
2272:
2267:
2266:
2241:
2236:
2235:
2191:
2186:
2185:
2164:
2163:
2136:
2117:
2080:
2055:
2030:
2029:
2025:
1992:
1987:
1964:
1960:
1944:
1925:
1906:
1901:
1900:
1878:
1877:
1858:
1857:
1830:
1825:
1824:
1789:
1770:
1751:
1750:
1746:
1712:
1696:
1691:
1690:
1660:
1644:
1639:
1638:
1614:
1609:
1608:
1607:conditional on
1581:
1576:
1575:
1549:
1530:
1529:
1525:
1514:
1513:
1494:
1493:
1472:
1471:
1437:
1418:
1417:
1413:
1384:
1362:
1344:
1339:
1317:
1298:
1279:
1278:
1274:
1240:
1224:
1219:
1218:
1191:
1186:
1185:
1166:
1165:
1146:
1145:
1114:
1098:
1093:
1092:
1067:
1066:
1050:
1047:
1039:
1029:
1016:
1013:
1003:
1002:
986:
983:
975:
959:
956:
948:
927:
911:
910:
906:
890:
881:
880:
862:
829:
824:
823:
796:
791:
790:
766:
761:
760:
738:
728:
727:
702:
701:
673:
668:
667:
643:
638:
637:
618:
617:
598:
597:
578:
577:
547:
542:
541:
537:discount factor
510:
506:
495:
494:
473:
472:
453:
452:
428:
423:
422:
393:
388:
387:
384:state variables
359:
354:
353:
307:
291:
290:
286:
270:
246:
245:
241:
225:
220:
176:
170:
166:
125:
121:
95:
91:
83:
82:
56:
55:
52:
36:discrete choice
12:
11:
5:
6093:
6091:
6083:
6082:
6077:
6072:
6067:
6057:
6056:
6053:
6052:
6038:
6017:
5977:
5949:
5911:
5908:
5906:
5905:
5854:
5842:
5823:(3): 497–529.
5807:
5788:(1): 365–370.
5761:
5707:
5688:
5676:
5663:
5661:
5658:
5657:
5656:
5649:
5646:
5632:
5629:
5625:
5624:
5609:
5606:
5603:
5600:
5597:
5594:
5591:
5587:
5584:
5580:
5577:
5573:
5568:
5564:
5561:
5558:
5555:
5551:
5548:
5544:
5540:
5537:
5533:
5530:
5527:
5524:
5521:
5518:
5515:
5512:
5508:
5505:
5501:
5498:
5495:
5492:
5489:
5486:
5483:
5473:
5465:
5462:
5458:
5453:
5449:
5446:
5442:
5438:
5435:
5432:
5429:
5426:
5423:
5420:
5417:
5414:
5411:
5408:
5404:
5397:
5396:
5392:
5389:
5386:
5383:
5379:
5377:
5374:
5373:
5340:
5337:
5334:
5331:
5328:
5325:
5322:
5319:
5307:
5304:
5298:, such as the
5283:
5280:
5277:
5274:
5271:
5268:
5265:
5262:
5231:
5211:
5208:
5205:
5202:
5182:
5157:
5153:
5149:
5146:
5143:
5140:
5137:
5134:
5114:
5111:
5108:
5105:
5102:
5099:
5096:
5074:
5071:
5068:
5064:
5041:
5038:
5035:
5031:
5019:
5018:
5007:
5004:
5001:
4998:
4995:
4990:
4987:
4984:
4981:
4977:
4973:
4968:
4965:
4962:
4959:
4955:
4951:
4946:
4943:
4939:
4935:
4932:
4929:
4926:
4923:
4920:
4917:
4914:
4911:
4908:
4903:
4900:
4896:
4892:
4887:
4884:
4880:
4876:
4873:
4870:
4867:
4864:
4857:
4853:
4847:
4844:
4841:
4837:
4831:
4826:
4823:
4820:
4816:
4812:
4809:
4806:
4803:
4800:
4786:log-likelihood
4773:
4753:
4750:
4747:
4744:
4741:
4738:
4735:
4732:
4712:
4709:
4706:
4703:
4700:
4697:
4694:
4691:
4688:
4676:
4673:
4660:
4657:
4654:
4651:
4648:
4628:
4604:
4601:
4598:
4595:
4592:
4589:
4586:
4583:
4580:
4556:
4536:
4533:
4530:
4527:
4524:
4521:
4518:
4515:
4512:
4490:
4486:
4470:
4469:
