Model with Habit Persistence

The key feature of the models with structural state dependence is that occupancy of a state in another period determines current choices, controlling for the effect of unmeasured heterogeneity. The model considered in this section ignores this form of dependence but permits relative utility evaluations in other periods (Y(i, f'), t ^ t') to determine current choices. Models with habit formation have been considered by Pollak (1970) and are implicit in Coleman's latent Markov model (1964). The model considered here is the discrete data analogue of the classical distributed lag model in econometrics.

The basic idea of habit persistence can be captured by the following model for current relative utility, Y(i, t),

where (7(0) = 0, and G(L) is a polynomial lag of order K. (G(L) = gxL + g2L2 + ■ ■ ■ + gKLK, LK Y(i, t) = Y(i, t - K).) One can introduce distributed leads as well, but this is not done here. Assuming that (1 — G(L)) is invertible (e.g., see Granger and Newbold 1977), the model may always be rewritten as r(i,/) = [i -G(LTlz(i,t).

If the e(i, t) are iid, the coefficients of G(L) may be estimated (up to an unknown factor of proportionality) by multivariate probit analysis, provided the available panel is of suitable length (T>K) and that the initial conditions for Y(i, t'), t' < 0, are specified. If the £(/", t) are not iid, and the process determining s(i, t) is unknown, the model is not identified. This identification problem is exactly the same problem that arises in

27. Indeed the fixed cost model provides a rationalization for a first-order Markov model.

estimating a distributed lag model in the presence of serial correlation (see Griliches 1967, p. 35).

Introduction of exogenous variables into the model aids in identification. If the model of equation (3.25) is augmented to include exogenous variables, it is possible to estimate (variance normalized) elements of G(L) and fi as well as the correlations among the disturbances. This is so because in reduced form so that from the estimated coefficients on the lagged values of the Z(i, t) variables it is possible to solve for the normalized coefficients of G(L), provided that the Z(i, t),t = 1, . . . , T, are not linear combinations of each other for all i, and initial conditions Y(i\ f')> t -c 0, are specified.28

It is interesting to note that, if at least one variable in Z(i, 0 changes over time, and exact linear dependency among the Z(i, t) does not exist, a probit model fit on one cross section can be used to test for habit persistence. The test consists of entering lagged values of Z(z', t) into the probit model based on equation (3.27). If the lagged values of Z(i, t) are statistically significantly different from zero, one can reject the hypothesis of no habit persistence. Cross section probit models can be used to estimate the normalized coefficients of G(L), provided the analyst has access to lagged values of the Z(/, /).

The model for habit persistence may be grafted onto the models with structural state dependence developed earlier. General conditions for identification in this model are presented elsewhere (Heckman 1978a, p. 956). The important point to note is that subject to exclusion (or other identification) restrictions, even though Y(i, /') is never observed, its effect on current choice can be estimated and distinguished from the effect of structural state dependence. Thus one can separate the effect of past propensities to occupy a state on current choices from the effect of past occupancy of a state on current choices.

Y(i,t) = [1 - G(L)]~lZ{i,t)fi + [1 - G(L)]~1E{i,t),

28. A model with lagged latent variables appears in Heckman (1978a, pp. 932 and 956).

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