Green // Stats
Causal Order
One of the basic principles of statistics – or philosophy of science for that matter – is that
numbers themselves cannot adjudicate questions of causality.
Just because variables X
and Y are strongly correlated does not mean that X causes Y or Y causes X.
Conversely,
it is possible to construct examples in which X causes Y, yet X and Y are uncorrelated.
Indeed, it is possible for X to be
positively
correlated with Y even though X exerts a
negative
causal effect on Y.
A few examples may help to drive these points home.
1.
Correlation without causation: National League World Series victories and
presidential election outcomes.
If one looks at enough variables, meaningless
correlations will pop up.
2.
Correlation without causation due to reversal of cause and effect: Basketball
players tend to be unusually tall.
Does joining a basketball team cause one to
grow?
3.
Causation without correlation: Studies in New Haven have found little or no
correlation between voting rates and communication with getoutthevote
canvassers.
Yet the very same experimental studies have shown that this
communication substantially increases voter turnout.
4.
Positive causation despite negative correlation: Incumbent members of Congress
who lose their bids for reelection tend to spend much more than incumbents who
win reelection.
Given this pattern, should one infer that incumbents hurt their
reelection prospects by spending money on their campaigns?
Concept of “Observational Equivalence”
One of the most important ideas in statistics is the notion that different “data generation
processes” can produce the same observable outcomes.
In other words, the mere fact that
a given model is consistent with what we observe does not rule out the possibility that
other models would do equally well, if not better.
In order to see how this principle plays
out numerically, let’s take a step back and think about how regression results are
produced.
Recall that the OLS estimate of the effect of X on Y is obtained using the
formula
(
29
(
29
(
29
(
29
(
29
∑
∑
∑
∑


=





=
=
2
2
1
1
1
1
)
(
)
,
(
ˆ
X
X
Y
X
X
X
X
N
Y
Y
X
X
N
X
Var
Y
X
Cov
b
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentIn other words, the OLS estimate is just the ratio of a covariance and a variance.
So, in
order to formulate examples of observational equivalence, we need to understand how
covariances and variances are shaped.
Here are the basic rules of “covariance algebra”:
1.
Var(b)=0.
Interpretation: the variance of a constant is zero.
Algebra:
∑
=


=
0
)
(
1
1
)
var(
2
b
b
N
b
2.
Cov(X,b)=0.
Interpretation: there is no covariance between a constant and a variable.
Algebra:
∑
=


=
0
)
(
1
1
)
,
(
X
b
b
N
b
X
Cov
3.
Cov(aX,bZ)=abCov(X,Z).
Comment: covariances are easily factored.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '05
 JonathanReuningSchererDonaldGreen
 Statistics, Causality, Cov, Kerry Bush, Kerry Bush Total

Click to edit the document details