4458:
4455:
4452:
4449:
4446:
4443:
4440:
4437:
4433:
4430:
4426:
4423:
4419:
4414:
4410:
4407:
4404:
4401:
4397:
4394:
4390:
4386:
4383:
4379:
4376:
4373:
4370:
4367:
4364:
4361:
4358:
4354:
4351:
4347:
4344:
4341:
4338:
4335:
4332:
4329:
4319:
4311:
4308:
4304:
4299:
4295:
4292:
4288:
4284:
4281:
4278:
4275:
4272:
4269:
4266:
4263:
4260:
4257:
4254:
4228:
4225:
4222:
4219:
4216:
4213:
4210:
4207:
4204:
4193:
4192:
4178:
4175:
4172:
4169:
4165:
4162:
4158:
4155:
4152:
4149:
4146:
4143:
4140:
4137:
4134:
4131:
4127:
4124:
4120:
4117:
4114:
4111:
4108:
4105:
4102:
4097:
4094:
4091:
4088:
4085:
4081:
4078:
4073:
4067:
4064:
4061:
4058:
4055:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4028:
4025:
4022:
4019:
4016:
4013:
4010:
4007:
4004:
4001:
3995:
3992:
3989:
3986:
3983:
3980:
3977:
3974:
3971:
3948:
3928:
3925:
3922:
3918:
3915:
3911:
3907:
3904:
3900:
3897:
3894:
3866:
3863:
3860:
3856:
3853:
3849:
3845:
3842:
3838:
3835:
3832:
3812:
3809:
3806:
3803:
3800:
3797:
3794:
3790:
3787:
3783:
3780:
3777:
3766:
3765:
3753:
3749:
3746:
3743:
3740:
3737:
3734:
3731:
3727:
3724:
3720:
3717:
3714:
3711:
3708:
3705:
3701:
3698:
3694:
3690:
3687:
3683:
3680:
3677:
3674:
3671:
3668:
3664:
3661:
3657:
3653:
3650:
3646:
3643:
3640:
3637:
3632:
3628:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3596:
3585:
3577:
3574:
3570:
3566:
3563:
3560:
3557:
3554:
3551:
3548:
3545:
3542:
3511:
3507:
3484:
3481:
3478:
3475:
3472:
3468:
3445:
3442:
3439:
3435:
3403:
3400:
3396:
3368:
3365:
3361:
3333:
3330:
3326:
3298:
3295:
3291:
3270:
3259:
3258:
3245:
3240:
3232:
3229:
3217:
3215:
3205:
3202:
3198:
3194:
3193:
3190:
3182:
3179:
3167:
3165:
3155:
3152:
3148:
3144:
3143:
3141:
3136:
3133:
3130:
3127:
3124:
3121:
3116:
3112:
3108:
3105:
3102:
3097:
3091:
3081:
3078:
3074:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3036:
3032:
3022:
3019:
3015:
3011:
3008:
3005:
3002:
2997:
2993:
2989:
2986:
2983:
2980:
2979:
2977:
2972:
2969:
2966:
2963:
2960:
2957:
2952:
2948:
2944:
2939:
2935:
2931:
2928:
2901:
2881:
2878:
2858:
2838:
2835:
2832:
2827:
2823:
2819:
2816:
2796:
2770:
2766:
2723:
2720:
2718:
2715:
2706:
2703:
2692:Artelys Knitro
2668:
2665:
2638:
2635:
2634:
2633:
2619:
2614:
2611:
2608:
2604:
2600:
2597:
2594:
2589:
2584:
2581:
2578:
2574:
2568:
2563:
2560:
2557:
2553:
2549:
2546:
2543:
2537:
2532:
2529:
2526:
2522:
2488:
2485:
2482:
2478:
2433:
2430:
2427:
2423:
2399:
2379:
2363:
2360:
2345:
2342:
2339:
2335:
2312:
2309:
2306:
2302:
2279:
2275:
2248:
2244:
2232:
2231:
2210:
2207:
2204:
2201:
2198:
2194:
2172:
2160:
2159:
2148:
2143:
2139:
2135:
2130:
2127:
2124:
2120:
2116:
2113:
2110:
2105:
2099:
2096:
2093:
2090:
2087:
2083:
2079:
2076:
2071:
2068:
2065:
2062:
2058:
2054:
2049:
2046:
2043:
2040:
2037:
