# The Dynamic Multinomial Choice Model

In this section we present an example of Bayesian inference for dynamic discrete choice models using the Geweke-Keane method of replacing the future component of the value function with a flexible polynomial function. The discussion is based on a model that is very similar to ones analyzed by Keane and Wolpin (1994, 1997).

In the model we consider, i = 1,…, N agents choose among j = 1,…, 4 mutually exclusive alternatives in each of t = 1,…, 40 periods. One can think of the first two alternatives as work in one of two occupations, the third as attending school and the fourth alternative as remaining at home. One component of the current period payoff in each of the two occupational alternatives is the associated wage, wijt (j = 1, 2). The log-wage equation is:

ln wijt = P0j + e1jXi1t + ^2jXi2t + Рз/Sit + P4jX% + £ijt (j = 1, 2)

= Y’jtPj + Eijt (j = 1, 2), (22.6)

where Y% is the obvious vector, p% = (P0j,…, p4j)’, Sit is the periods of school completed, is the periods of experience in each occupation j, and the

е% are serially independent productivity shocks, with (ei1t, ei2t)’ ~ N(0, Xe). Each occupational alternative also has a stochastic nonpecuniary payoff, v%, so the complete current period payoffs are

Uijt = Wijt + Vijt (j = 1, 2). (22.7)

The schooling payoffs include tuition costs. Agents begin with a tenth-grade education, and may complete two additional grades without cost. We assume there is a fixed undergraduate tuition rate a1 for attending grades 13 through 16, and a fixed graduate tuition rate a2 for each year of schooling beyond 16. We assume a "return to school" cost a3 that agents face if they did not choose school the previous period. Finally, school has a nonstochastic, nonpecuniary benefit a 0 and a mean zero stochastic nonpecuniary payoff vi3t. Thus we have

Ui3t = a0 + a1x(12 < Sit < 15) + a2%(SU > 16) + a3%(di, t-1 Ф 3) + vRi = Arta + vai,

(22.8)

where % is an indicator function that takes value one if the stated condition is true and is zero otherwise, Ait is a vector of zeros and ones corresponding to the values of the indicator functions, a = (a0,…, a3)’, dit Є {1, 2, 3, 4} denotes the choice of i at t. Lastly, we assume that option four, home, has both a nonstochastic nonpecuniary payoff ф and a stochastic nonpecuniary payoff vijt, so

Ui4t = Ф + Vi4t. (22.9)

We will set uijt = uijt + Vijt, (j = 1,…, 4). The nonpecuniary payoffs (vi/t)/=14 are assumed serially independent.

The state of the agent at the time of each decision is

Iit = {(Xijt)j=1,2, Sit, t, di, t-1, (ei/t)j=1,2, (vijt)j=1, .. .4}. (22.10)

We assume di0 = 3. The laws of motion for experience in the occupational alternatives and school are: Xim = Хщ + %(dit = j), j = 1, 2, S^ = Sit + %{dit = 3). The number of "home" choices is excluded from the state-space as it is linearly dependent on the level of education, the period, and experience in the two occupations.

It is convenient to have notation for the elements of the state vector whose value in period t + 1 depends nontrivially on their value in period t or on the current choice. The reason, as we note below, is that these elements are the natural arguments of the future component. We define

= {(Xijt)j=1,2, Sit, t, di, t-1}.

The value of each alternative is the sum of its current period payoff, the stochastic nonpecuniary payoff and the future component:

Vijt(Iit) = ЩДи) + vijt + F(Xi1t + X( j = 1), Xi2t + X( j = 2),

Sit + x( j = 3), t + 1, x( j = 3)) (j = 1,… 4) (t = 1,…, 40)

= Uijt(Iit) + vijt + F(I*t, j) (22.11)

The function F represents agents’ forecasts about the effects of their current state and choice on their future payoff stream. The function is fixed across alternatives, implying that forecasts vary across alternatives only because different choices lead to different future states, and it depends only on the choice and the state variables in I*+1.4

Since choices depend only on relative alternative values, rather than their levels, we define for j Є {1, 2, 3}:

Zijt = Vijt – Vi4t

= Ujt + Vj + F(I*, j) – йш – Vi4t – F(I*, 4)

= Bijt + f (I*, j) + Щ t, (22.12)

where = йщ – йі41, {nijt}j=i,2,3 = (vijt – ~ N(0, Xn) and f(I*, j) = F(I*, j) –

F(I*, 4). Importantly, after differencing, the value ф of the home payoff is subsumed in f the relative future component. Clearly, if an alternative’s future component has an intercept (as each of ours does) then it and the period return to home cannot be separately identified.

The value function differences Zit are latent variables unobserved by the econometrician. The econometrician only observes the agents’ choices {dit} for t =

1,. .., 40, and, in the periods when the agent works, the wage for the chosen alternative. Thus, payoffs are never completely observed, both because wages are censored and because the nonpecuniary components of the payoffs (vijt) are never observed. Nevertheless, given observed choices and partially observed wages, along with the functional form assumptions about the payoff functions, it is possible to learn both about the future component F() and the structural parameters of the payoff functions without making strong assumptions about how agents form expectations. Rather, we simply assume that the future component lies along a fourth-order polynomial in the state variables. After differencing to obtain {f(I*, j )}/=1/2,3, the polynomial contained 53 terms of order three and lower (see Appendix A). We express the future component as

f(I*, j) = j (j = 1, 2, 3), (22.13)

where yijt is a vector of functions of state-variables that appear in the equation for fI*, j) and n is a vector of coefficients common to each choice. Cross-equation restrictions of this type are a consequence of using the same future component function F for each alternative and reflect the consistency restrictions discussed earlier.

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