Two-State Models with Exogenous Variables

We shall consider a two-state Markov model with exogenous variables, which accounts for the heterogeneity and nonstationarity of the data. This model is closely related to the models considered in Section 9.7.2. We shall also discuss an example of the model, attributed to Boskin and Nold (1975), to illustrate several important points.

This model is similar to a univariate binary QR model. To make the subse­quent discussion comparable to the discussion of Chapter 9, assume that/ = 0 or 1 rather than 1 or 2. Let yu = 1 if the tth person is in state 1 at time t and ya = 0 otherwise. (Note that yu is the same as the y[(t) we wrote earlier.) The model then can be written as

P{yit= lyi. t-i) = F(Fxit + a’xttyUt-l), (11.1.34)

where Fisa certain distribution function. Note that (11.1.34) is equivalent to Пі(і) = №«) (H-1.35)

P‘n(t) – F[(a+ )?)%].

Writing the model as (11.1.34) rather than as (11.1.35) makes clearer the similarity of this model to a QR model. The model defined by (9.7.11) is a special case of (11.1.34) if {uIf} in (9.7.11) are i. i.d.

Using (11.1.4), we can write the likelihood function of the model condi­tional on Ую as

L = П П ріп(0%,-‘лЛо(Оп’-,(1-уі,) (11-1-36)

г «

X Pl0l (/)< і. )yl. p‘0Q (/)(і – л,-1Xі -»>.

However, the similarity to a QR model becomes clearer if we write it alterna­tively as

L = П П W – Fuy-y (11.1.37)

і t

where Fu = F(fi’xit + ).

Because we can treat (/, t) as a single index, model (11.1.34) differs from a binary QR model only by the presence of уц – j in the argument of F. But, as mentioned earlier, its presence causes no more difficulty than it does in the continuous-variable autoregressive model; we can treat y4t_, as if it were an exogenous variable so far as the asymptotic results are concerned. Thus, from

(9.2.17) we can conclude that the MLE of у — (/?’, a’)’ follows

Шіу-n^N{„, [plim jL? 2 – фщ-.-і]"},

(11.1.38)

where fu is the derivative of Fu and

The equivalence of the NLWLS iteration to the method of scoring also holds for this model.

Similarly, the minimum chi-square estimator can also be defined for this model as in Section 9.2.5. It is applicable when there are many observations of уit with the same value of хй. Although a more general grouping can be handled by the subsequent analysis, we assume хй = x, for every /, so that yu, і = 1,2,. . . , N, are associated with the same vector xr Define

N

-Уи,-1)

Подпись: P*=Подпись: (11.1.39)i-l____________

So-*-.)

image784 Подпись: (11.1.40)

and

Then we have

Подпись: (11.1.41) (11.1.42) F-‘(P?)=fi% + t„ /=1,2,…, T

and

F~P}) = (a +fi)% + r,„ t – 1,2,. . . , T.

image787 Подпись: (11.1.43)

The error terms £ and у, approximately have zero means, and their respective conditional variances given yi>t_x, і = 1, 2,. . . , N, are approximately

and

p in — pi)

V(m)——- ——^———— , (11.1.44)

/4F-4P})1 g

where P? = P‘oi (/) and P = P, (/). The MIN x2 estimator of у is the weighted least squares estimator applied simultaneously to the heteroscedastic regres-

image789

sion equations (11.1.41) and (11.1.42). As in QR models, this estimator has the same asymptotic distribution as the MLE given in (11.1.38), provided that N goes to oo for a fixed T.

To illustrate further the specific features of the two-state Markov model, we shall examine a model presented by Boskin and Nold (1975). In this model, j = 1 represents the state of an individual being on welfare and j= 0 the state of the individual being off welfare. Boskin and Nold postulated

Подпись: (11.1.45)P[0(t) = A(a) = Ai

РШ = МР%) = В„

where A(x) = (1 + e~x)~l, a logistic distribution. The model can be equiva­lently defined by

Подпись: (11.1.46)Р(Уи = 1 ІУ(,і-і) = Л[у?’х, – (a + P)%yut-, ].

Thus we see that the Boskin-Nold model is a special case, of (11.1.34).

Note that the exogenous variables in the Boskin-Nold model do not depend on t, a condition indicating that their model is heterogeneous but stationary. The exogenous variables are dummy variables characterizing the economic and demographic characteristics of the individuals. The model is applied to data on 440 households (all of which were on welfare initially) during a 60-month period (thus t denotes a month).

Because of stationarity the likelihood function (11.1.36) can be simplified for this model as

Подпись: (11.1.47)L = J] Afal ~1 – В(уЬ.

