Opportunity and Choice in Socially Structured Labor Markets

by John Allen Logan
Opportunity and Choice in Socially Structured Labor Markets
John Allen Logan
The American Journal of Sociology
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Opportunity and Choice in Socially Structured Labor ~arkets'

John Allen Logan

University of Wisconsin-Madison

Employment outcomes depend on opportunity and choice, both of which are subject to structural influences. This article presents a new approach to studying the factors that determine employment outcomes. It develops a statistical technique, two-sided logit (TSL), directly from a model of the preferences and resources of employers and workers. Opportunity and choice are functions of these prefer- ences and resources. Application of the TSL model to 1972-90 GSS data shows substantial variation in the importance of the worker characteristics of education, race, and age for the opportunity for employment in different occupational categories. The relationships of TSL to other sociological and economic models are also dis- cussed.


Opportunity and choice determine employment outcomes in a free labor market. Opportunity is provided by employers, who make jobs available. Workers choose among the opportunities provided, that is, among available jobs. The factors influencing opportunity and the factors influencing choice among opportunities are distinct in principle since they are determined by different types of actors. Furthermore, the factors influencing opportunity are a main object of public policy, while the factors influencing choice among opportunities, though not irrelevant to policy, are, within broad limits, considered to be a

I thank the following for helpful comments on earlier versions of this paper or for other assistance: Adam Gamoran, Charles Halaby, Robert Hauser, Jeffrey Hayes, Michael Hout, Robert Mare, Nadine F. Marks, Andrew Michener, Judith Seltzer, William H. Sewell, Jr., William H. Sewell, Sr., Donald Treiman, John Robert War- ren, Erik Olin Wright, Yu Xie, members of ISA Research Committee 28, and partici- pants in the summer 1995 Research Workshop of the Institute for Research on Pov- erty. Computing support was provided by the Center for Demography and Ecology and the Social Science Computing Cooperative of the University of Wisconsin- Madison. Direct. correspondence to John Allen Logan, Department of Sociology, 1180 Observatory Drive, University of Wisconsin, Madison, Wisconsin 53706.

01996 by The University of Chicago. All rights reserved. 0002-9602/97/10201-0004$01.50

114 AJS Volume 102 Number 1 (July 1996): 114-160

matter of private freedom. Thus any discussion of the openness of the

occupational system, of fairness, of systematic limits on mobility or

achievement would seem almost to require some distinction between

the determinants and consequences of opportunity and the determi-

nants and consequences of individual choice.

The models used most often by sociologists to study these topics, how- ever, fail to make such a distinction. Linear regression models simply estimate the average effects of individual characteristics on occupational outcomes (SES or income, e.g.) without distinguishing whether a particu- lar effect is due to opportunity or choice or both. Log-linear models, even as augmented with exogenous variables (Yamaguchi 1983; Logan 1983; Hout 1984; DiPrete 1990; Breen 1994), make no fundamental distinction between the two types of effect. Some less widely applied models do represent opportunity clearly (e.g., Simon [I95 11 1982; White 1970; Bou- don 1974; Thurow 1975; Sgrensen 1977; Stewman and Konda 1983; Boy- lan 1992) but fail to develop choice fully, presuming that workers will uniformly strive to "rise" in the system according to some universally accepted definition of "up," even though voluntary unemployment and retirement-not to mention alternative definitions of success-are obvious features of working life. Empirical economic models, on the other hand, have their own characteristic limitation: a ready assumption that either employers or workers or both act in a forward-looking, self-interested, "rational" manner (see, e.g., Devine and Kiefer 1991; Sat- tinger 1993), an assumption that facilitates estimation while shrugging off sociological arguments that action is structured in a way not reducible to self-interested rationality (e.g., Giddens 1984; Sewell 1992).

The present article proposes a new multivariate method for simulta- neously investigating the determinants of individual opportunity and the determinants of choice given opportunity, while giving equal weight to both aspects. The parameters estimated by the model represent the pref- erences of employers that determine workers' opportunities and the pref- erences of workers that determine their choices among opportunities. The data requirement is any random sample of individuals, both working and not working, that records their employment-relevant characteristics and the characteristics of jobs they hold. Estimating the preferences of employers using such data requires controlling for selection effects due to individuals' preferences, and vice versa. The model that is developed to make such doubly controlled estimations is called a two-sided logit (TSL) model and is based on an underlying random matching model of the labor market, which itself is a stochastic variant of deterministic models studied in game theory (e.g., Roth and Sotomayor 1990). Though the underlying model represents individual jobs being evaluated by work- ers, actual estimation makes use of the average characteristics of jobs within broad occupational categories rather than the characteristics of particular jobs.

The TSL model is constructed in such a way that estimates of employers' and workers' preferences are insensitive to changes in demand for jobs among occupational categories. Instead of directly estimating the effect of a worker characteristic on his or her likely employment outcome (as in linear regression), the model estimates the effect of the characteristic on the rankings of workers that are made by the employers hiring in the different occupational categories. The rankings are determined by utility functions in which the employers' preference parameters appear. Only by specifying levels of demand across occupational categories, assumed to be functions of exogenous economic causes, are the rankings of workers by employers translated into definite probabilities of their employment within occupations. This very clearly separates economic demand from socially structured preference effects, a result long desired but not convincingly achieved by log-linear or earlier models (see Harrison 1988).

The preferences of employers and workers determine opportunities and choices only in conjunction with the resources that workers and employ- ers bring to the labor market, since it is these resources or characteristics to which the preferences of potential employment partners will apply. Nonetheless the preferences themselves have clear interpretations that do not require explicit reference to resource holdings. Employers' preference coefficients estimate the change in opportunity, measured by the log odds of an offer, that is associated with a one-unit change in a worker charac- teristic. The same coefficients are also estimates of the chance, again measured by a log odds, that two workers who differ by one unit on a characteristic are ranked concordantly with that difference by employers in the relevant occupational category. Finally, the set of all preference coefficient estimates is used to estimate the proportion of variance in employers' utilities for workers that is explained by the model within each category, when the model fits well by an independent criterion. These various interpretations of the model coefficients can provide new insights into the structure of opportunity, compared with estimates from prevailing models.

Application of the TSL model, including appropriate variable selec- tion methods and tests for goodness of fit and structural change, is demonstrated using the 1972-90 waves of the General Social Survey (GSS). The relatively simple model specification used here suggests substantial differences in the preferences of employers within broad occupational categories and the overall stability of these patterns across the two decades.

The organization of the article is as follows. The first section develops the TSL model and discusses its estimation as well as discussing certain special cases corresponding to other, existing models. The second section then applies the model to study the determinants of occupational opportu- nity and attainment in the United States during 1972-90 while devel- oping parameter interpretations and statistical tests in the context of this example. The third section reviews a variety of other sociological and economic models of the determinants of employment outcomes in order to contrast them clearly with the model developed here. The fourth section considers generalizations and extensions of the model that may be helpful for other sociological research questions.


Several principles underlie the model. First, the employment opportunity of a worker is essentially synonymous with the set of jobs that employers would be disposed to offer him or her if approached.' Studying the factors affecting employers' preferences or dispositions is therefore necessary to understanding opportunity. However, it is also necessary to study the factors influencing workers' employment preferences, so that these are not confounded with the effects of employers' dispositions. A two-sided approach explicitly combining models of employers' and workers' prefer- ences, together with data on the characteristics or resources that each side values in the other, therefore provides an attractive and direct repre- sentation of the determinants of employment opportunity and choice. Such an approach will represent each individual's opportunity by the chance of obtaining job offers in different occupations. Seen from the other side, each employer has an opportunity to hire that corresponds to the chance of obtaining acceptances of its offers from particular types of workers.

It is important to emphasize at once that such an approach does not reduce the study of the determinants of employment outcomes to the study of preferences. Employers' and workers' preferences are only effec- tive in constraining others to the degree that either commands resources desired by others. Conversely, the opportunities of any particular actor

The terms "worker" and "employer" are used as a convenience. Strictly speaking, the model includes all persons who may or may not decide to work; the model allows workers to choose or be forced into nonemployment. Similarly, "employers" include private firms as well as public-sector actors like government agencies who may decide whether to make employment or employmentlike benefits such as welfare available to "workers." Self-employed workers are also considered within the model frame- work; for them the "employer" is thought of as the financiers, franchisers, or relatives who typically make self-employment possible.

are the result of its own resource holdings and of the distribution of

preferences and resource holdings among all other actors. Understanding

opportunity also involves understanding the existing distribution of re-

sources and the principles that may govern that distribution's develop-

ment, stability, and change through time.

A second principle is that mutual constraint on employers' and work-

ers' choices, derived from the distribution of preferences and resources, is

not a synonym for structure in any useful sociological sense. Sociological

inquiry should ultimately go beyond the description of constraint into the

causes of its particular form in a given historical situation. The way in

which the model preserves a place for structural effects will be discussed

after it has been developed and contrasted with economic alternatives.

A final principle guiding the model development is that practical tech-

niques require obtainable data. A practical approach to the idea of model-

ing opportunity as the outcome of a multitude of discrete choice problems

must forgo data on all the choice situations each actor faces and consider

how estimation can be done with much more modest data, such as the

widely available samples of workers that contain characteristics only (or

principally) of the individual and the job currently held. Such data from

the GSS are used below.

