Encyclopedia of Sociology

CAUSAL INFERENCE MODELS

In many instances, however, one cannot col-
lect the kinds of data necessary to ascertain these
lag periods. Furthermore, especially in the case of
aggregated data, the lag periods for different ac-
tors may not coincide, so that macro-level changes
are for all practical purposes continuous rather
than discrete. Population size, literacy levels, ur-
banization, industrialization, political alienation,
and so forth are all changing at once. How can
such situations be modeled and what additional
complications do they introduce?

In the general case there will be k mutually
interdependent variables Xi that may possibly each
directly affect the others. These are referred to as
endogenous variables, with the entire set having
the property that there is no single dependent
variable that does not feed back to affect at least
one of the others. Given this situation, it turns out
that it is not legitimate to break the equations
apart in order to estimate the parameters, one
equation at a time, as one does in the case of a
recursive setup. Since any given variable may af-
fect the others, this also means that its omitted
causes, represented by the disturbance terms εi
will also directly or indirectly affect the remaining
endogenous variables, so that it becomes totally
unreasonable to assume these disturbances to be
uncorrelated with the ‘‘independent’’ variables in
their respective equations. Thus, one of the critical
assumptions required to justify the use of ordinary
least squares cannot legitimately be made, mean-
ing that a wide variety of single equation tech-
niques discussed in the statistical literature must
be modified.

There is an even more serious problem, how-
ever, which can be seen more readily if one writes
out the set of equations, one for each of the k
endogenous variables. To this set are added anoth-
er set of what are called predetermined variables,
Zj, that will play an essential role to be discussed
below. Our equation set now becomes as shown in
equation system 5.

The regression coefficients (called ‘‘structural
parameters’’) that connect the several endogenous
variables in equation system 5 are designated as ßij
and are distinguished from the γij representing the
direct effects of the predetermined Zj on the rele-
vant Xi. This notational distinction is made be-
cause the two kinds of variables play different roles

( 5 )

...

x 1 = β 12 x 2 + β 13 x 3 +...+ βlkxk + y 11 z 1 + y 12 z 2

+...+ y1mzm + ε 1

x 2 = β 21 x 1 + β 23 x 3 +...+ β2kxk + y 21 z 1 + y 22 z 2

+...+ y2mzm + ε 2

x 3 = β 31 x 1 + β 32 x 2 +...+ β3kxk + y 31 z 1 + y 32 z 2

+...+ y3mzm + ε 3

xk = βk1x 1 + βk2x 2 +...+ βk,k-1xk-1 + yk1z 1 +

yk2z 2 +...+ ykmzm + εk

in the model. Although it cannot be assumed that

the disturbances εi are uncorrelated with the en-

dogenous X’s that appear on the right-hand sides

of their respective equations, one may make the

somewhat less restrictive assumption that these

disturbances are uncorrelated with the predeter-

mined Z’s.

Some Z’s may be truly exogenous, or distinct

independent variables, that are assumed not to be

affected by any of the endogenous variables in the

model. Others, however, may be lagged endoge-

nous variables, or prior levels of some of the X’s. In

a sense, the defining characteristic of these prede-

termined variables is that they be uncorrelated

with any of the omitted causes of the endogenous

variables. Such an assumption may be difficult to

accept in the case of lagged endogenous variables,

given the likelihood of autocorrelated disturbances,

but we shall not consider this complication fur-

ther. The basic assumption regarding the truly

exogenous variables, however, is that these are

uncorrelated with all omitted causes of the X’s,

though they may of course be correlated with the

X’s and also possibly each other.

Clearly, there are more unknown parameters

than was the case for the original recursive equa-

tion system (1). Turning attention back to the

simple recursive system represented in equation

system 1, one sees that the matrix of betas in that

equation system is triangular, with all such coeffi-

cients above the main diagonal being set equal to

zero on a priori grounds. That is, in equation

system 1, half of the possible betas have been set

equal to zero, the remainder being estimated us-

ing ordinary least squares. It turns out that in the

more general equation system 5, there will be too