情報論的学習理論テクニカルレポート 2009
Technical Report on Information-Based Induction Sciences 2009 (IBIS2009)

Extending the Use of Instrumental Variables for the Identiﬁcation of
Direct Causal Eﬀects in SEMs
Hei Chan ∗Manabu Kuroki †Abstract: In this paper, we present an extended set of graphical criteria for the identiﬁcation of direct causal eﬀects in linear Structural Equation Models (SEMs). Previous
methods of graphical identiﬁcation of direct causal eﬀects in linear SEMs include methods
such as the single door criterion, the instrumental variable and the IV-pair, and the accessory set. However, there remain graphical models where a direct causal eﬀect can be
identiﬁed and these graphical criteria all fail. We present a few of these examples, andpresents an extended set of graphical criteria which uses descendants of the cause variable
as “path-speciﬁc instrumental variables”. The results can be used to identify the directcausal eﬀect as long as an certain conditions based on an extended set of graphical criteria
and the identiﬁability of other causal eﬀects are satisﬁed.
Keywords: Structural Equation Models, Causality, Parameter Identiﬁcation

1

Introduction

Structural Equation Models (SEM) is a useful tool forcausal analysis, and is widely used in areas of social
science such as economics [Bollen 1989, Duncan 1975].
Research by scientists, social scientists, and computerscientists in this area has allowed the problem to be
applied in real-life models.
In a linear SEM, the relationships between observed
variables are expressed in linear equations. The structure of the equations is such that they not only express
the linear relationships between the variables, together
with a stochastic error term for unobserved factors,
but also the causal dependence among the observed
variables. For each variable Y , its structural equation
where it appears on the left-hand side, the presence(and absence) of a variable X on the right-hand side
speciﬁes that X is (or is not) a direct cause of Y .
A fundamental problem in linear SEMs is to estimate the strength of a certain direct causal eﬀect fromone variable to another from a combination of observed
data and model structure. This is called the identiﬁcation problem [Fisher 1966]. Although many methods,∗ Data Based Modeling Research Team,

Center for Service
Research, AIST Japan,2-41-6, Aomi, Koto-ku, Tokyo 135-0064,
e-mail hei.chan@aist.go.jp† Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, 1-3, Machikaneyama-cho,
Toyonaka, Osaka 560-8531, e-mail mkuroki@sigmath.es.osakau.ac.jp

both algebraic and graphical, have been developed over
the years, this problem is still not solved. Currently
there are no suﬃcient or necessary criteria for deciding
whether a causal eﬀect can be identiﬁed from observed
data. Current identiﬁcation methods based on graphical criteria, including the single door criterion, thefront door criterion and the back door criterion, theinstrumental variable and the IV-pair, and the accessory set, can be used to identify direct causal eﬀects
only when certain conditions are met. However, there
exist direct causal eﬀects that can be found using algebraic methods by solving for a set of equations involving the causal eﬀects and the covariances but cannot
be identiﬁed using these methods.
The aim of this paper is thus to provide an extended set of graphical criteria for the identiﬁcationof direct causal eﬀects in linear SEMs. We will show
examples where the desired causal eﬀect can be identiﬁed using algebraic methods but cannot be identiﬁed
by current graphical methods. Then based on the algebra involved in solving for these causal eﬀects, we willpropose this new set of graphical criteria, where themotivation is as follows. We will treat descendants ofthe cause variable as “path-speciﬁc instrumental variables”, and ﬁnd certain path-speciﬁc total eﬀects which
can be used to compute the desired causal eﬀect. Thisextended set of graphical criteria we develop will thus
check if these path-speciﬁc eﬀects can be computedfrom covariance values that are available from observed
data based on the graphical model.

The outline of this paper is as follows. First, we
provide the preliminary deﬁnitions for this paper, including linear SEMs, graphical models, statistical terms
such as covariances, Wright’s method of analysis, andthe identiﬁcation problem. Then, we recap previous results in solving the parameter identiﬁcation problem,
most notably IV-pairs [Brito and Pearl 2002a] and accessory sets [Tian 2007b]. Next, we present our new
results for the identiﬁcation of direct causal eﬀects inlinear SEMs based on an extended set of graphical criteria, which are able to identify direct causal eﬀects
which are not possible using previous results based ongraphical criteria. Finally, we conclude our paper andprovide a few discussions on future work on this topic.