The Modeling Strategy: Separate Discrete Choice Submodels

The extensive discrete choice literature in economics (reviewed in Ben- Akiva and Lerman [1985]; Pudney 1989) has derived models of the proba- bilities that individuals will make particular choices from sets of discrete alternatives, under a wide range of assumptions. The most useful of these models in practice has been the conditional logit model, derived by assuming that individuals evaluate their alternatives using linear utility functions, which in practice must contain random disturbance terms to represent relevant factors unknown to the researcher. The model is ob- tained by assuming that the random disturbances have independent type I extreme value (or Gumbel) distributions. That is, by assuming the utility of alternative j for individual i is V,(j)= aizij + vij,where ai is a row vector of i's preferences, zijis a column vector of the observed characteristics of alternative j available to i, and vijis a random Gumbel error, it can be shown that the probability that i will choose alternative

j from the set of all J available alternatives is the polytomous conditional logit model:

This model, in which the observed data are characteristics of the alternatives, is often simply called the conditional logit model to distinguish it from the so-called multinomial logit model in which the data are characteristics of the decision makers. I, however, call it the polytomous conditional logit model to distinguish it from the binary conditional logit model, where only two alternatives are available. Understand-

It should be understood that the linear model for the utility V,(j)is the basic model under consideration and that model (1)is merely its implica- tion at the level of observable, discrete choices. In practice, the a preference coefficients are assumed to be numerically equal across all individu- als, or else within demographic subgroups of individuals, to achieve identification.

Use of the conditional logit model in either binary or polytomous form requires only modest behavioral assumptions about the decision makers. It is assumed that the preferences of individuals are consistent (cf. Ben- Akiva and Lerman 1985, p. 38). That is, when confronted with the same set of alternatives with identical characteristics on separate occasions, individuals will make the same choices4 The assumption of consistency does not, however, mean that individuals will never be observed to make apparently inconsistent choices, since the model assumes that some rele- vant characteristics of alternatives are unobservable (and are represented in the error terms). Neither does the use of a conditional logit model assume that decision makers have complete knowledge of all aspects of the labor market or that they can or do make complex calculations of the optimal choice given market conditions. The modeling strategy I will use is to adopt a utility-based, conditional logit formulation both for the decisions of employers to make jobs available to workers and for the decisions of workers regarding which jobs to accept of those they find available. Doing this assumes consistency of preferences for both parties.

ing the derivation of the discrete probability model (1) from the assumption of an underlying, continuous-valued utility function is important for a full understanding of the proposed model. Readers are directed to Ben-Akiva and Lerman (1985, pp.

101-8) for a clear exposition. In brief, the derivation proceeds by integrating over the joint distribution of the utilities of all alternatives, subject to the condition that the utility for the chosen alternative, j, is the maximum among alternatives; the resulting probability is the cdf of the logistic distribution (see also Pudney 1989, pp. 293-97). The assumption of Gumbel disturbances allows the computationally simpler logit rather than probit form of the model to be derived when normal errors are theoretically appropriate. Without loss of generality, all Gumbel errors introduced in this article will be in standard form, that is with distribution function exp(-e-"), mode "0," mean 0.57722, and variance .rrz/6 (Johnson and Kotz 1970). McFadden (e.g., 1974) is given primary credit for the development of this model in the economic context.

Because the model below will imply the possibility that each individual's choice might have occurred in the context of an arbitrarily small choice set, the assumption of consistency can be shown to be tantamount to one of transitivity as well; this is of no practical importance in the present context (cf. Sen 1987).

The Employer's Decision

An underlying random utility model is defined to describe the decisions of employers whether or not to make jobs available to particular workers. For an employer j, the utility of hiring an individual i is defined as

while j's utility of not hiring i is

with the definitions of right-side quantities given below. When expression

(2) is greater in value than expression (3)' employer j makes a job avail- able:

= 1 if Uj(i) > Uj(ii)


= 0 otherwise.

The dummy variable oij is "1" when employer j makes an offer to i, and "0" otherwise.

The right-side terms in (2) and (3) are defined to give a behaviorally realistic characterization of the decision facing each employer. In equa- tion (2)'

Pj = a row vector of employer j's preferences for relevant characteristics of employees; = a column vector of i's measured values on the characteristics


employers value; mj = a scalar representing systematic contributions to j's utility for making a hire that are unrelated to i's characteristics; two principal factors affecting mj are the level of demand for the products of the employer and the level of staffing desired by the employer, relative to current demand; and el,; = a random error representing factors that are not known to the observer but which influence j's utility of hiring i.

In equation (3)'

= the baseline utility that employer j derives from its present


state of staffing; = a strategic increment to its baseline utility that j may require sj of an individual before making a decision to offer employment, representing j's reluctance to hire the first worker who would increase its utility; and

eoij = a random error representing factors which are not known to the observer but which influence j's utility of not hiring i.

This model of employer j's decision is called a random utility model because it represents the decision rule, (4), as a function of utilities that have random components, the error terms in (2) and (3). The systematic scalar quantities mj, bj, and sj are important for behavioral realism and for linking the random utility matching model to the game theoretic matching and economic job search literatures (see Logan 1996a) but do not complicate the estimation of the model when using standard data, as will be shown.

The exact probability that j will make an offer depends on the distribu- tion of the differences between the two error terms, as well as on the nonstochastic parts of (2) and (3). If the two distributions are indepen- dent, standard Gumbel distributions, then the distribution of the differ- ence between the errors is logistic, and the probability that j will make an offer under the model is the binary conditional logit (cf. Ben-Akiva and Lerman 1985):

where xT = (1, xT)~is the original vector of individual characteristics

augmented by an entry of unity in the first position, and = (Pjo, Pj) is the original vector of employer j's preferences, augmented by an inter- cept parameter in its first position. This intercept, Pjo, accounts for the net influence of all the systematic scalar terms in (2) and (3):

In particular, Pjo accounts for all the exogenous influences on i's chances of being offered a job by j that are not associated with his or her own characteristics; these include changes in market demand and staffing lev- els (m,), baseline characteristics of employers (bj), and strategic consider- ations of employers (sj) For brevity, Pjo can be termed a demand inter- cept, since the net effect of these factors determines the level of demand for workers.

The formal conception is that each employer obtains access to informa- tion about workers in sequence and makes any of them an offer when its evaluation of the individual's qualifications, shown in (2), exceeds the threshold given in (3).' I assume that employers act independently of one another, conditional on the individual's characteristics in xi. The result is that each individual is presented some set 0,of offers from the employ- ers as a whole. The possible sets, containing the indexes of employers making offers, are 0, = {O), 0, = (0, I), 0, = (0, 2), 0, = (0, 1, 2), 0, = (0, 3), . . . , OR = (0, 1, 2, . . . , 5).Set 0, = (0) represents no job offers, which forces ~nemployment;~

set OR, at the other extreme, represents offers from all employers, giving the worker complete freedom of selection. The event that i experiences offering set 0,is denoted Sik. The probability that i experiences any particular opportunity set 0, is found from the multiplication rule for (conditionally) independent events:

where Dkrepresents the complement of the employer indexes in 0,.

The Individual's Decision

The individual's choice of his or her most preferred offer from the avail- able set is in turn specified as a second random utility model. The utility that i would obtain from the job offered by employer j is defined as

Vector z,. contains the characteristics of the offered job, ai contains the preferences of the individual, and uijis a random error representing unknown influences on the utility. The decision rule for the individual is to select the single job that offers the highest utility. If the uij are specified as independent Gumbel disturbances, then the probability that i will select j from a set 0,of offers is this polytomous conditional logit:

The success of the model does not depend on a literal circumstance in which all employers examine the qualifications of all individuals. The actions of employers that have a negligible chance of making an acceptable offer to an individual are irrelevant in practice and in the estimation method employed.

In this article, "unemployment" will be used to mean any state of nonemployment, rather than a condition of searching for employment.

Here, prob(AijISik) is read as the probability i will accept an offer from employer j,conditional on the event Sik that a particular set 0, of offers has been obtained. The formula specifies a zero probability of accepting an offer from j when j $E Ok, since this means the offering set does not include an offer from employer j.Note that zi,represents the characteris- tics of unemployment available to i, measured in the same terms as the characteristics of the offers; all sets Ok were defined to contain the integer zero, representing an "offer" of unemployment, since unemployment is always available to any individual, as implied by equation (5).

Summary of the TSLiRandom Matching Model Relationship

The random matching model is specified by describing a matching proce- dure with reference to formulas (2), (3), (4), and (8), in the following steps:

  1. Employers evaluate workers according to (2) and (3) and make offers according to decision rule (4);
  2. Workers evaluate any offers and the possibility of unemployment ac- cording to (8) and choose the highest-utility alternative.

It is assumed that employers have complete knowledge regarding the measured characteristics of workers in xi,i = 1, 2, . . . , n. Relaxation of this assumption will be considered later.

The TSL model is specified by a corresponding combination of the preceding probabilities for each step. The probability of any single em- ployer's offer in step 1 is given by (5) and the probability of any offering set across employers is given by (7). The probability of accepting any offer in step 2, given an offering set, is equation (9). Assuming that the distributions of error terms in (2), (3), and (8) are independent Gumbel, the final, unconditional probability that i will accept an offer from j is

This formula, which completes the definition of the TSL model, takes into account all the possible offering sets that i may have experienced, weighting each by its probability of having occurred. It is a simple combi- nation of the probabilities of offers and the probabilities of acceptances conditional on offers, using elementary rules of conditional probability.

Under the model, the conditional probability that individual i will accept an offer from employer j, given that it is made, is

the ratio of formulas (10) and (5). Note that (10) depends on the offering set probabilities, prob(Si,), k = 1, 2, . . . ,R, and that these probabilities, given in formula (7), depend in turn on the probabilities of offers from all employers. Therefore the implication of (11) is that the probability that an offer from any particular employer will be accepted depends on the dispositional states of all other employers, an illustration of the sys- temic nature of the model.

The model defined in (10) is termed the "two-sided logit" model be- cause it models the two sides of the job market separately and is made up of J binary logits on the employers' side and a single polytomous logit on the workers' side; these components are seen in (5) and (9). A two- sided probit model is of course another possible statistical manifestation of the underlying matching model.'

Special Cases of the TSL Model

Three well-known models are special cases of the TSL model and help reveal its characteristics. The polytomous conditional logit model of McFadden (1974) is equivalent to a TSL model in which the offering probability from each employer is specified as prob(oij = 1) = 1, all i and j, rather than as (5). This in turn corresponds to a random matching model in which the condition in (4) is always true: all employers always make jobs available to all workers. Such a model seems unconvincing for employment matches.'