2033:
2028:
2022:
2018:
2013:
2005:
2002:
1999:
1995:
1990:
1986:
1983:
1979:
1974:
1971:
1967:
1963:
1957:
1954:
1951:
1947:
1943:
1940:
1935:
1932:
1928:
1924:
1919:
1916:
1913:
1909:
1885:
1865:
1843:
1840:
1837:
1833:
1821:
1820:
1808:
1802:
1799:
1796:
1792:
1788:
1785:
1780:
1777:
1773:
1769:
1764:
1761:
1758:
1754:
1749:
1743:
1739:
1734:
1730:
1727:
1722:
1719:
1715:
1711:
1706:
1703:
1699:
1675:
1670:
1667:
1663:
1659:
1654:
1651:
1647:
1621:
1617:
1594:
1591:
1588:
1584:
1562:
1556:
1552:
1548:
1543:
1540:
1537:
1533:
1528:
1524:
1521:
1512:'s, and where
1501:
1480:
1468:
1467:
1455:
1450:
1444:
1440:
1436:
1431:
1428:
1425:
1421:
1416:
1412:
1409:
1405:
1400:
1397:
1394:
1391:
1387:
1383:
1378:
1375:
1372:
1369:
1365:
1357:
1354:
1351:
1347:
1342:
1338:
1335:
1330:
1327:
1324:
1320:
1316:
1313:
1308:
1305:
1301:
1297:
1292:
1289:
1286:
1282:
1277:
1271:
1267:
1262:
1258:
1255:
1250:
1247:
1243:
1239:
1234:
1231:
1227:
1201:
1198:
1194:
1173:
1153:
1129:
1124:
1121:
1117:
1113:
1108:
1105:
1101:
1089:
1088:
1081:
1080:
1063:
1060:
1057:
1053:
1049:
1045:
1041:
1036:
1032:
1026:
1023:
1019:
1015:
1011:
1007:
1004:
999:
996:
993:
989:
985:
981:
977:
972:
969:
966:
962:
958:
954:
950:
946:
940:
937:
934:
930:
926:
921:
918:
914:
909:
903:
900:
897:
893:
889:
888:
870:
869:
861:
858:
857:
856:
842:
839:
836:
832:
809:
806:
803:
799:
776:
773:
769:
747:
744:
741:
736:
724:
709:
699:
686:
683:
680:
676:
653:
650:
646:
625:
605:
585:
560:
557:
554:
550:
539:
523:
519:
516:
513:
509:
505:
502:
492:
480:
460:
438:
435:
431:
420:
403:
400:
396:
369:
366:
362:
347:
346:
335:
331:
326:
320:
317:
314:
310:
306:
301:
298:
294:
289:
283:
280:
277:
273:
268:
264:
261:
256:
253:
249:
244:
238:
235:
231:
228:
223:
217:
212:
209:
206:
202:
196:
191:
188:
183:
179:
174:
169:
164:
156:
151:
148:
145:
140:
135:
132:
128:
124:
118:
114:
110:
105:
102:
98:
94:
90:
63:
51:
48:
32:utility theory
13:
10:
9:
6:
4:
3:
2:
6092:
6081:
6078:
6076:
6073:
6071:
6068:
6066:
6063:
6062:
6060:
6049:
6045:
6041:
6035:
6031:
6027:
6023:
6018:
6014:
6010:
6006:
6002:
5998:
5994:
5990:
5986:
5982:
5978:
5974:
5970:
5966:
5962:
5958:
5954:
5950:
5946:
5942:
5938:
5934:
5930:
5926:
5919:
5914:
5913:
5909:
5901:
5897:
5893:
5889:
5885:
5881:
5877:
5873:
5869:
5865:
5858:
5855:
5851:
5846:
5843:
5838:
5834:
5830:
5826:
5822:
5818:
5811:
5808:
5803:
5799:
5795:
5791:
5787:
5783:
5779:
5772:
5770:
5768:
5766:
5762:
5757:
5753:
5748:
5743:
5739:
5735:
5731:
5727:
5723:
5720:Su, Che-Lin;
5716:
5714:
5712:
5708:
5703:
5699:
5692:
5689:
5685:
5680:
5677:
5673:
5668:
5665:
5659:
5655:
5652:
5651:
5647:
5645:
5643:
5639:
5630:
5628:
5604:
5601:
5598:
5595:
5592:
5589:
5585:
5582:
5575:
5571:
5566:
5556:
5553:
5549:
5546:
5542:
5538:
5535:
5528:
