Empirical Examples of Markov Models without Exogenous Variables

In this subsection we shall discuss several empirical articles, in which Markov chain models without exogenous variables are estimated, for the purpose of illustrating some of the theoretical points discussed in the preceding subsec­tion. We shall also consider certain new theoretical problems that are likely to be encountered in practice.

Suppose we assume a homogeneous stationary first-order Markov model and estimate the Markov matrix (the matrix of transition probabilities) by. the MLE PJk given by (11.1.22). Let Pfk be a transition probability of lag two; that is, Pfk denotes the probability a person is in state к at time t given that he or she

Подпись:was in state j at time t — 2. If we define п? Лі) = 2fLiyj(t — 2)y‘k(t) and n(? =

-………………………………. (11.1.25) where P2 = PP. Many studies of the mobility of people among classes of income, social status, or occupation have shown the invalidity of the approxi­mate equality (11.1.25). There is a tendency for the diagonal elements of P® to be larger than those of?2 (see, for example, Bartholomew, 1982, Chapter 2). This phenomenon may be attributable to a number of reasons. Two empirical articles have addressed this issue: McCall (1971) explained it by population heterogeneity, whereas Shorrocks (1976) attributed the phenomenon in his data to a violation of the first-order assumption itself.

McCall (1971) analyzed a Markov chain model of income mobility, where the dependent variable is classified into three states: low income, high income, and unknown. McCall estimated a model for each age-sex-race combina­tion so that he did not need to include the exogenous variables that represent these characteristics. Using the mover-stayer model (initially proposed by Blumen, Kogan, and McCarthy, 1955, and theoretically developed by Good­man, 1961), McCall postulated that a proportion Sj of people,./ =1,2, and 3, stay in state j throughout the sample period and the remaining population follows a nonstationary first-order Markov model. Let Vjk(f) be the probabil­ity a mover is in state к at timet given that he or she was in state/ at time/ — 1. Then the transition probabilities of a given individual, unidentified to be either a stayer or a mover, are given by

Подпись: (11.1.26)W-$+(l-3)^(0 and

Pjk(t) = (1 – Sj)VJk(t) if j + k.

McCall assumed that Vjk(t) depends on t (nonstationarity) because of eco­nomic growth, but he got around the nonstationarity problem simply by estimating VJk(t) for each t separately. He used the simplest among several methods studied by Goodman (1961). Stayers in state j are identified as people who remained in stated throughout the sample periods. Once each individual is identified to be either a stayer or a mover, Sj and VJk(f) can be estimated in a straightforward manner. This method is good only when there are many periods. If the number of periods is small (in McCall’s data T— 10), this method will produce bias (even if n is large) because a proportion of those who stayed in a single state throughout the sample periods may actually be movers. After obtaining estimates of Vjk(t), McCall regressed them on a variable repre­senting economic growth to see how economic growth influences income mobility.

For a stationary Markov model, Goodman discussed several estimates of 5} and VJk that are consistent (as n goes to infinity) even if Г is small (provided T> 1). We shall mention only one simple consistent estimator. By defining matrices V = {VJk), P = {Pjk}, and a diagonal matrix S = D(S)), we can write (11.1.26) as

P = S + (I-S)V. (11.1.27)

The matrix of transition probabilities of lag two is given by

P<2> = S + (I — S)V2. (11.1.28)

Now, P and P(2) can be consistently estimated by the MLE mentioned earlier. Inserting the MLE into the left-hand side of (11.1.27) and (11.1.28) gives us 2M(M — 1) equations. But since there are only M2 parameters to estimate in S and V, solving M1 equations out of the 2M(M — 1) equations for S and V will yield consistent estimates.

The empirical phenomenon mentioned earlier can be explained by the mover – stayer model as follows: From (11.1.27) and (11.1.28) we obtain after some manipulation

P™ – p2 = S + d – S)V2 – [S + (I – S)V][S + (I – S)V]

= (I – SKI – V)S(I – V). (11.1.29)

Therefore the diagonal elements of P<2) — P2 are positive if the diagonal elements of (I — V)S(I — V) are positive. But the jth diagonal element of (I — V)S(I — V) is equal to

Sjd-Vtf + ^SbVjbVq,

k+j

which is positive.

Shorrocks (1976) accounted for the invalidity of (11.1.25) in a study of income mobility by postulating a second-order Markov model. Depending on the initial conditions and the values of the parameters, a second-order Markov model can lead to a situation where the diagonal elements of I*2) are larger than the corresponding elements of P2.

The likelihood function of a second-order Markov model conditional on the initial values yj(— 1) and yj(0) is given by

where PjkiU) is the probability the і th person is in state l at time t given that he or she was in state j at time t — 2 and in state к at time t— 1. If homogeneity and stationarity are assumed, then Pjw(t) = PJkl. Even then the model con­tains MM — 1) parameters to estimate. Shorrocks grouped income into five classes (M= 5), thus implying 100 parameters. By assuming the Champer – nowne process (Champemowne, 1953), where income mobility at each time change is restricted to the three possibilities—staying in the same class or moving up or down to an adjacent income class, he reduced the number of parameters to six, which he estimated by ML. We can see this as follows: Let t],£ = —1,0, 1 represent the three possible movements. Then the Champer- nowne process is characterized by P^, the probability a person moves by £ from t — 1 to t given that he or she moved by rj from t — 2 to t — 1. Notice that the model is reduced to a first-order Markov chain with M = 3.

For the purpose of illustrating the usefulness of equilibrium probabilities in an empirical problem and of introducing the problem of entry and exit, let us consider a model presented by Adelman (1958). Adelman analyzed the size distribution of firms in the steel industry in the United States using the data for the periods 1929 -1939 and 1945 -1956 (excluding the war years). Firms are grouped into six size classes according to the dollar values of the firms’ total assets. In addition, state 0 is affixed to represent the state of being out of the industry. Adelman assumed a homogeneous stationary first-order model.

Movements into and from state 0 are called exit and entry, and they create a special problem because the number of firms in state 0 at any particular time is not observable. In our notation it means that is not observable, and

hence, MLE Pok = йол/2/Ио/ cannot be evaluated. Adelman circumvented the
problem by setting 2/.0По/ = 100,000 arbitrarily. We shall see that although changing changes the estimates of the transition probabilities, it does not affect the equilibrium relative size distribution of firms (except relative to sizeO).

Let p = p(o°) be a vector of equilibrium probabilities. Then, as was shown earlier, p can be obtained by solving

image780(11.1.31)

where * means eliminating the first row of the matrix. Now, consider pj/pk for j, кФ 0, where pj and pk are solved from (11.1.31) by Cramer’s rule. Because affects only the first column of [I — P’]* proportionally, it does not affect pj/pk.

Duncan and Lin (1972) criticized Adelman’s model, saying that it is unreal­istic to suppose a homogeneous pool of firms in state 0 because a firm that once goes out of business is not likely to come back. Duncan and Lin solved the problem by treating entry and exit separately. Exit is assumed to be an absorbing state. Suppose j = 1 is an absorbing state, then Pu = 1 and Plk = 0 for к = 2, 3,. . . , M. Entry into state к—mk(t) firms at time t, k = 2,3,. . . ,M—is assumed to follow a Poisson distribution:

(11.1.32)

Then PJk and fik are estimated by maximizing the likelihood function

Подпись: T M (11.1.33)

where L is as given in (11.1.4). This model is applied to data on five classes of banks according to the ratio of farm loans to total loans. Maximum likelihood estimates are obtained, and a test of stationarity is performed following the Anderson-Goodman methodology.

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