If equation (8) is such that all workers always prefer higher-numbered employers to lower-numbered employers, where the numbering might correspond to a prestige ranking, for example, the TSL model simplifies to the sequential logit model, discussed in Amemiya (1981) and closely related to survival models (Allison 1982). In this model, the probability of any lower-numbered outcome is equal to one minus the probability that all higher-numbered employers have failed to make offers. This

'The idea of a two-sided logit or probit model should not be confused with the familiar bivariate probit model, in which two binary outcomes are modeled simultaneously. In the TSL model there are J binary outcomes on the employers' side, plus one polyto- mous outcome over J + 1 alternatives on the individual's. The use of the term "two-sided" is consistent with Roth and Sotomayor's (1990) usage concerning match- ing processes in such markets.

McFadden's (1974) model also allows for the possibility of incorporating direct knowledge of whether each employer would make an offer to each individual, but such data is impractical for employment studies.

model pertains if workers all share a strict preference ranking of employ- ers. In such a situation, outcomes are then completely dependent on employers' decisions to make offers.

The partial observability probit model of Abowd and Farber (1982; Sakamoto and Powers 1995) is closely related to TSL as the probit version of its single-employer special case. Abowd and Farber modeled the allo- cation of workers to union and nonunion jobs as the outcome of a sequen- tial decision process in which workers first decide whether to join a queue for union jobs, and employers (represented as a single actor) then decide whether to select them from the queue. Both are of course binary deci- sions. In the corresponding single-employer TSL model, the sequential decision process consists of the employer deciding, for each individual, whether to offer a job and the individual then deciding whether to accept it or to remain unemployed. In each model the outcome of the first deci- sion is assumed to be unobserved.

Showing the correspondence requires some additional notati~n.~

For the first decision in either the Abowd-Farber or the TSL case, a decision indicator dummy variable must be defined I, = 1 if the decision is positive and "0" otherwise, and a similar indicator I, must be defined for the second decision. Then the TSL probability that an offer is made is

using the notation Liw) for the logistic cumulative distribution function. There are only two possible offering sets for workers in the single- employer case: 0, = (0) for no offer, and 0, = (0, 1) when an offer is made. The TSL probabilities that the offer is accepted for these respec- tive sets are prob(~il Sio) = 0, and


where z; = zil -ziO is a vector of the employer's job characteristics

I am following the notation and exposition of Maddala (1983, pp. 278-81) and am omitting the underlying wage equations that correspond approximately to the utility equations underlying TSL.

measured as differences from tile characteristics of unemployment. Then the unconditional probabilities of employment and unemployment are

from (10) above, and

by complementation.

Defining a second-level indicator I = Ill2,equal to one only when the employer has offered and the individual has accepted, the likelihood corresponding to the probabilities can then be written as

where the product ranges are over all i such that I = 1 and 0, respec- tively. The single-employer TSL model is thus the logit equivalent of the normal-based partial observability probit model of Abowd and Farber, whose likelihood is given in equation (9.56) of Maddala (1983):

represented here with the obvious substitutions for parameter and data vectors and using @(w)to mean the cumulative normal distribution function.

That a logit version of Abowd and Farber's partial observability probit model is a special case of TSL gives an additional perspective on the model. TSL is seen to be a partial observability logit model, as well as a two-sided, multiactor, discrete choice model. The partial observability is induced by decisions of the workers and employers that are not ob- served in full. TSL, like Abowd and Farber's model, is an estimation method that compensates for the censoring of information caused by two kinds of selection. In the TSL model, these are the self-selection of work- ers into available occupations and the selection by employers of which workers are to have particular occupations from which to choose. From this point of view, TSL is estimating a linear model of preferences (namely, the coefficients in the linear utility functions [2] and [8]) while correcting for sample selectivity.


Practical use of the TSL model requires the solution of certain problems. First, the shape of likelihood of the model is such that quasi-Newton algorithms seem very likely to diverge rather than to find a maximum. My solution to the estimation problem has been to derive an EM maxi- mum-likelihood algorithm (Logan 1994~). EM algorithms (Dempster, Laird, and Rubin 1977) break complex estimation problems into simpler pieces by iteratively estimating unobserved data values and are inher- ently stable insofar as the simpler pieces are themselves stable, which is the case here.'' The EM algorithm for TSL can be implemented by iterative use of standard conditional logit programs, but a much faster custom program has been developed for the purpose.

Aside from the technical difficulties due to the shape of the likelihood, a very severe computational problem arises from the combinatorial com- plexity of the model. The model specifies that each worker chooses the best alternative from the set of jobs that employers are disposed to make available but that the actual set of available jobs is not observed by the researcher. The likelihood function of the model requires the evaluation of the probabilities of accepting jobs under all possible sets of offers that contain each job, as can be seen from (lo), while the number of such sets increases geometrically with the number of jobs. There are R = 2' sets, where J is the number of jobs: four sets for two jobs, 32 for five, 1,024 for 10, over a million for 20. Clearly, there is no prospect of using the TSL model directly on job-level data in broad labor markets, and some form of data reduction must be used. A practical solution is to use the mean characteristics of jobs within a few, broad occupational catego- ries rather than using characteristics of individual jobs directly. Since some reduction in the number of separate sets of preference coefficients across employers is also necessary, it is convenient to use the same occu- pational categories to classify the employers into sets with equivalent preferences according to the occupational categories in which they are hiring.

lo The likelihood of the TSL model is not globally concave, so that local maxima are possible results of the algorithm. Experiments with random starting values have so far failed to find alternative solutions on test data sets. However, a local maximum was found in the data analyzed below, where it was recognizable by virtue of inflated SEs and a reversal in the sign of a coefficient, compared to similar subsamples. An alternate set of starting values located a higher-likelihood maximum qualitatively similar to the estimates in other subsamples; these are the estimates reported below. See Logan (1996~)for further discussion.

The strategy of introducing occupational categories involves two types of categorization that should be carefully distinguished. With regard to the workers' evaluations of jobs, represented in (8), the characteristics of all jobs within an occupational category j are represented by a category mean vector Ej which takes the place of zij.With regard to the employers' evaluations of workers, represented in (2), the utility functions of all employers offering jobs within a given occupational category j are consid- ered to be identical, so that Pj represents the shared preferences of the employers in the category. The latter type of categorization is actually desirable for interpretative purposes, since there is little point in attempting to estimate separate preference structures for each of a vast number of employers. The former type of categorization, replacing the job characteristics in (8) by mean occupational characteristics, is less desirable because it does not make optimal use of all the information in the sample concerning the characteristics of each worker's job and be- cause it can be shown to induce a bias in the estimates of the preferences of the workers (cf. Ben-Akiva and Lerman [I9851 regarding the one-sided case). However, this form of categorization does not directly bias esti- mates of the shared preferences of employers within the categories, which become the main interest in studies of the occupational opportunities of individuals. By using mean characteristics of jobs within occupational categories and by estimating shared preferences of employers offering jobs within the same categories, the TSL model becomes computationally tractable for studies of large labor markets."

Estimation of the TSL model can be performed with data from a random sample of individuals (both working and not working) that re- cords their employment-relevant characteristics and the characteristics of the job (if any) held by each. Characteristics of jobs held by the sample individuals are averaged within each occupational category to obtain the mean vectors described above. The EM algorithm produces maximum- likelihood estimates, which make possible standard likelihood-ratio and BIC tests of model fit, to be described in the next section.


This section applies the TSL model to the structure of opportunity for younger men and women in the United States during 1972-90 and also develops parameter interpretations and test statistics in the context of this example. Data from the GSS were divided into two periods, 1972-80

l1 Other approaches to estimation are discussed in "Generalization and Extensions" below.

and 1982-90, which I will refer to approximately as the seventies and eighties. Separate analysis files of men and women ages 25-44 were created for each period.'' The restriction to younger workers is intended to sharpen the focus on a more homogeneous group than all adults. Table 1 reports descriptive statistics for all variables to be used in the analysis. Education and age are measured in years, while nonwhite is a dummy variable. Prestige is the Hodge-Seigel-Rossi score, the derivation of which is described in the GSS codebook. Supervisor is a dummy variable indi- cating that the respondent reported having a supervisor on his or her current or most recent job.

A relatively simple TSL specification is estimated for each of the four combinations of gender and period. Occupations are classified into five categories: professional; managerial (including salaried and self-employed managers, non-retail sales workers, farmers, and farm managers); sales (retail), clerical, and service; manufacturing blue collar (crafts, opera- tives, and laborers); and other blue collar (including farm laborers). Em- ployers are specified as evaluating the education, age, and race of workers in making their offering decisions. Workers are specified as evaluating the prestige and autonomy of offered jobs. Following Hout (1988, p. 1367), autonomy in each category is measured as the odds of having a supervisor (multiplied by -1 so that higher scores mean higher auton- omy). Prestige is measured as the category mean. Note that prestige and autonomy data are not available for unemployed individuals: the mean prestige and supervisor scores shown for the unemployed in table 1 are values for the last jobs held among those who had held previous jobs. Prestige for the nonemployed was therefore set somewhat arbitrarily to the value 18, about 50% of the mean prestige of the last job held. Auton- omy for the nonemployed was set to minus the average odds of having a supervisor, as calculated across all employed sample members for both genders and periods.13

Tables 2 and 3 present simultaneous maximum-likelihood estimates of

l2 The analysis files include all individuals in the main yearly samples with nonmissing values of education, marital status, race, and age, and, if employed at the time of the survey, nonmissing occupational category data. Black oversample cases from 1982 and 1987 were omitted.

l3 It seems unclear to me whether the unemployed as a whole are high or low on autonomy, considering the possible dependency implied by their status. Experiments show that estimates of employers' preferences (the most direct determinants of work- ers' opportunities) are relatively insensitive to differences in the values assigned to the characteristics of unemployment. Estimates of workers' preferences are of course more strongly affected. It would be desirable to collect direct measures of prestige, autonomy, and similar characteristics of the unemployed.