5525:
5522:
5519:
5513:
5510:
5506:
5503:
5499:
5496:
5490:
5484:
5481:
5471:
5463:
5460:
5456:
5451:
5447:
5444:
5440:
5436:
5433:
5427:
5424:
5421:
5418:
5415:
5409:
5406:
5387:
5381:
5364:
5363:
5362:
5360:
5355:
5335:
5332:
5329:
5326:
5323:
5317:
5305:
5303:
5301:
5297:
5278:
5275:
5272:
5269:
5266:
5260:
5253:to calculate
5252:
5247:
5245:
5229:
5206:
5200:
5180:
5171:
5155:
5151:
5147:
5144:
5141:
5138:
5135:
5132:
5112:
5109:
5106:
5103:
5100:
5097:
5094:
5072:
5069:
5066:
5062:
5039:
5036:
5033:
5029:
5005:
4996:
4993:
4988:
4985:
4982:
4979:
4975:
4971:
4966:
4963:
4960:
4957:
4953:
4949:
4944:
4941:
4937:
4930:
4924:
4921:
4918:
4909:
4906:
4901:
4898:
4894:
4890:
4885:
4882:
4878:
4871:
4865:
4862:
4855:
4851:
4845:
4842:
4839:
4835:
4829:
4824:
4821:
4818:
4814:
4810:
4804:
4798:
4791:
4790:
4789:
4787:
4771:
4748:
4745:
4742:
4739:
4736:
4730:
4707:
4704:
4701:
4698:
4695:
4689:
4686:
4674:
4672:
4655:
4652:
4649:
4626:
4618:
4599:
4596:
4593:
4590:
4587:
4581:
4578:
4570:
4554:
4531:
4528:
4525:
4522:
4519:
4513:
4510:
4488:
4484:
4475:
4456:
4450:
4447:
4444:
4441:
4438:
4435:
4431:
4428:
4421:
4417:
4412:
4402:
4399:
4395:
4392:
4388:
4384:
4381:
4374:
4371:
4368:
4365:
4359:
4356:
4352:
4349:
4345:
4342:
4336:
4330:
4327:
4317:
4309:
4306:
4302:
4297:
4293:
4290:
4286:
4282:
4279:
4273:
4270:
4267:
4264:
4261:
4255:
4252:
4245:
4244:
4243:
4242:
4223:
4220:
4217:
4214:
4211:
4205:
4202:
4170:
4167:
4163:
4160:
4156:
4153:
4147:
4144:
4141:
4138:
4132:
4129:
4125:
4122:
4118:
4115:
4109:
4103:
4100:
4092:
4086:
4083:
4079:
4076:
4071:
4059:
4056:
4053:
4050:
4047:
4041:
4038:
4035:
4032:
4026:
4023:
4020:
4017:
4014:
4008:
4002:
3999:
3993:
3987:
3984:
3981:
3978:
3975:
3969:
3962:
3961:
3960:
3946:
3923:
3920:
3916:
3913:
3909:
3905:
3902:
3898:
3892:
3883:
3881:
3861:
3858:
3854:
3851:
3847:
3843:
3840:
3836:
3830:
3807:
3804:
3801:
3798:
3795:
3792:
3788:
3785:
3781:
3775:
3751:
3744:
3741:
3738:
3735:
3732:
3729:
3725:
3722:
3718:
3712:
3706:
3703:
3699:
3696:
3692:
3688:
3685:
3681:
3675:
3669:
3666:
3662:
3659:
3655:
3651:
3648:
3641:
3638:
3635:
3630:
3626:
3622:
3616:
3613:
3610:
3607:
3604:
3598:
3594:
3583:
3575:
3572:
3564:
3558:
3555:
3552:
3549:
3546:
3540:
3533:
3532:
3531:
3530:
3525:
3509:
3505:
3482:
3479:
3476:
3473:
3470:
3466:
3443:
3440:
3437:
3433:
3424:
3401:
3398:
3394:
3366:
3363:
3359:
3331:
3328:
3324:
3296:
3293:
3289:
3268:
3238:
3230:
3227:
3213:
3203:
3200:
3196:
3188:
3180:
3177:
3163:
3153:
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2038:
2035:
2031:
2026:
2020:
2003:
2000:
1997:
1993:
1988:
1984:
1981:
1977:
1972:
1969:
1965:
1961:
1955:
1952:
1949:
1945:
1941:
1933:
1930:
1926:
1917:
1914:
1911:
1907:
1899:
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1897:
1883:
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1800:
1797:
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1269:
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1241:
1232:
1229:
1225:
1217:
1216:
1215:
1214:is revealed:
1199:
1196:
1192:
1171:
1151:
1143:
1122:
1119:
1115:
1106:
1103:
1099:
1087:
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1082:
1061:
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1055:
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1043:
1034:
1030:
1024:
1021:
1017:
1009:
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994:
991:
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901:
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875:
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866:
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840:
837:
834:
830:
807:
804:
801:
797:
774:
771:
767:
745:
742:
739:
725:
723:
707:
700:
684:
681:
678:
674:
651:
648:
644:
623:
603:
583:
576:
558:
555:
552:
548:
540:
538:
521:
517:
514:
511:
507:
503:
500:
493:
478:
458:
436:
433:
429:
421:
419:
401:
398:
394:
385:
367:
364:
360:
352:
351:
350:
333:
329:
324:
318:
315:
312:
308:
304:
299:
296:
292:
287:
281:
278:
275:
271:
266:
262:
259:
254:
251:
247:
242:
236:
233:
229:
226:
221:
215:
210:
207:
204:
200:
194:
189:
186:
177:
172:
167:
154:
149:
146:
143:
138:
133:
130:
126:
122:
112:
108:
103:
100:
96:
92:
88:
81:
80:
79:
77:
61:
49:
47:
44:
39:
37:
33:
29:
28:present value
25:
22:
18:
6021:
5988:
5985:Econometrica
5984:
5964:
5960:
5928:
5924:
5867:
5863:
5857:
5845:
5820:
5816:
5810:
5785:
5782:Econometrica
5781:
5729:
5726:Econometrica
5725:
5701:
5691:
5679:
5667:
5634:
5626:
5353:
5309:
5248:
5243:
5172:
5020:
4678:
4471:
4194:
3959:is given by
3884:
3767:
3526:
3260:
2755:
2745:. For every
2732:
2725:
2712:
2708:
2700:
2696:
2680:Kenneth Judd
2677:
2670:
2660:
2656:
2654:
2643:
2640:
2467:
2412:
2367:
2365:
2263:Markov chain
2233:
2161:
1822:
1636:
1492:is over the
1469:
1184:just before
1141:
1090:
871:
863:
722:time horizon
575:flow utility
416:the agent's
348:
53:
40:
20:
16:
15:
5967:(1): 1–22.
5747:10419/59626
5702:Unpublished
3425:, and that
2783:denote the
2728:Rust (1987)
2463:mixed logit
2234:The states
451:represents
46:increase.)
34:upon which
6059:Categories
5981:Rust, John
5660:References
5638:simulation
5400:subject to
4615:is also a
4571:holds, so
3880:stationary
2637:Estimation
2390:in period
1164:in period
1091:Define by
616:in period
6048:1573-4412
6005:0012-9682
5945:0304-4076
5884:0034-6527
5802:0012-9682
5756:1468-0262
5684:Rust 1987
5605:θ
5590:∣
5557:θ
5523:β
5514:θ
5485:
5457:∑
5448:
5437:∫
5428:θ
5388:θ
5361:problem:
5336:θ
5327:∣
5279:θ
5270:∣
5230:θ
5207:θ
5181:θ
5170:periods.
