From: Judea Pearl (UCLA)

Subject: Counterfactuals in linear systems

**Question to author:**

What do we know about counterfactuals
in linear models?

**Author's reply**

Glad you asked.

**Here is a neat result concerning the testability
of counterfactuals in linear systems.**

We know that counterfactual queries of
the form *P*(*Y _{x}*=

This note shows that things are much friendlier in linear analysis:

Claim A. Any counterfactual query of the form
*E*(*Y _{x}* |

Claim B. *E*(*Y _{x}*|

Thus, whenever the causal effect

Claim A is not surprising. It has been established in generality by Balke and Pearl (1994b) where expressions involving the covariance matrix were used for the various terms in (1).

Claim B offers an intuitively compelling interpretation of (1) that reads
as follows: Given evidence *e*, to calculate *E*(*Y _{x}* |

Note:
Eq. (1) can also be written in *do*(*x*) notation as

**Proof:**

(with help from Ilya Shpitzer)

Assume, without loss of generality, that we are dealing with a
zero-mean model. Since the model is linear, we can write
the relation between *X* and *Y* as:

It is always possible to bring the function determining *Y* into
the form (3) by recursively substituting the functions for
each rhs variable that has *X* as an ancestor, and grouping all
the *X* terms together to form *TX*.
Clearly, *T* is the Wright-rule sum of the path
costs originating from *X* and ending in *Y*
(Wright, 1921).

From (3) we can write:

The last term in (5) can be evaluated by taking expectations on both sides of (3), giving:

and, substituted into (5), yields

and proves our target formula (1).

-------------------- QED

**Some Familiar Problems Cast in Linear Outfits**

Three Special cases of *e* are worth noting:

Example-1. *e*: *X =x', Y = y'*

(The linear equivalent of the probability of causation)
From (1) we obtain directly

This is intuitively compelling.
The hypothetical expectation of *Y* is simply
the observed value of *Y, y'*, plus the anticipated
change in *Y* due to the change *x-x'* in *X*.

Example-2. *e*: *X = x'* (effect of treatment on treated)

=

=

Example-3. *e*; *Y = y'*

(Gee, my temperature is *Y=y'*, what if I had taken
*x* tablets of aspirin. How many did you take? Don't remember.)

=

where

Example-4. Let us consider the non-recursive, supply-demand model of page 215 in

Our counterfactual problem (page 216) reads:
Given that the current price is *P=p _{0}*, what would be the
expected value of the demand

=

where

Eq. (8) replaces Eq. (7.17) on page (217). Note that the parameters of the price equation

**Remark 1:**

Example 1 is not really surprising; we know that
the probability of causation is empirically identifiable
under the assumption of monotonicity (*Causality*, p. 293).
But examples 2 and 3 trigger the following conjecture:

**Conjecture**

Any counterfactual query of the form
*P*(*Y _{x}* |

It is good to end on a challenging note.