Encyclopedia of Sociology

CAUSAL INFERENCE MODELS

Z 2

X 2

X 3

X 1

Z 1

Neighborhood

delinquency

Parental

education

Guilt

Delinquent acts

Self-esteen

β 12

β 21

β 13

β 32

y 11 y 32

Figure 4. Nonrecursive Modification of Figure 1

many unknowns unless additional restrictive as-
sumptions are made. In particular, in each of the k
equations one will have to make a priori assump-
tions that at least k − 1 coefficients have been set
equal to zero or some other known value (which
cannot be estimated from the data). This is why
one needs the predetermined Zi and the relevant
gammas. If one is willing to assume that, for any
given endogenous Xi, certain direct arrows are
missing, meaning that there are no direct effects
coming from the relevant Xj or Z variable, then one

may indeed estimate the remaining parameters.
One does not have to make the very restrictive
assumptions required under the recursive setup,
namely that if Xj affects Xi, then the reverse cannot
hold. As long as one assumes that some of the
coefficients are zero, there is a chance of being
able to identify or estimate the others.

It turns out that the necessary condition for

identification can be easily specified, as implied in
the above discussion. For any given equation, one
must leave out at least k − 1 of the remaining
variables. The necessary and sufficient condition is
far more complicated to state. In many instances,
when the necessary condition has been met, so will
the sufficient one as well, unless some of the

equations contain exactly the same sets of vari-

ables (i.e., exactly the same combination of omit-

ted variables). But since this will not always be the

case, the reader should consult textbooks in econo-

metrics for more complete treatments.

Returning to the substantive example of delin-

quency, as represented in Figure 1, one may revise

the model somewhat by allowing for a feedback

from delinquent behavior to guilt, as well as a

reciprocal relationship between the two internal

states, guilt and self-esteem. One may also relabel

parental education as Z 1 and neighborhood delin-

quency as Z 2 because there is no feedback from

any of the three endogenous variables to either of

these predetermined ones. Renumbering the en-

dogenous variables as X 1 , X 2 , and X 3 , one may

represent the revised model as in Figure 4.

In this kind of application one may question

whether a behavior can ever influence an internal

state. Keeping in mind, however, that the concern

is with repeated acts of delinquency, it is entirely

reasonable to assume that earlier acts feed back to

affect subsequent guilt levels, which in turn affect

future acts of delinquency. It is precisely this very

frequent type of causal process that is ignored

whenever behaviors are taken, rather simply, as

‘‘dependent’’ variables.