5145:…
5107:…
4997:θ
4986:−
4964:−
4950:∣
4925:
4910:θ
4891:∣
4866:
4836:∑
4815:∑
4805:θ
4772:θ
4749:θ
4740:∣
4708:θ
4639:for each
4627:θ
4600:θ
4555:θ
4532:θ
4451:θ
4436:∣
4403:θ
4369:β
4360:θ
4331:
4303:∑
4294:
4283:∫
4274:θ
4224:θ
4171:θ
4142:β
4133:θ
4104:
4084:∈
4072:∑
4060:θ
4036:β
4027:θ
4003:
3988:θ
3979:∣
3924:θ
3910:∣
3903:ξ
3862:θ
3848:∣
3841:ξ
3808:θ
3793:∣
3745:θ
3730:∣
3707:θ
3693:∣
3686:ξ
3670:θ
3660:ξ
3639:∬
3627:ξ
3617:θ
3559:θ
3553:ξ
3483:∙
3474:−
3467:ξ
3444:∙
3434:ξ
3395:ξ
3360:ξ
3325:ξ
3290:ξ
3197:ξ
3147:ξ
3129:θ
3073:ξ
3063:θ
3048:−
3039:−
3014:ξ
3004:θ
2982:−
2965:θ
2947:ξ
2900:β
2857:θ
2834:θ
2673:John Rust
2596:
2573:∑
2545:
2505:from the
2477:ε
2422:ε
2341:−
2308:−
2261:follow a
2193:ε
2134:∣
2082:ε
1989:∫
1985:β
1791:ε
1547:∣
1500:ε
1435:∣
1341:∫
1337:β
1319:ε
1193:ε
1052:ε
1031:α
988:ε
929:ε
798:ε
743:⋅
675:ε
504:∈
501:β
309:ε
234:−
222:β
201:∑
182:′
173:∑
5900:55199895
5648:See also
5586:′
5550:′
5539:′
5507:′
5246:(NFXP).
4547:for any
4432:′
4396:′
4385:′
4353:′
4164:′
4126:′
4080:′
3917:′
3906:′
3855:′
3844:′
3789:′
3726:′
3700:′
3689:′
3663:′
3652:′
2785:odometer
2717:Examples
2325:and not
1876:at time
230:′
6013:1911259
5892:2297981
5837:2298122
5476:replace
4322:replace
3588:replace
3406:replace
3336:replace
3235:replace
3208:replace
3084:replace
1142:ex ante
789:'s and
720:is the
573:is the
535:is the
386:, with
349:where
6046:
6036:
6011:
6003:
5943:
5898:
5890:
5882:
5835:
5800:
5754:
5021:where
4784:. The
4195:where
3768:where
3261:where
2751:engine
1823:where
822:'s in
54:Agent
6009:JSTOR
5921:(PDF)
5896:S2CID
5888:JSTOR
5833:JSTOR
2461:, or
6044:ISSN
6034:ISBN
6001:ISSN
5941:ISSN
5880:ISSN
5798:ISSN
5752:ISSN
5468:keep
5294:and
5054:and
4314:keep
3823:and
3580:keep
3386:and
3371:keep
3316:and
3301:keep
3185:keep
3158:keep
3025:keep
2912:the
2756:Let
2659:and
2648:and
1140:the
382:are
6026:doi
5993:doi
5969:doi
5933:doi
5929:156
5872:doi
5825:doi
5790:doi
5742:hdl
5734:doi
5482:exp
5445:log
5376:max
4922:log
4863:log
4619:of
4328:exp
4291:log
4101:exp
4000:exp
3569:max
2747:bus
2741:in
2652:.
2593:exp
2542:exp
2503:iid
2448:iid
2017:max
1738:max
1266:max
117:max
74:'s
6061::
6042:.
6032:.
6007:.
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5989:55
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5868:61
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5786:84
5784:.
5780:.
5764:^
5750:.
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1787:+
1784:)
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