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1
Properties of OLS residuals
The aim of regression analysis is to partition y into
systematic and nonsystematic
components. A
desirable estimator generates u
i
which are not
systematic
For OLS
• the mean value of u
e
(the regression residuals) is
zero if the regression contains a constant
• the correlation of u
e
with any x variable included
in
the regression is zero: in simple regression
Cov(u
e
,x) = Cov(y 
α
e

β
e
x, x)
= Cov(y,x) 
β
e
Cov(x, x)
= Cov(y,x)  {Cov(x,y)/V(x)}V(x) = 0
Analysing the variance of Y
The withinsample variance of Y is
Var(Y)
= Var(
α
e
+
β
e
X+
U
e
)
= (
β
e
)
2
Var(X) + 2
β
e
Cov(X,U
e
) + Var(U
e
)
which can be decomposed into two components
Var(Y) = (
β
e
)
2
Var(X) + Var(U
e
)C
o
v
(
X
,
U
e
) = 0
Total
Sum of Squares (SST) =
Explained
Sum of
Squares (SSE) +
Residual
Sum of Squares (SSR)
R
2
= (
β
e
)
2
Var(X)/Var(Y) = SSE/SST
R
2
=
(SST  SSR)/SST = 1  SSR/SST
This decomposition works for all
OLS regressions
The ANOVA table
Source
Sum of squares
(around mean)
Degrees of
freedom
Mean square
Total
SST
(N1)
SST/(N1)
{Var(Y)}
Explained
SSE
{(N1)*
β
2
*Var(X)}
1S
S
E
Residual
SSR
{SST  SSE}
(N2)
SSR/(N2)
{s
2
}
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Regression and causality
Correlation is symmetric
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 Spring '10
 Cowell
 Economics

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