2

Preliminaries

In statistical causal analysis, a directed acyclic graph(DAG) that represents cause-eﬀect relationships is called
a path diagram. A directed graph is a pair G = (V , E),
where V is a ﬁnite set of vertices and the set E of directed edges is a subset of the set V × V of orderedpairs of distinct vertices. Regarding the graph theoretic terminology used in this paper, for example, refer
to Pearl [Pearl 2000] and Spirtes et al. [Spirtes 2000].
Suppose a directed acyclic graph G = (V , E) witha set V = {V1 , · · · , Vn } of variables is given. The graph
G is called a path diagram, when each child-parent
family in the graph G represents a linear structural
equation model (SEM):Vi =αvi vj Vj + vi
i = 1, . . . , n, (1)Vj ∈pa(Vi )where pa(Vi ) is a set of parents of Vi and αvi vj (= 0)
is called a direct causal eﬀect. In addition random disturbances v1 , . . . , vn are assumed to be normally distributed with mean 0. Here, when vi is correlatedwith vj (i = j), this relationship is represented bya bi-directed (dashed) arc between Xi and Xj in thegraph G.
Given a graph G, we deﬁne a path between thevariables X and Y as a sequence of variables, (V0 =X, V1 , . . . , Vn = Y ), where there is an edge between
each Vi and Vi+1 and each variable appears only oncein the sequence. A path is a directed path from X to
Y if all edges between Vi and Vi+1 are directed edges
from Vi to Vi+1 .
We say that Vi is a collider in the path if both theedges between Vi−1 and Vi and between Vi and Vi+1

point into Vi . If there are no colliders in the path, we
say that the path is an unblocked path between X and
Y . Given a set of variables Z, we say the path is an
open path if for all variables Vi which are not colliders
on the path, Vi ∈/ Z, and for all variables Vi which arecolliders on the path, V ∈ Z where V is either Vi or adescendant of Vi . If there are no open paths between
X and Y given Z, we say that X and Y are d-separated
given Z. Otherwise, we say that they are d-connected.
The conditional independence induced from a setof equations in the form of Equation 1 can be obtained from the graph G according to d-separation[Pearl 2000], that is, when Z d-separates X from Y ina path diagram G, X is conditionally independent of Y
given Z in the corresponding linear SEM [Spirtes 2000].
In this paper, it is assumed that a path diagram G
and the corresponding linear SEM are faithful to eachother; that is, the conditional independence relationships in the linear SEM are also reﬂected in G, andvice versa [Spirtes 2000].
Here, we denote some notations for further discussion. Let σxy·z = cov(X, Y |Z = z), σyy·z = var(Y |Z =z) and βyx·z = σxy·z /σxx·z be a conditional covariance
between X and Y given Z = z, a conditional varianceof Y given Z = z and the regression coeﬃcient of x
in the regression model of Y on x and z, respectively.
When Z is an empty set, they are omitted from thesearguments.
A total eﬀect τyx of X on Y is deﬁned as the total sum of the products of the direct causal eﬀects on
the sequence of directed edges along all directed paths
from X to Y . In addition, γyx = σxy − τyx is called aspurious correlation between X and Y .
A path-speciﬁc total eﬀect is deﬁned as the totalsum of the product of the direct causal eﬀects on thesequence of directed edges along directed paths of ourinterests from X to Y . For example, we deﬁne τyx·z
as the path-speciﬁc eﬀects where all paths that passthrough any variable in Z are not counted. Similarterms such as a path-speciﬁc correlation and a pathspeciﬁc spurious correlation can be deﬁned similarly.
Wright’s method of path analysis [Wright 1934], whichplays an important role in this paper, can be used tocompute the covariance of two variables X and Y given
a graph G. If the set S contains all paths path = (V0 =X, V1 , . . . , Vn = Y ) that are unblocked paths between

X and Y , we have:σxy =pv0 v0path

pvi vi+1 ,(2)

i=0,...,n−1

where pvi vi+1 is the parameter of the edge between Vi
and Vi+1 , which is either αvi+1 vi (or αvi vi+1 ) if it is adirected edge, or γvi+1 vi if it is a bi-directed edge. We
deﬁne pv0 v0 as σv0 v0 if all edges in path are directededges, or 1 otherwise.
Given the matrix of observed covariances, we saythat a causal parameter, such as a total eﬀect and a
direct causal eﬀect, is identiﬁed if there is a uniquesolution of this parameter given the covariances. If all
direct causal eﬀects can be identiﬁed, we say that themodel is identiﬁed.
The single door criterion is one of the famous graphical identiﬁcation conditions for the direct causal effects, that is, the direct causal eﬀect αyx of X on Y is
identiﬁable and is equal to βyx·z , if there exist a set Zof variables such that (i) Z contains no descendant of
Y , and (ii) Z d-separates X from Y in GX→Y , formedby removing X → Y from G [Pearl 2000]. A set Zof variable satisfying both (i) and (ii) is said to satisfy
the single door criterion relative to (X, Y ).

