CAUSAL INFERENCE MODELSequation system 1 requires the naive assumption
that all variables have been perfectly measured, an
assumption that is, unfortunately, frequently ig-
nored in many empirical investigations using path
analyses based on exactly this same type of causal
system. Measurement errors require one to make
an auxiliary set of assumptions regarding both the
sources of measurement-error bias and the causal
connections between so-called true scores and
measured indicators. In principle, however, such
measurement-error assumptions can be explicitly
built into the equation system and empirical esti-
mates obtained, provided there are a sufficient
number of multiple indicators to solve for the
unknowns produced by these measurement er-
rors, a possibility that will be discussed in the final
section.
In many instances, assumptions about one’s
disturbance terms are even more problematic but
equally critical. In verbal statements of theoretical
arguments one often comes across the phrase
‘‘other things being equal,’’ or the notion that in
the ideal experimental design all causes except
one must be literally held constant if causal infer-
ences are to be made. Yet both the phrase ‘‘other
things being equal’’ and the restrictive assumption
of the perfect experiment beg the question of how
one can possibly know that ‘‘other things’’ are in
fact equal, that all ‘‘relevant’’ variables have been
held constant, or that there are no possible sources
of measurement bias. Obviously, an alert critic
may always suggest another variable that indeed
does vary across settings studied or that has not
been held constant in an experiment.
In recursive causal models this highly restric-
tive notion concerning the constancy of all possi-
ble alternative causes is relaxed by allowing for a
disturbance term that varies precisely because they
are not all constant. But if so, can one get by
without requiring any other assumptions about
their effects? Indeed not. One must assume, essen-
tially, that the omitted variables affecting any one
of the X’s are uncorrelated with those that af-
fect the others. If so, it can then be shown that
the disturbance term in each equation will be
uncorrelated with each of the independent vari-
ables appearing on the right-hand side, thus justi-
fying the use of ordinary least-squares estimating
procedures. In practical terms, this means that if
one has had to omit any important causes of a
given variable, one must also be willing to assume
that they do not systematically affect any of its
presumed causes that have been explicitly includ-
ed in our model. A skeptic may, of course, be able
to identify one or more such disturbing influenc-
es, in which case a modified model may need to be
constructed and tested. For example, if ε 3 and ε 4
contain a common cause that can be identified and
measured, such a variable needs to be introduced
explicitly into the model as a cause of both X 3 and
X 4.Perhaps the five-variable model of Figure 1
will help the reader visualize what is involved. To
be specific, suppose X 5 , the ultimate dependent
variable, represents some behavior, say, the actual
number of delinquent acts a youth has perpetrat-
ed. Let X 3 and X 4 , respectively, represent two
internal states, say, guilt and self-esteem. Finally,
suppose X 1 and X 2 are two setting variables, paren-
tal education and delinquency rates within the
youth’s neighborhood, with the latter variable be-
ing influenced by the former through the parents’
ability to select among residential areas.The fact that the disturbance term arrows are
unconnected in Figure 1 represents the assump-
tion that they are mutually uncorrelated, or that
the omitted variables affecting any given Xi are
uncorrelated with any of its explicitly included
causes among the remaining X’s. If ordinary least
squares is used to estimate the parameters in this
model, then the empirically obtained residuals ei
will indeed be uncorrelated with the independent
X’s in their respective equations, but since this
automatically occurs as a property of least-squares
estimation, it cannot be used as the basis for a test
of our a priori assumptions about the true
disturbances.If one is unwilling to accept these assumptions
about the behavior of omitted variables, the only
way out of this situation is to reformulate the
model and to introduce further complexities in
the form of additional measured variables. At
some point, however, one must stop and make the
(untestable) assumption that the revised causal
model is ‘‘closed’’ in the sense that omitted vari-
ables do not disturb the patterning of relation-
ships among the included variables.Assuming such theoretical closure, then, one
is in a position to estimate the parameters, attach