Men, 1972-80: ...........Education
Age .....................
Prestige ...................
Supervisor ................
CL G, CL Total cases ............... % ........................... Men, 1982-90: ................Education Age .........................
  Nonwhite .................
  Prestige ...................
  Supervisor .............
  Total cases ............... % ...........................
  NOTE.-NOS.in parentheses are SDs.


1972-80 1982-90

Sales1   Other     Sales1   Other
Clerical1 Manufacturing Blue     Clerical1 Manufacturing Blue
Professional Managerial Service Blue Collar Collar Professional Managerial Service Blue Collar Collar

Women: Intercept ............. Education ........... Age ....................

Nonwhite ............

R: ..................... C ....................... N ......................

Men: Intercept ............. Education ........... Age ....................

Nonwhite ............ RE ..................... C ....................... N ......................

NOTE.-Absolute values of t = estimatelSE are in parentheses

* Very strong evidence of effect is demonstrated by It1 2\'ln .Z + 10. t Strong evidence of effect is demonstrated by It( 2din.\-+ 6.



Prestige ................. .037* .059* .099* .094*
      (5.22) (9.08) (9.95) (10.78)
Autonomy .............. .143* .136* ,034 ,037
      (11.64) (10.12) (3.21) (3.60)

NOTE.-Absolute values of t = estimate1SE are in parentheses

* Very strong evidence of effect is demonstrated by Itl 2 dln S + 10

the preferences of employers and workers for the characteristics of their employment partners.14 There are separate sets of estimates for each of the five categories of employers, as well as for workers, who are modeled as sharing a single preference structure in each of the gender-period sam- ples. Asymptotic t-statistics are shown in parentheses, with daggers indicating effects for which there is strong evidence according to the BIC criterion and asterisks indicating very strong evidence (i.e., It 1 2 dln LV + 6, and It 1 2 vln LV + 10, respectively; see Raftery 1995).15

The estimates in table 2 show several interesting patterns. First, each gender-period section shows that education is most highly valued by professional employers, as would be expected; their preference coeffi- cients range from .674 to .988, with an average t-statistic of 15.1 (BIC indicates very strong evidence for each effect). The values placed on education by managerial employers are substantially smaller, ranging from .228 to ,431 (though BIC indicates very strong evidence that they are nonzero) and overlap the range of estimates for sales/clerical/service employers, which is .138-.3 70. Perhaps surprisingly, manufacturing blue-collar employers seem to place a negative value on formal education,

l4 All reported estimates and supplementary statistics were obtained with TSLogit, version 1.0, a Fortran program written by the author to implement the EM maximum- likelihood algorithm developed in Logan (19966). The program has many features intended to increase its flexibility and ease of use. It and a user's guide are available from the author.

'' The asymptotic "t-statistics" are the estimates divided by their estimated SEs. The EM algorithm, unlike other estimation methods, does not automatically produce estimates of SEs. Though iterative methods are available for finding SEs with the EM method, the estimates shown here were calculated using analytical formulas for the first and second derivatives in order to save considerable computer time. BIC calculations are justified for the TSL model since it satisfies classical regularity condi- tions: the likelihood is twice continuously differentiable in the parameters, and the range of the data values is not restricted by the parameters.

with preference coefficients ranging from -0.691 to -1.068, an average absolute t-statistic of 4.4, and very strong evidence of each effect ac- cording to BIC. Among other blue-collar employers, there is weaker evidence of the same pattern. The negative coefficients imply that the probability of manufacturers' offers of blue-collar employment decline with workers' years of education. This may be a reflection of the average blue-collar employer's distaste for worker traits associated with formal education, or it may reflect an omission of other relevant variables in the model.l6

Age is positively valued by managerial employers in the two male samples but in no other categories, using BIC as the criterion. This finding for men is plausible, since promotion to managerial positions often comes with experience and therefore age. The managerial category also includes self-employed managers (entrepreneurs as well as farmers) for whom the acquistion of sufficient capital to become self-employed may take substantial time, that is, age. The age effects for women in this category are significant by the conventional standard of t 2 2.0 but not according to BIC.

l6 It may be tempting to conclude that the negative coefficients in the less desirable blue-collar occupations are simply an implication of the positive value of education for obtaining white-collar work, but this is not the case. Though the positive valuation of education by white-collar employers does imply that low education will be associ- ated with blue-collar employment, unemployment, or both, it does not imply that the estimated preference coefficients of blue-collar employers will be negative. It is entirely possible in the TSL framework for employers in each category to have the same, positive valuation of education. Then, assuming that workers value the characteristics of white-collar work over blue-collar work, white-collar employers will obtain more of the highly educated workers, with blue-collar employers obtaining the most highly educated among the workers not taken as white-collar employees, and with the least educated workers winding up unemployed. Lack of education would then be associ- ated with blue-collar employment and unemployment, but the TSL model would correctly estimate the positive coefficients for education among both blue-collar and white-collar employers. The actual estimates suggest blue-collar employers dislike education. Note in this connection that the employers' preference parameters are not multinomial or polytomous logit coefficients, but binary logit coefficients (of the component binary logit models implied in eq. [S]). Unlike polytomous logit estimates, there is no common reference category that gives the magnitudes in the J sets of binary logit estimates their meanings. Instead, the reference alternative giving meaning to the coefficients differs for each set: it is the possibility in that set that the employer will not offer a job to the individual (i.e., [3] is the reference alternative for [2]). This reference alternative for each employer's decision cannot be associated simply with the unemployment outcome, since an employer's decision not to offer a job does not necessarily lead to unemployment for the individual but may instead lead to employ- ment elsewhere. In thinking about the TSL estimates, it is important not to mechani- cally transfer rules for interpreting multinomial logit coefficients or polytomous condi- tional logit coefficients over to the new model; the best way of understanding TSL estimates is to consider the lower-level utility functions, eqq. (2), (3), and (8), directly.

The coefficients for the nonwhite dummy variable are strongly negative for managerial occupations in the male samples: the coefficients are -1.570 and -1.693, respectively, in the seventies and eighties, with BIC indicating very strong evidence. For women, there is strong evidence of a negative effect in the eighties but not in the seventies; the magnitudes of the two estimates are also lower for the women. In no other occupa- tional category is there strong evidence for an effect of the nonwhite dummy. These results suggest that qualified nonwhites' opportunities for advancement to higher-level positions are most seriously blocked in managerial occupations, while the professions are relatively open (assum- ing sufficient education). The estimated effects of race on managerial opportunity are very strong: the eighties' estimated coefficient of -1.693 transforms the 69% chance of a managerial offer for a college-educated white male to a 29% chance for the corresponding nonwhite male." There is no strong evidence for apparent racial discrimination in other occupational categories; the effect appears to be localized in management.

Regarding the simultaneously estimated preferences of workers shown in table 3, it appears that men and women place a positive value on prestige but that men value it more strongly. Women appear to value autonomy more than prestige, while there is no strong evidence of an autonomy preference for men in this model.

The demand intercept values in table 2 were obtained after centering the continuous characteristics of workers on their means during estima- tion. This allows the intercepts to be transformed directly to the estimated probabilities of offers within occupations for individuals with gender- period mean values of education and age and zero values on nonwhite. So, for example, the probability that a white female with average values of education (13.25 years) and age (33.65 years) would be able to obtain

I' Calculations of implied offering probabilities are described below. How much of this apparent racial effect among managerial employers may be due to the preferences of banks and other financers of the self-employed managers rather than to the prefer- ences of the employers of the salaried managers was examined in a separate run combining the males' data for the two decades and removing self-employed managers and farmers from the managerial category to form a sixth category. (Here I am using double asterisk and asterisk to mean very strong and strong evidence, respectively, by the BIC criterion [Raftery 19951.) The estimated coefficient for nonwhite was then -1.876 (1 tl = 7.03"") for the remaining, salaried managers (and nonretail salesper- sons), compared to -1.747 (1 t 1 = 4.91**) for the self-employed managers and farmers; strong racial discrimination is suggested in both subcategories. Age was also found to be significant in both categories (salaried = ,082, with It1 = 6.30**; self-employed

= ,065, with It1 = 5.0**). The effect of education was stronger for the salaried than for the self-employed (.579, It1 = 12.1**, vs. .120, It1 = 4.3"). The constant term was much more negative for the self-employed, indicating less overall opportunity at comparable levels of qualifications (-1.527, SE = ,097, vs. -,205, SE = ,122, for the salaried). All of these results seem plausible.

a professional offer in the 1982-90 period is given by a simple transforma- tion of the relevant demand intercept: the probability is exp(- 1.563)/ [l + exp(- 1.563)] = 0.173, as implied by equation (5). To find the estimated probability of an offer for an otherwise similar woman with 16 years of education, it is necessary to subtract the overall mean from 16 before applying the formula: this probability is exp[ -1.563 + .791(16 -13.25)]/{1 + exp[-1.563 + .791(16 -13.25)]) = 0.648. The differ- ence indicates the strength of the preference for formal education among employers of professional women. For employers of professional men in the period, the corresponding offering probabilities are 0.15 7 and 0.5 17. (In comparing these estimates it is important to keep in mind that many of the women's professional opportunities are in traditionally women's professions, which are generally controlled by a different set of employers than those who grant access to what are traditionally men's professions.)