3

Previous Results on Parameter Identification

Figure 1: A path diagram where a multiple IV-pair arenecessary to identify the direct causal eﬀects of X1 (orX2 ) to Y .
2. In the graph G, W and X are d-connected given
Z, or W = X.
If the above conditions are satisﬁed, the direct causal
eﬀect αyx is given by:
αyx =

σwy·z
.
σwx·z

(3)

The pair (W, Z) can also be called an IV-pair.
Some identiﬁable direct causal eﬀects cannot befound using a single IV-pair, but by the collective action of a multiple IV-pair, such as the example in Figure 1. We now deﬁne the conditions where a set of
direct causal eﬀects can be identiﬁed using multiple IVpairs. The following lemma is adapted from previous
work, where the multiple IV-pair are called accessory
sets [Tian 2007a].

Lemma 2 Let the graph G contain the directed edges
X1 → Y, . . . , Xk → Y , and let ({W1 , . . . , Wk } be a list
There have been many work done on the problems ofof variables. Given Z, a (possibly empty) set of varimodel identiﬁcation and parameter identiﬁcation usables which consists of non-descendants of Y and dising graphical test [Pearl 2000, Brito and Pearl 2002a,
tinct from W1 , . . . , Wk , we say that W1 , . . . , Wk can be
Brito and Pearl 2002b, Brito and Pearl 2002c, Tian 2004,
used as instrumental variables given Z if the following
Tian 2005]. Here we will focus only on the problem ofthree conditions are satisﬁed:
parameter identiﬁcation, in particular the identiﬁca1. In the graph GX1 →Y,...,Xk →Y , formed by removtion of direct causal eﬀects.
ing {X1 → Y, . . . , Xk → Y } from G, {W1 , . . . , Wk }The previous most general result for the graphicaland Y are d-separated given Z;
identiﬁcation of direct causal eﬀects is the use of an
IV-pair [Brito and Pearl 2002a], which embraces both
2. In the graph G, each pair of Wi and Xi are dinstrumental variables [Bowden and Turkington 1984]
connected given Z, or Wi = Xi , where we denoteand regression methods.
this path as pathi ;Lemma 1 Let the graph G contain the directed edge
X → Y , and let W be a variable. Given Z, a (possibly empty) set of variables which consists of nondescendants of Y and distinct from W , we say that
W can be used as an instrumental variable given Z if
the following two conditions are satisﬁed:
1. In the graph GX→Y , formed by removing X → Y
from G, W and Y are d-separated given Z;

3. If two diﬀerent paths, pathi (from Wi to Xi ) andpathj (from Wj to Xj ), have a common variable
U , then either both pathi [Wi , . . . , U ] and pathj [U,. . . , Xj ] point into U , or pathj [Wj , . . . , U ] andpathi [U, . . . , Xi ] point into U , but not both, i.e.,
U cannot be a collider. Here, pathi [Wi , . . . , U ] isa sub-path between Wi and U included in pathi ,and the similar notation is used for other paths.

Therefore, σwy − σyx βwx·v = γwv αyx αxv , and σwx −σxx βwx·v = γwv αxv . Therefore, αyx can be computedby:σwy − σyx βwx·v
αyx =.
σwx − σxx βwx·v
Figure 2: Path diagram for Example 1.
If the above conditions are satisﬁed, the direct causal
eﬀects αyx1 , . . . , αyxk can be solved by a system of equations involving equation coeﬃcients:

where Qi =Pi,j= σwi xj ·z ,Qi

= σwi y·z ,j Pi,j αyxj .