It is interesting to evaluate the equilibrium probability—the probability that a given individual is on welfare after a sufficiently long time has elapsed since the initial time. By considering a particular individual and therefore omitting the subscript i, the Markov matrix of the Boskin-Nold model can be written as

image793(11.1.48)

Solving (11.1.15) together with the constraint Гр(°°) = 1 yields

image794(11.1.49)

person is on welfare. It increases proportionally with the transition probability

P0l (-5).

image795 Подпись: -A-к A image797

It is instructive to derive the characteristic roots and vectors of P’ given in (11.1.48). Solving the determinantal equation

image798

yields the characteristic roots = 1 and Я2 = 1 — A — В. Solving

image799

yields (the solution is not unique) the first characteristic vector h, = (В, A)’. Solving

yields the second characteristic vector h2 = (— 1, 1)’. Therefore

H‘|j ~i‘] and н“-7Тв[Л і]’

Using these results, we obtain

Подпись: lim H J ‘ 0 ;-»oo Ц ) (1-А (11.1.50)

-H[i 2]»-‘

Подпись: В В A+B A + B A A A + B A + B

which, of course, could have been obtained directly from (11.1.16) and (11.1.49). л л

Although the asymptotic variance-covariance matrix of the MLEaand/lin the Boskin-Nold model can be derived from (11.1.38), we can derive it directly using the likelihood function (11.1.47). We shall do so only for a because the derivation for fi is similar. We shall assume for this assumption en­

ables us to obtain a simple formula.

We need to consider only the part of (11.1.47) that involves A,; its loga­rithm is

Подпись: (11.1.51)log L = 2 и’ю log At + ^ и’,, log (1 – A,). Differentiating (11.1.51) with respect to a, we obtain

Подпись: d log L da _ nо 2d A І Ai dA,_ da Y «її dAt 1 -A( da d2 log L - y| [и'ю . w‘n 1 dAt dAt dada' A}* (1 -A,)2J da da' +?l ~n[0 n‘u 1 &Ai ,At 1 -A,] dada'■ (11.1.52)

(11.1.53)

Tо take the expectation of (11.1.53), we need to evaluate En’10 and En‘u. We have

En[0 = E2 У – 1)у8(0 (11.1.54)

image804

<-i

= (1>0)[х(р’)’]р(0)р’ю-

If we define D to be the 2 X 2 diagonal matrix consisting of the characteristic roots of P’, we have from (11.1.50)

Подпись: r-i (PT image806(11.1.55)

image807 Подпись: En io —

where the approximation is valid for large T. Inserting (11.1.55) into (11.1.54) yields

(11.1.60)

Next, let us consider the consistency of the estimator of a and (i obtained by maximizing L defined in (11.1.17). Using (11.1.50), we can write the loga­rithm of (11.1.17) in the Boskin-Nold model as

Подпись:Подпись:log ц= 2 [ і log [BMi+Bi)]

і L/-o J

і Ь-о J

Defining y(t) = [/,(/), уШ)> we have

phm ^ Y Y(t) = plim ^ 5) (P’)’y'(O)

Г-.» / Г—« 1 /Го

-J_W.

I® + Вю

Подпись: where the subscript 0 denotes the true value. Therefore we obtain 1Подпись: (11.1.63)

image813
image814

&шІоеЦ

T—mo

= fe N % L + ВЮ l0g (її7+в;) + Аю + Вю log from which we can conclude the consistency of the estimator.

We can now introduce in the context of the Boskin-Nold model an impor­tant concept in the Markov model called duration. Let be the number of months a particular individual stays on welfare starting from the beginning of the sample period. It is called the duration of the first spell of welfare. Its probability is defined as the probability that the individual stays on welfare up to time tx and then moves off welfare at time t{. Therefore

P(*i)-(1 – АУ’-‘А. (11.1.64)

The mean duration can be evaluated as

Подпись: (11.1.65)Еі,= ^х{-А)^А=А^^

т-1 t-J or

Подпись:image817d r dr 1 — r

A _ 1
(1 — r)2 A’

where r = 1 — A. In words, this equation says that the mean duration on welfare is the inverse of the probability of moving off welfare.

Suppose the ith person experiences H welfare spells of duration t[, t‘2, • • • , t’H and К off-welfare spells of duration s, . . . , s’#. If we

generalize the Boskin-Nold model and let the і th person’s transition probabil­ities At and Bj vary with spells (but stay constant during each spell), the likelihood function can be written as

L – Ц {ft i’1 – Лі(й)ґ*-Ч(Л) ft u – (і 1.1.66)

Equation (11.1.66) collapses to (11.1.’47) if A, and Bt do not depend on h and к and therefore is of intermediate generality between (11.1.36) and (11.1.47). Expressing the likelihood function in terms of duration is especially useful in the continuous-time Markov model, which we shall study in Section 11.2.

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