Interpretation of Parameter Estimates

The estimated preference coefficients in tables 2 and 3 can be interpreted most directly and naturally as the relative weights that employers and workers place on each other's measured characteristics. For example, in the estimates of preference coefficients for managerial employers of women in the eighties, the values of .3 11 for education and -.864 for nonwhite imply that being white was equivalent to .864/.3 11 = 2.78 years of education in the eyes of the employers. To managerial employers of men in the same period, the estimated value of being white was equal to 1.693 / .43 1 = 3.93 years of education. It is also permissible to compare the same coefficients across types of employers, to say, for example, that professional employers of women valued education approximately two and one-half times as much as managerial employers of women during the eighties (.791/.311 = 2.54).18

Aside from these direct interpretations of the coefficient estimates as evaluative weights, various indirect interpretations are helpful for under- standing the implications of the model. One interpretation involves a parallel with linear regression, where a coefficient's value gives the abso- lute change in the observed, dependent variable that results from a unit change in an independent variable. The TSL model does not allow this exact interpretation, since there is no observed quantitative dependent variable. However, a similar interpretation can be made for the change in opportunity implied by a unit change in an independent variable.

This interpretation is valid because the error terms in the utility functions of all employers are constrained to have the same variances. See the explained-variance interpretation of the parameters below.

Occupational opportunity is defined in the TSL model as the chance that an individual would be offered work in a given occupational cate- gory, if he or she desired it. This chance can be expressed either as a probability or an odds without any fundamental change in concept. Thus the 0.173 probability calculated above for a professional offer to a white women of average age and education in the eighties corresponds to odds of 0.1731(1 -0.173) = 0.210.'~ With a single additional year of educa- tion the probability becomes exp[ -1.563 + .791(1)]/{1 + exp[-1.563

+ .791(1)]) = 0.316, corresponding to odds of 0.462. The log of the ratio of these two odds, ln(0.46210.210) = 0.791, gives a measure of the in- crease in opportunity associated with a increase of one year in education. It is easy to confirm that this log odds is mathematically equal to the estimated preference coefficient for education reported in table 2. The general rule is that, for any worker characteristic, the magnitude of the preference coefficient is an estimate of the change in opportunity associ- ated with a one-unit change in the characteristic, where the change in opportunity is measured by the log odds of offers with and without the change in the characteristic.

A second indirect interpretation of the preference coefficients concerns the relative rankings of different workers made by employers. Under the model, the employers within each occupational category share a utility function that serves to rank all potential workers from low to high as desirable employees. Consider two workers who differ on their observed characteristics only in that the first has one more year of education than the second. Then equation (2) gives the respective utilities for the two workers, Uj(l) and Uj(2). Since each utility contains a random distur- bance, it is not clear whether Uj(l) actually exceeds Uj(2) in any particular case. But since the disturbances in the utilities for the two workers are independent, the mathematics that leads from equations (2) and (3) to equation (5) can be adapted to show that the probability the unobserved utility for the first worker exceeds that for the second is:

where is the coefficient of education in the employer's utility function. The log of the odds that the first, unobserved utility exceeds the second

l9 Calculations were carried out with more digits than reported here

is equal to log[exp(P ,)I = P , = 0.79 1. A one-year difference in education between two otherwise similar women in the eighties thus implies a log odds of 0.791 that the woman with the greater educational attainment actually ranks higher than the other in the estimation of professional employers; this is the second indirect interpretation of the numerical value of the education coefficient.

The interpretation of the preference coefficients as the log odds that observed differences in characteristics indicate concordant rankings by employers also has an implication concerning the stratification of workers by measured characteristics. If we consider the difference in employers' utilities between otherwise similar women who have a four-year differ- ence in years of education, for example, the log odds that the more highly educated woman is preferred by professional employers is 4(0.791) = 3.164, corresponding to a probability of 0.959; at eight years of difference the probability becomes 0.998. Thus it becomes increasingly unlikely that women of such different levels of education will be substantially interspersed in the preference rankings of professional employers. Simi- larly, if the preference coefficient itself were to rise over time, the same calculation would show that workers of different levels of education were increasingly less likely to be interspersed in the preference rankings of the employers; there would be increasing stratification of offers of em- ployment according to levels of workers' educations and thus increasing stratification of the obtained professional positions as well. If the educa- tional preference coefficients of employers in all occupational categories were observed to rise over time, and if workers maintained a constant set of preferences across the occupational categories, then increased strat- ification by education levels across occupational categories would be im- plied.

A third and final indirect interpretation of the coefficients follows from observing that when workers' preferences are constant, occupational out- comes in the model depend only on the employers' staffing requirements and their rankings of workers. The staffing requirements being deter- mined by economic factors, a key question for sociological research is how completely the employers' rankings can be determined from the available data. Since the employers' rankings are completely determined by their unobserved utilities for the workers, the question of the com- pleteness of the determination of rankings by the data is answered in practice by calculating the proportion of the variance in the employers' utilities that can be explained by worker characteristics. Such a calcula- tion is easily made. If the systematic part of the employers' utility for a worker is defined as Vj(i) = Pjxi + mj, then the total utility given in

(2) is Uj(i) = Vj(i) + elij. Call the sample variance of Vj(i) across all

cases sj .20 Since the variance of the standard Gumbel disturbance is fixed at .ir2/6 and the disturbance is assumed independent of the systematic part of the utility, the variance of the total utility is estimated by sj + .ir2/6, and the proportion of the total variance in utilities that is attribut- able to the measured characteristics of workers is estimated by R; = sj/(sj + .ir2/6). Thus R; provides a summary measure of the joint impor- tance of the complete set of measured worker characteristics for the em- ployers in a particular occupational category. Its advantage over a simple examination of the magnitudes of the preference coefficients is that it simultaneously considers the contribution of all measured worker charac- teristics, while accounting for the observed variability of each. However, it should be emphasized immediately that R; is not a measure of the goodness of fit of the model as a whole or in part, and its values are not meaningful if the model does not fit well for a particular occupational category. Goodness of fit for the model as a whole and with respect to particular categories will be considered below.

The R; values shown in table 2 indicate that, for men, the worker characteristics included in the model-education, age, and race-explain over 70% of the variance in the utilities for employers of professional and manufacturing blue-collar workers, about 50% for employers of managerial and other blue-collar workers, and a negligible amount for employers of clerical workers. The patterns are somewhat similar for the women's samples, but evidence is presented below indicating a lack of fit in some occupational categories for these samples, so it is inappropriate to assert that the R; values are meaningful measures of the proportions of variance explained for employers' decisions regarding women workers.

Selection of Variables

Since the EM algorithm produces maximum-likelihood estimates, it is a simple matter to apply likelihood ratio tests for variable selection and nested testing of alternative models, just as in structural equation or log-linear models. Alternatively, if the researcher has no strong theoreti- cal reasons to prefer a particular set of independent variables, the use of BIC statistics may be preferable (Raftery 1995). Both approaches are illustrated here.

20 The actual estimate of s: used below is obtained with a formula that weights each case by the probability that an offer from an employer in the category would be accepted; this appropriately discounts contributions from cases whose outcomes are relatively less informative about the utilities in the relevant category. The effects of this weighting procedure are not large, however.

Table 4 presents statistics relevant to determining whether any of the variables in the model in tables 2 and 3 might be dropped. This model itself is labeled model 1 in table 4, with succeeding rows of the table corresponding to models dropping one or more job or worker characteris- tic. The number of estimated parameters is indicated for each model, as well as the identity of the model within which each successive model will be tested when using the likelihood ratio approach.

The final row of the table, model 8, concerns the null TSL model, in which workers and employers have no preferences for any measured characteristics of their potential partners and the demand intercept pa- rameters are fixed at "0," implying a 50% chance of an offer from each category for each case. The log likelihood for the null model is Lo = No log Po+ (N -No)log[(l -Po)IJ], where N is the total sample size; No is the number of unemployed in the sample; J is the number of occupational categories (not counting unemployment); and Po = (2/+' -1)/[2/(J + I)] is the probability that unemployment will be the outcome (the appen- dix shows the derivation). This model is used as the baseline for calculat- ing the BIC' test statistics shown in the table (see Raftery 1995).

The columns labeled 2LL report twice the log likelihoods for the vari- ous models; differences in entries between nested models in the same columns are distributed as X2 under the null hypothesis that the coeffi- cients for variables omitted from each nested model are "0" in the nest- ing model. Asterisks beside the likelihood statistics indicate that all nested models can be rejected in favor of model 1, using P < 0.05 as the crite- rion, except in the case of the seventies women's sample, where model 6 is not rejected. Thus the likelihood ratio tests suggest that all variables in model 1 should be retained in both the men's samples and in the eighties women's sample but that the age variable should be omitted in the seventies women's sample.

The BIC statistics give different results. In both women's samples, model 7, which omits age and nonwhite, is preferred to any other model, though the evidence against including nonwhite is not strong for the eighties. BIC suggests the full model 1 for the seventies men's sample but indicates, though not strongly, that age should be dropped for the eighties sample. Thus the BIC results are also somewhat ambiguous, suggesting that neither age nor race is important in the women's models, but that race should be included in the men's models while age may be less important (see Raftery [I9951 for the definitions of strong and very strong evidence used here).

The likelihood ratio and BIC procedures are presented here to demon- strate their application to the TSL model. Since I have a relatively strong prior belief that race and age are relevant to hiring decisions, and since


LR 1972-80 1982-90 1972-80 1982-90
MODEL Worker Employer INTERCEPTS PAKAMETERS MODEL (+6,380) (+1,850) (+9,210) (+2,320) (+7,010) (+1,520) (+8,710) (+2,030)
  Prestige and Education, age, Yes    
  autonomy nonwhite      
  Prestige Education, age, Yes    
  Autonomy Education, age, Yes    
  Neither Education, age, Yes    
  Prestige and Education, age Yes    
  Prestige and Education, non- Yes    
  autonomy white      
  Prestige and Education Yes    
  Neither None No    

NOTE -LR = likelihood ratio, 2LL = twlcr the log likelihood Nos in parentheses show quantities that were added to the respective columns of statistics to make them easier to examine in the table Indicates statistics pertaming to models preferred elthcr by the likelihood ratio or BIC procedures

* Llkcllhood ratlo chi-square prababllity of fit wlthin nesting model is 5 0 05

the number of variables under consideration is not excessive, I will retain model 1 for further analysis in all samples.