The last criterion is adapted from the G criterion[Brito and Pearl 2006] by adding the conditions necessary to deal with colliders. This criterion ensures thatthe system of equations we use to solve for the parameters αyx1 , . . . , αyxk are linearly independent. It is a
suﬃcient condition for parameter identiﬁability.
However, there are many cases where a single directcausal eﬀect is identiﬁable even though neither IV-pair
nor a set of variables satisfying the single door criterion
can be found. We will illustrate this in the next section
with a few examples, and extend previous results ongraphical identiﬁcation to ﬁnd the direct causal eﬀects.

4

New Results on Parameter Identification

Example 1 Assume that cause-eﬀect relationships between variables can be described as the DAG shown in
Figure 2 and the corresponding linear SEM. The directcausal eﬀects of αxv and αwx can be identiﬁed easily
by the single door criterion [Pearl 2000]:
αxv= βxv ,

αwx

= βwx·v .

We now want to ﬁnd the direct causal eﬀect of αyx .
However, no IV-pair can be used to ﬁnd αyx using
our previously shown results. Instead, we use Wright’s
method of path analysis (Equation 2). In particular,
we have:
σwy=(γwv αyx + γyv αwx )αxv + (σxx αyx + γyx )αwx ,σwx=

γwv αxv + σxx αwx ,σyx=

γyv αxv + σxx αyx + γyx .

Notice that in this example, neither W nor V canbe used in any IV-pair to identify αyx . In particular for
W , the presence of an open path W ← X ↔ Y in thegraph GX→Y makes it invalid to be used as an instrumental variable. However, if we consider a latent variable U along the bi-directed edge between W and X,
this latent variable, if observable, can be used as an instrumental variable to identify αwx [Cai and Kuroki 2008],
meaning we need to aim to ﬁnd a method to “indirectly” estimate the correlations both between U andX, and U and Y . To do this, we use W , which isa descendant of the cause variable X, as a “pathspeciﬁc instrumental variable”, and estimate the correlations both between W and X, and between W and
Y , through certain paths in the graph. To proceedwith our discussion, we make the following deﬁnition.
Definition 1 Let the graph G contain the directed edge
X → Y , and let W be a descendant of X (but not Y ).
Given Z, a (possibly empty) set of variables which consists of non-descendants of X and Y distinct from W ,we say that W can be used as a “path-speciﬁc instrumental variable” given Z if the following two conditions are satisﬁed:
1. In the graph G, W and X are d-connected given
Z, where at least one open path between W and
X given Z is not a directed path from X to W .
2. In the graph GX→Y , formed by removing X → Y
from G, W and Y are either d-separated given Z,
or d-connected such that all open paths between
W and Y given Z must contain a sub-path which
is a directed path from X to W .
The path diagram in Figure 2 satisﬁes the conditions in Deﬁnition 1. In particular, in the graph
GX→Y , formed by removing X → Y from G, W and Y
are d-connected only through the path W ← X ↔ Y ,where it contains a directed path from X to W .
Now we are ready to compute the direct causal effect αyx . We classify all open paths between W and Y
given Z in the graph G into two subsets:
1. Swy·z: Paths that contain a sub-path which isa directed path from X to W (a path-speciﬁc
correlation between Y on W , denoted as swy·z );2. Swy·z : Paths that do not contain a sub-path which
is a directed path from X to W (a path-speciﬁc
spurious correlation between Y and W , denotedas swy·z ).
Similarly, we classify all open paths between W and X
given Z in the graph G into two subsets:
1.


Swx·z
:

Paths that are directed paths from X to
W (a path-speciﬁc total eﬀect of X on W , denoted as swx·z );2. Swx·z : Paths that are not directed paths from X
to W (a path-speciﬁc correlation between X and
W , denoted as swx·z ).
First, the subsets Swy·z
and Swy·z are mutually ex
clusive, and so are the subsets Swx·z
and Swx·z . Therefore, we have:swy·z + swy·z= σwy·z ;σxx·z swx·z + swx·z= σwx·z .

contains all open directed paths from
Second, Swx·z
X to W given Z in the graph G. Since Z does notcontain any descendants of X, this is equivalent to all
open directed paths from X to W . Therefore, we have:swx·z = τwx .
Third, any open path between X and Y given Z in thegraph G must not pass through W . Otherwise, as the
sub-path between W and Y does not pass through X,
this would violate the second condition of Deﬁnition 1.
Moreover, as all paths in Swx·z
are directed paths fromX to W , they do not point into X. This means a pathin Swy·z
can be (and must be) formed by extending apath in Swx·z
by an open path between X and Y given
Z. Therefore, we have:swy·z = σyx·z swx·z = σyx·z τwx .
Fourth, as all paths in Swy·z do not contain a sub-path
which is a directed path from X to W , they must contain the directed edge X → Y , otherwise this would violate the second condition of Deﬁnition 1. This meansa path in Swy·z can be (and must be) formed by extending a path in Swx·z by the directed edge X → Y .
Therefore, we have:swy·z = αyx swx·z .
Finally, the ﬁrst condition of Deﬁnition 1 (and thefaithfulness condition) ensures that swx·z = 0. Combining all equations, we have the following theorem.