Goodness of Fit

Neither the likelihood ratio nor BIC provides a suitable test of the abso- lute goodness of fit of individual-level models such as TSL (cf. Hosmer and Lemeshow 1989, pp. 136-40; Raftery 1995, n. 6); they are tests of relative fit. Applying the Hosmer-Lemeshow test within each occupa- tional category provides a reasonable check on the absolute goodness of fit of the TSL model.'l The Hosmer-Lemeshow statistic, 6, allows a chi-square test comparing the proportions of outcomes actually observed in each decile of risk with the proportions that are expected according to a model. For each occupational outcome, deciles of risk are formed across the observations by collecting the first 10% of cases that have the lowest predicted probabilities of accepting a job within the occupation, followed by the 10% with the next lowest probabilities, and so on. The test statistic should be distributed approximately as X2(8)when calculated as just described.

c statistics are shown in table 2 for each sample and.occupation. For 8 df, the expectation of x2is 8.0, and critical values for the null are 13.36 for P = 0.10, and 15.51 for P = 0.05. The range of e values for the men's samples, from 3.49 to 15.47, suggests that the fit is acceptable. On the other hand, four of the 10 statistics for the women's samples are unacceptable. Though introducing, for example, nonlinear terms in education does improve the fit for women, I prefer to avoid arbitrary curve fitting and conclude instead that the models do not include suffi- cient relevant variables to fit women's observed outcomes very well in an absolute sense.

Tests for Structural Change

Taking both the men's and women's models in the two decades as provi- sionally acceptable, it is interesting to ask how much evidence the data give for changes in either the workers' or employers' preferences between

21 I am following the spirit of Hosmer and Lemeshow's own recommendation for assessing the fit of the multinomial logit model, which is to calculate separate binary model test statistics for each outcome category, even though neither theoretical nor simulation justification is given for their recommendation. I believe the results should be approximately correct here as well.

the seventies and eighties. Here again either likelihood ratio or BIC meth- ods can be applied. Table 5 shows relevant statistics.

All the models in table 5 contain the full set of preference coefficients from the preferred model 1 of table 4: that is, they all contain employers' preferences for education, age, and nonwhite status, and workers' prefer- ences for prestige and autonomy. The models in table 5 differ in the specification of which preferences will be held fixed across the seventies and eighties and which will be allowed to differ. (Technically, interac- tions with a period dummy are introduced to free the estimates across periods.) Abbreviating the consideration of table 5 to conserve space, the fact that the smallest BICt statistic for the men's data is found in model 9 suggests that there is no evidence for any change in the circumstances of opportunity of men between the seventies and eighties, including changes in the overall levels of demand in the five categories. For women, the minimal value of BICt in model 10 indicates very strong evidence of changes in overall demand in the categories (demand rises strongly except in blue-collar manufacturing occupations); there is no evidence of change in the preferences for the measured characteristics of education, age, and race, however. That model 10 for women is only weakly favored over model 11 does suggest that an increase in women's preferences for high prestige jobs should not be ruled out entirely.


Having developed the TSL model and applied it to questions of occupa- tional opportunity and choice in the United States, I include in this sec- tion some comments about the differences between TSL and other mod- els, comments that would have been less understandable if made earlier in the article. The models to be reviewed in this section share the goal of understanding how jobs with differing characteristics come to be held by different types of individuals, though they differ in the ways in which the job outcomes are characterized, measuring them variously by socio- economic status, occupational category, or income. Both economic and sociological models are considered, with some emphasis on the long- standing sociological goal of estimating relevant effects independently of economic influences on overall demand for particular occupations. Read- ers less interested in the relationships of TSL to other models may skip to the next section ("Generalizations and Extensions") without loss of continuity.

Linear Regression and Status Attainment Models

By far the most productive method for understanding individual-level determinants of occupational outcomes has been direct, linear regression

of employment outcomes on individual characteristics. This type of re- gression appears in structural equations models as well as in the single equation models often used by economists. The proposed TSL method can be used to evaluate the importance of the same types of variables studied with regression-such as education, experience, gender, and race- which raises the question of the relative benefits of the two approaches.

There is no doubt that linear regression is appropriate for obtaining descriptive knowledge of the background factors differentiating subjects on a quantitative dependent variable in a given set of labor market condi- tions. Linear regression has much to recommend it in its well-developed treatment of measurement issues, indirect causation, and many other technical points. However, there are certain advantages of TSL over linear regression. TSL estimates have a direct, behavioral interpretation as the preferences of employers and workers and, consequently, a direct relationship to relevant sociological ideas of structure, briefly discussed below. TSL estimates of the importance of various individual characteris- tics are also differentated by occupational category, so that a trait like formal education, as was seen above, can have different effects on chances of employment in different occupations. In contrast, in linear regression the effect of a variable like education appears uniformly bene- ficial or harmful no matter what the range of occupations an individual may be destined for; TSL estimates can and do contradict this ap- pearan~e.'~

Because of its construction from an underlying random matching model, TSL estimates have an invariance property under conditions of change in the demand for occupational categories that is not shared by linear regression estimates. The lack of this property in linear regression means that estimates of the importance of individual traits will change when demands for occupations change, even if employers maintain the same preference coefficients for individual traits. Logan (1996b) presents a simulation study that demonstrates the sensitivity of linear regression estimates to demand changes and the approximate stability of TSL esti- mates.

Log-Linear and Related Multinomial Models Log-linear and related multinomial models have been sociologists' main tools for research on structural factors in occupational mobility for 25 years. Though primarily restricted to the analysis of categorical variables alone, methods for extending log-linear models to include continuous or

22 See the earlier elaboration of this point in n. 6 above.

categorical background variables have long been available (e.g., Yama- guchi 1983; Logan 1983; Hout 1984; DiPrete 1990; Breen 1994). The ability to combine continuous and categorical variables is shared by TSL, and TSL could also be applied to a multivariate analysis of mobility table patterns. This makes the relative merits of TSL and log-linear and related multinomial models important to consider. An advantage of log-linear and related models is their high state of technical development and the ease with which they can be estimated using standard software. Like a simple regression, these models can be regarded as providing a description of the bivariate distribution of data in a specific set of histori- cal circumstances. When augmented with background variables, the models describe the bivariate categorical distribution of parents' and offspring's occupations conditional on these variables.

However, log-linear models have no mathematical relationship to any behavioral model of actors. This makes it hard to determine what the mechanisms of the various structural effects proposed in the models might be. Is a "barrier" in a log-linear model to be associated with the actions (or inactions) of persons or employers blocking the individual's mobility? If so, it is not clear how the parameters are related to these actions. By comparison with the TSL model, the structures of log-linear models are vague and allusive: disembodied "barriers," "levels," and symmetries abound, untied mathematically to any behavioral referents. In the TSL model, all parameters have behavioral referents.

The lack of behavioral referents for the log-linear parameters makes it difficult to justify the claims of log-linear modelers that the parameters describing association within tables are insensitive to exogenous changes in occupational structure. In fact, the intercept parameters of the TSL model, pj, in equation (6) above, are able to fit changes in marginals due to changes in demand for occupations while the preference parameters of the model remain constant, just as the marginal parameters of log-linear models can accommodate changing outcome distributions while the asso- ciation parameters stay constant. Thus the mere fact that a parametric structure accommodates shifting marginals does not automatically justify it as a model of opportunity or mobility, since both TSL and log-linear models satisfy this requirement using completely different structures. Simulation with the behavioral matching model shows that exogenous changes in demand do not significantly affect TSL estimates of prefer- ences, while the parameters of comparable multinomial models are strongly affected (Logan 1996b; see Harrison [I9881 for related observa- tions). On the other hand, it is not possible to simulate a system of workers and employers in which the log-linear/multinomial association parameters are constant under changes in demand: the reason is that log-linear/multinomial models have no behavioral implications that can be simulated.

Discrete Choice Models

Economic discrete choice models, briefly considered in "The TSL Model" section above, seem to provide a natural framework for the study of occupational outcomes. Boskin (1974) provides an early example of such an application, using the polytomous conditional logit model. The problem with the direct application of such a simple model to occu- pational choice is that the standard models assume either that all alterna- tives are freely available to all workers, or that the researcher can deter- mine which alternatives are available to which workers a priori. As Pudney (1989, p. 17) puts this assumption, "In general in applied work [the set of available alternatives] either is observed directly or can be constructed without statistical estimation." While certain alternatives can reasonably be ruled out for decision makers in certain types of stud- ies, as when households not owning cars are excluded from the private automobile alternative in transportation studies (e.g., Swait and Ben- Akiva 1986), the breadth and imprecision of occupational categories make this strategy impractical in the present context.

Manski (1977), in an article developing the foundation of discrete choice models, introduced the idea of random choice set generation as a formal means of accounting for unobservable differences in the alterna- tives available to different workers. In this formalism, workers are repre- sented as having a probability distribution over access to all the possible sets that can be generated from the basic list of alternatives. Swait and Ben-Akiva (1986, 1987), building on this work and on Ben-Akiva (1977), developed and estimated such a model empirically in a study of transpor- tation mode choices. Their Independent Availability Logit (IAL) model assumes that there is a constant probability across workers that each alternative will be available and that these probabilities are independent across alternatives. However, though Ben-Akiva (1977) had suggested making the probabilities of the alternative availabilities functions of indi- vidual characteristics, and though Andrews and Srinivasan (1994) have recently estimated a model with such individually varying probabilities in a marketing study, no known studies using random choice set genera- tion have specified the choice set availabilities as dependent on another set of actors, as is done here.23

23 The IAL model was unknown to me when I first developed and estimated the TSL model, as was Manski's (1977) theoretical analysis. Though he did not develop a choice-set generation formulation that explicitly represented other actors making alter- natives available, Manski did suggest the possibility verbally (p. 234). Swait and

Game-Theoretic Matching and Job Search Models

The addition of a second set of actors in the random matching model produces what can be considered a stochastic specification of a determin- istic two-sided game called the college admissions model in the game theoretic literature (see Roth and Sotomayor [I9901 for a review of rele- vant game theory models). However the college admissions model is a centralized matching game, while matches are formed in a decentralized manner in the random matching model described above. Formal analysis of the random matching model as a game produces new insights into the behavioral characterization of the model, reveals strategies that employ- ers and workers may try to implement, and leads to a related process interpretation that helps justify the TSL estimation method (Logan 1996~).