Figure 3: Path diagram which satisﬁes the conditionsof Deﬁnition 1.
Theorem 1 Given a path diagram which satisﬁes theconditions in Deﬁnition 1, the direct causal eﬀect αyx
is given by:
αyx =σwy·z − σyx·z τwx.
σwx·z − σxx·z τwx(4)

Theorem 1 (Equation 4) expands on the the useof the IV-pair in Lemma 1 (Equation 3), by allowing
the use of the “path-speciﬁc instrumental variable” W
which is a descendant of X, showing that the directcausal eﬀect αyx is identiﬁable as long as the total effect of X on W given Z, τwx , is identiﬁable.
For example, in the path diagram in Figure 2, thetotal eﬀect of X on W is given by τwx = αwx = βwx·v ,and we can use Equation 4 to compute αyx . For another example, in the path diagram in Figure 3, given
Z, W can be used as an “path-speciﬁc instrumental
variable” for the direct causal eﬀect of X → Y , andthe total eﬀect of X on W is given by τwx = βwx·vz .
We conclude this section by stating that we canuse a set of multiple “path-speciﬁc instrumental variables” to identify multiple direct causal eﬀects similar
to Lemma 2, as long as the G criterion is satisﬁed.
However, we do not state the details here.

5

Discussion and Conclusion

In this paper, we presented our new results for the identiﬁcation of direct causal eﬀects in linear SEMs basedon an extended set of graphical criteria, which are able
to identify direct causal eﬀects which are not possibleusing previous results based on graphical criteria. The
results are useful in the sense that they allow us toeﬀectively identify the desired direct causal eﬀect, by
ﬁrst checking whether the model satisﬁes the graphicalcriteria, then using the necessary covariance values andother total eﬀects to compute the causal eﬀect, insteadof solving the whole set of equations for all variables
given by Wright’s method of analysis.
In future work, there are three main areas which
we will pursue:

Algorithm We need to ﬁnd an algorithm where we
can easily test for the satisfaction of our graphical criteria given in this paper. Previous work on IV-pairs
and accessory sets use ﬂow analysis to ﬁnd the necessary variables, and we look to adapt these algorithms
for our extended set of graphical criteria.
Completeness It remains to be seen if the graphical criteria given in this paper, combined with previous methods such as IV-pairs, are complete, i.e., they
are necessary conditions for the identiﬁcation of causal
eﬀects in linear SEMs. If not, we will look for counterexamples, and if possible, extend further our graphicalcriteria. In particular, our new graphical criteria aredependent on a latent variable U , if observable, being avalid instrumental variable for our direct causal eﬀect.
Recent work has looked at the role of a latent variable
in identifying causal eﬀects [Cai and Kuroki 2008].
Robustness Even in models where previous graphical criteria, such as instrumental variables, can be used
to identify a certain causal eﬀect, our new set of criteria may provide another distinct function for computing this causal eﬀect. The concept of robustness[Pearl 2004] deals with whether a function for computing a causal eﬀect is still valid when certain independence relations are relaxed in a model (by addingedges between variables). If for all super-graphs of ourcurrent graph, function A is valid whenever function
B is valid, then we say that function A is at least as
robust as function B, and should at leat be more preferred than function B, because function A will remain
valid even in cases where function B is no longer valid
when some of our current independence relations arerelaxed. Moreover, if we are given two functions thatare no more robust than one another, then the computation of the desired causal eﬀect using these two
diﬀerent functions (and the fact that the two computations agree) will greatly conﬁrm the correctness of
our model. Therefore, we would like to compare therobustness quality of our new set of criteria comparedwith previous graphical criteria.

was supported by the Mazda Foundation, the JIST
Foundation, and the Ministry of Education, Culture,
Sports, Science and Technology of Japan.

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