Game theoretic models have been developed as abstract accounts of behavior in labor markets, with limited comparisons of theoretical pre- dictions to the behavior of real markets (see Mortensen [I9881 for an additional review). Various existing refinements of the deterministic col- lege admissions model offer possible directions of extension for the ran- dom utility matching model. However, the game theoretic literature sometimes tends toward hyperrational analyses in which workers must make difficult calculations of optimal strategies; sociological development of the random matching model should avoid such tendencies. The TSL model, nonetheless, is, so far as I know, the first applied random utility specification of a two-sided matching game and could be extended in a hyperrational direction as such. Use of the TSL model allows para- meter estimation under the assumption that a formal game is being played.

Economic job-search models closely related to game-theoretic match- ing models (see Mortensen 1986, 1988; Devine and Kiefer 1991) also offer potential directions for extending the random matching model. In job search models, the quality of matches between employers and workers is assumed to be difficult for the actors to evaluate initially, which leads to the formation of poor initial matches. Explicit models of information flow among actors then lead to plausible deductions about the rates of

Ben-Akiva (1986, p. 76) express reservations about the implication of their IAL model, expressed in its name, that the availabilities of the various alternatives to individuals are statistically independent. Under IAL there should not be an association between, say, the probability of being offered a professional and a managerial job. However, the TSL model does not share this property, even though it is assumed that the disturbance terms are independent across categories. In TSL, the systematic depen- dence of the offering probabilities on the characteristics of the individuals receiving offers can be expected to induce cross-individual correlations among the offers ob- tained from different categories of employers.

match formation and dissolution and the properties of equilibrium states the market should approach through time. Refinements along these lines could lead to improved random matching specifications that still have a bounded rationality character.

In many game and job search models a principal resource that employ- ers offer in jobs-namely, wages-is not considered to be a fixed prop- erty of the job, but rather to be negotiated between potential employment partners (Roth and Sotomayor 1990; Mortensen 1988). This type of speci- fication is rejected in the sociological random matching model for reasons familiar to readers of Thurow (1975) and Sgrensen (1983, 1986).

Economic Assignment Models of the Distribution of Earnings

A broad category of economic models, reviewed by Sattinger (1993), considers the problem of individual occupational outcomes by taking the idea of employers negotiating wages to its logical conclusion. In these economic assignment models, the distribution of workers into jobs, which are typically classified by industrial sector rather than occupati~n,~~

is assumed to occur not by virtue of any rejection of potential workers by employers, but by employers optimally setting the wages that each individual would be offered to perform each particular job. In Sattinger's words, these models derive "the wage differentials that are consistent with an equilibrium assignment of workers to jobs. The equilibrium wage differentials are those that yield equality between amounts of labor supplied and demanded in each [sectoral] submarket of labor" (p. 832). The models produce predicted distributions of workers across sectors as a function of the preferences and resources of workers and employers, just as the random matching model does." Some have also been devel- oped statistically so that the preferences of workers and employers (classi- fied by sector) can be estimated, as in the TSL model.

The key difference between these models and the random matching model is the assumption that employers adjust wages to clear the market, as workers engage in wage competition for jobs. That is, to make employ- ment in a sector just sufficiently remunerative to attract the necessary number of workers, and no more, employers raise or lower wages. In contrast, the random matching model assumes that the demand for em- ployment in an occupation at prevailing wages may exceed the supply of jobs, and that employers may choose among an excess of willing candi-

24 The TSL model could also be estimated with a sectoral classification of employers,

but the occupational classification is more relevant to sociological questions. 25 The terms "preference" and "resource" are my characterizations and not the econ- omists', who use a much more detailed and theoretically grounded terminology.

dates rather than lowering wages until the supply of labor just balances demand. As a result, queuing and job competition, as in Thurow (1975) and Sgirensen (1983, 1986), may be induced. It is perhaps worth saying that such an exercise of choice by employers seems rather common, even though it is removed by assumption from the economic assignment models.

Sattinger (1993, p. 842 n. 13) nicely differentiates the type of model I am proposing from the economic assignment models, when he refers the reader to another article: "Dale Mortensen (1988) reviews matching problems related to the assignment problem. In this literature, matches are formed through the voluntary actions of agents rather than as the solution to an aggregate maximization problem." He also (p. 834) distin- guishes the economic assignment models from "structuralist theories in sociology . . . [which] do not assume competitive access to jobs," citing Granovetter (1981) and Kalleberg and Berg (1987) as examples. The ran- dom matching model underlying TSL is a structuralist model in this sense, since it excludes (wage) competitive access to jobs; TSL is proposed as suitable for use in empirical studies of ideas developed in sociological structuralist theories.

Actor-Oriented Sociological Models A relatively formal research tradition has mathematically modeled the competition of workers for advancement within systems of opportunity structured either according to the needs of employers or as a result of power struggles between workers and employers. White's (1970) vacancy chains, Boudon's (1974) box model, Thurow's (1975) job competition, Sgirensen's (1977) vacancy competition, and Stewman and Konda's (1983) venturi models are examples. The theorizing underlying these models is actor oriented, and systemic properties of the theorized systems are clearly retained in the translation to mathematics. A key idea is that opportunities for employment or advancement are contingent on the dis- positions or needs of employers, often discussed in terms of vacancies in fixed job positions. Certain of these models have been faulted for provid- ing too rigid a representation of structure, in which nothing but a vacancy can lead to increased achievement (Rosenbaum 1984),'~ but a main prac- tical drawback has been a lack of a flexible statistical technique suitable for weighing the relative importance of measured variables in the alloca- tions of jobs. Coleman (1991) developed a model that represents occupational out-

26 This criticism does not apply to TSL.

comes as a system result of the distributions of preferences and resources among employers and workers, as does the random matching model. He emphasizes the importance of a system approach to opportunity, in which outcomes are determined "by the resources of the person, relative to those of other persons in the market, the resources of the job, relative to those of other jobs, the interest of the employer in the person's resources, and the interest of the person in the job's resources" (p. 3), and discusses very lucidly the advantages of such an approach in making calculations of expected changes in occupational distributions when demographic or other presumptively exogenous changes in circumstances occur. These sorts of calculations can also be made with the random matching model underlying TSL.

However, Coleman makes very different assumptions in order to achieve a model, assumptions that are economistic in flavor. He assumes a perfect market with an equal number of jobs and workers, discarding the possibility of unemployment. He assumes that all resources held by workers and employers have an equilibrium, a per unit, monetary, mar- ket value that prevails for all members of the system. Thus, unlike the random matching model underlying TSL and unlike assignment models such as Heckman and Sedlacek's (1985, 1990), there is no possibility of different valuations of resources by employers or workers in different occupational or sectoral markets. Because Coleman's model assumes a perfect market, and therefore a lack of sociological structuralist or even frictional explanations of intermarket differences in the values of resources, it is hard to see how it might be developed into a more differenti- ated and structural model. It is also unclear how Coleman's model could be used to generate a proper statistical estimation procedure useful for hypothesis testing, rather than the ad hoc procedure demonstrated in Coleman and Hao (1989).


The TSL model as developed here is closely tailored to the research problem of occupational opportunity. However, the general approach may find application in other sociological areas, which may or may not have the same characteristic data limitations as in the occupational case. This section describes other possible uses and interpretations of the model, as well as planned extensions appropriate for the analysis of occu- pational opportunity and similar problems.

In principle, the TSL model could be applied wherever individuals are faced with choices among a few types of actions or outcomes, where there are some external constraints on the choices due to other actors, and where these constraints are not directly observed by the researcher.

Obviously, this is a vast potential scope of application. What problems can really be addressed with TSL depends on the nature of the available data, on whether the researcher is interested in micro- or macrocharacter- izations of constraint, and, of course, on computational resources.

Problems in Which the Characteristics of All Alternatives Can Be

Measured Directly

If the researcher has access to data on the actual characteristics of the nonaggregated alternatives potentially available to each individual, the TSL model of equation (10) is directly applicable without recourse to category means. Promotion opportunities within firms might provide this kind of data, if there are known rules that allow approximate calculations of the relevant benefits (wage and nonwage) that each individual would obtain upon promotion. Only a few job alternatives could be differenti- ated with present estimation methods, though this can be expected to improve. There may be other research questions where a few, nonaggre- gated alternatives with known characteristics are relevant.

Problems in Which a Purely Macrolevel Representation of Opportunity

Is Intended

TSL was developed as a microlevel model composed of individuals and employers. When mean occupational characteristics are substituted for job characteristics, as in the GSS analysis, two divergent interpretations seem reasonable. The first is that the micro model has been misspecified and that questions of resulting biases in estimates of the microlevel utility function parameters should be pursued. This interpretation was empha- sized previously and is discussed further in the next subsection. The second is that the researcher's interest has changed from determining the microlevel preference coefficients of employers to characterizing the influences on opportunity at the macro level of occupational categories. That is, the researcher may be interested in modeling opportunity in broad categories as though it were determined by utility functions applying to the categories themselves, making opportunity a probabilistic function of the individuals' characteristic^.^' This approach would reject the interpretation of coefficient estimates as employers' preferences, but would retain the attractive interpretation in terms of opportunity: that is, the coefficients for each occupational category would represent the

*'I thank a persistent AJS reviewer for pressing this interpretation on me

shift in opportunity associated with a unit change in an individual charac-

teristic, just as before. Exactly how this change in opportunity might

arise from employers' preferences would be left unspecified.

Probably the closest parallel to the use of TSL as a macrolevel opportu- nity model would be an augmented log-linear model such as discussed by Logan (1983) or Breen (1994); those models use individual characteris- tics to predict probabilities of particular outcomes but contain no mi- crobehavioral components. In the case of Breen (1994), characteristics of the outcome categories are also used in the predictions. Both of these models are essentially one-sided, however; there is no differentiation of opportunity from choice, given opport~nity.~~

Therefore it is hard to interpret the coefficient estimates obtained from these models, and the peculiar mixtures of concepts that are often used to label log-linear mod- els' effects are consequently en~ouraged.'~ In contrast, the a coefficients as individuals' preferences and the P coefficients as opportunity parame- ters would retain their distinct interpretations in a comparable TSL macro model.

The range of applications of TSL macro models might include employ- ment outcomes classified by broad occupational or industry categories, by market segments, by traditionally women's versus men's occupations, or by other schemes involving small numbers of alternatives. Adding structures of dummy variables reflecting the occupational classes of origin of respondents would make it possible to address mobility-table questions now analyzed primarily with log-linear and related models. Aside from employment outcomes, TSL macro models might be used to investigate determinants of entry into selective and nonselective colleges, housing or insurance availability in neighborhoods of varying exclusivity, or perhaps admission to informal social circles.30

TSL models could take advantage of variability in the characteristics of alternatives measured at some level above the individual but less than the population as a whole. Making these characteristics vary, for exam- ple, by geographic region or by the background of the respondent could be expected to aid identification of the a parameter^.^^

28 Mathematically, the two models are special cases of TSL in which the opportunity

side has been suppressed. 29 For example, Breen (1994) calls certain effects "desirability/barrier," an amalgam of two concepts that in a two-sided model would correspond to effects from different sides of the model, the desirability corresponding to workers' preferences and the barriers to employers' preferences.

30 Most macro-opportunity model applications could be undertaken with the software developed for the present article, though mobility table models require a more flexible set of constraints on parameters than has been programmed so far.

31 Thanks respectively to Rob Mare and Yu Xie for these suggestions.

Extensions Directly Addressing the Micro-Macro

Opportunity Connection

Though it seems reasonable to interpret TSL as a macro model when category means are used for estimation, one principal drawback of this approach is that the detailed characteristics of the particular jobs held by each individual in the sample are not used directly. Job characteristics that are only moderately collinear at the individual level can show ex- treme collinearity at the level of category means, making it impossible to estimate their separate effects on job preferences. In addition, the number of categories defined limits the number of job characteristics that can be used: the limit is J -1 characteristics, so long as one set of category means is used for all respondents. It seems possible however to remove these problems by deriving estimation methods that explicitly account for the microlevel origins of the mean job characteristics in occu- pational categories. Such an approach should allow the actual character- istics of the particular job held by an individual to enter the estimation along with the mean characteristics of occupational categories and should also alleviate the bias in the estimation of the workers' preference coeffi- cients, which is relevant from the micro perspective. With such an esti- mation method, the aand P estimates could both properly be interpreted as the utility function preference coefficients of particular types of actors. This approach to estimation remains to be developed as a practical method but could greatly increase the ability of the model to differentiate various influences on individuals' job choices, within the constraints im- posed by employers.


The mechanism by which the preferences and resources of employers and workers determine the allocation of workers to jobs in a free economy is this: workers choose the jobs whose resources they prefer from the sets of jobs available to them; and employers choose the workers whose resources they prefero from the sets of workers they find available. To transform such a truism into a useful model, the preferences of both types of actors have been represented here by utility functions containing measured and unmeasured characteristics. A direct development of this simple idea leads to a two-sided logit, or TSL, model that provides joint estimates of the preferences of both workers and employers, using data only on the characteristics of individuals and the jobs they hold.

The TSL model considers the basic problem of occupational attain- ment from a point of view different from prevailing approaches. By mathematically describing the matching process that determines out- comes, it has been possible to break into interrelated pieces an estimation problem that prevailing techniques have considered unitary. Instead of directly predicting outcomes from the characteristics of individuals, as in regression, the TSL model tries first to explain the rankings that em- ployers make over workers and the rankings that workers make over types of jobs.32 These rankings are functions of unobserved utilities whose systematic parameters are estimated. The actual outcomes, the assignments of workers to jobs, cannot be predicted by the model without external information on the level of demand for jobs of different types, represented by the demand intercepts in each category. Since the estima- tion of preferences is separated analytically from the estimation of de- mand effects, TSL estimates of preferences are insensitive to changes in demand, unlike regression estimates and (as simulation with the model shows [Logan 1996b1) also unlike log-linear/multinomial estimates, if the simple matching process is granted to be a reasonable mechanism. TSL estimation attempts to find constancy or change in preferences over time (or, potentially, place), net of changes in demand.

Application of the model to GSS data did indeed show constancy of preferences of both employers and workers over the seventies and eight- ies, at least according to the BIC criterion, which is more conservative than the corresponding likelihood ratio tests. For women, only levels of demand in occupations seem to have changed between the decades; there were large positive changes in the demand intercepts for women in pro- fessional, managerial, and sales/clerical/service occupations, but there was little evidence of change in the importance women's employers placed on education, age, or race. Nor was there much evidence of changes in women's evaluations of prestige and autonomy. For men, the situation was simpler still: there was no evidence of change, period. Even the demand intercept estimates for men showed little evidence of change. The implication of these results is that the opportunity of men, defined as the chances of employment in the various occupations, stayed stable over the period, while the opportunity of women increased in ways that did not reflect changing valuations of education, age, or race.

Within this general pattern of stability, there are still interesting com- parisons to be made across occupational categories. In general, education was valued most highly by professional employers, then managerial and sales/clerical/service employers, while blue-collar employers appeared to value years of education negatively. This is an indication that blue-collar offers (and not merely outcomes) become less likely with more education. Apparent discrimination against nonwhites was localized in managerial

32 However, the rankings of types of jobs by workers have been deemphasized in this presentation, for the sake of simplicity.

occupations for men, with very strong evidence for the effect by the BIC criterion. For women there was strong evidence of this effect in the 1980s, but not the 1970s. In no other category besides management was there strong evidence of such discrimination, net of other characteristics, against either nonwhite men or women. Age appeared to be a factor in obtaining employment offers only for men, and then only for managerial jobs. The positive valuation of age for managerial offers may reflect the time it usually takes to obtain relevant experience and/or capital. These findings of differences in the determinants of opportunity across occupa- tions (which were generally replicated across the independent samples from the two periods) give insights into the structure of opportunity that are not available either from linear regression or log-linear/multinomial models.

The basic theoretical approach itself should be considered in evaluating the TSL model and merits some closing comments. It is evident that TSL is based on a model of choice, but it may be less clear whether this is a "rational choice model," as this term is commonly used. Rational choice has connotations that many sociologists find unpalatable as a basic characterization of behavior. Though this is not the place to develop the point, rational choice is not a single concept (see Sen [I9871 for a concise review). The idea of rational choice begins with the idea of consistency of choice but often proceeds to include the self-interested, forward-looking, calculating behavior associated with "economic man" and the idea of optimization (though in fact neither consistency nor self-interest implies the other, as Sen points out).

TSL stops at consistency of choice as the basic characterization of behavior that is necessary to estimate preferences. Clearly, without some consistency of choice across similar circumstances, no estimation would be possible. But there is no implication in the model that the choices are made on account of any self-interested calculations; instead, the choice behavior may result for moral or normative reasons, to satisfy collective rather than individual goals, because of habit (Camic 1986), or through the operation of the rules (Giddens 1984) or schemas (Sewell 1992) that theorists have persuasively argued may structure action for reasons lying outside the immediate situations of apparently atomistic actors making isolated choices.

By contrast, economic models of choice often assume some self-interested, optimizing calculations in their subjects and are often then able to simplify the inferential problem as a result. Heckman and Sedla- cek's (1985, 1990) model is a good example. By assuming that employers will optimally adjust wage offers for each individual, it is possible to pretend that individuals choose among jobs that are freely available to them in all sectors, just as they would in a simple conditional logit model.

The labor market is assumed to clear, in that no excess demand for jobs occurs at the offered wages. Though Heckman and Sedlacek's model is still difficult to estimate, the simplifying assumption of this optimal be- havior on the part of the employers has removed the choice behavior of the employers (replacing it with optimal wage setting) and has thus sidestepped the natural complexity of modeling interrelated choices on both sides of the market.

The point of retaining this complexity in the TSL model is to avoid introducing any assumptions of optimizing calculation on the part either of employers or workers, though without denying that such calculation often occurs. The absence of such assumptions is from the sociological point of view an achievement rather than an omission, a choice taken for theoretical reasons just as economists have for their own reasons tended to take the opposite tack. The TSL model is designed to measure the empirical determinants of opportunity and choice among employers and workers, whether the choices are economically rational or sociologi- cally structured, or some combination of both. Further application of the model may help discover the separate determinants of opportunity and choice in a variety of situations where two types of actors mutually con- strain one another.


Derivation of the Null-Model Log Likelihood

In the null model, all parameters are set to "0" in equations (5) and (9) of the text. This implies that the probability of selecting a job from any given category is simply llNi, where N, is the number of categories from which individual i has obtained offers, including the unemployment category, and that the probability of obtaining an offer from any category except unemployment is exp(O)/[l + exp(0)l = 112 for all individuals (the probability of an "unemployment offer" remains 1.0). The probabil- ity that i accepts a job in category "0," unemployment, is then given by reformulating equation (10) in the text to reflect the simplifying role of Niin the null case: using the identity n!l[k!(n -k)!] = 2". This null model prob(Aio) is the value called Poin the text. By symmetry, the probabilitiesof select- ing any categories besides unemployment must all be equal to one another and thus equal to (1 -Po)IJ. The log likelihood in the text follows from substituting Poand (1 -Po)lJ, respectively, for the probabilities of cases observed in unemployment and the employed cate- gories.


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