Economics 245B
Spurious Regressions
Consider the linear regression model
Y
t
=
+
X
0
t
±
+
U
t
:
In our previous analysis of the model, we have not mentioned serial correlation
of the regressors and the dependent variable.
If both the regressors and the
dependent variable are highly correlated, then our standard regression inference
can be very misleading.
To understand why, recall that the OLS estimator of the coe¢ cients is
A
B
±
=
²
n
P
X
0
t
P
X
t
P
X
t
X
0
t
³
1
²
P
Y
t
P
X
t
Y
t
³
:
As long as
Y
t
and
X
t
are generated by stationary processes, our standard limit the
ory holds. If, however,
Y
t
and
X
t
are generated by nonstationary processes, then
we must make adjustments. To equate serial correlation with nonstationarity, it
is easiest to consider an autoregression
Y
t
=
²Y
t
1
+
V
t
:
If
j
²
j
<
1
, then the serial correlation is attenuated enough so that the series is
stationary. If
j
²
j
= 1
, then the serial correlation is so severe that the series is not
stationary. If
j
²
j
= 1
, then
Y
t
is a unitroot process. If, in addition,
V
t
is white
noise, then
Y
t
is a random walk. Consider estimating the intercept model
Y
t
=
+
U
t
:
The estimator of the intercept is
A
=
n
1
P
Y
t
. If
Y
t
contains a unit root, then
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 Fall '08
 Staff
 Economics, Normal Distribution, Regression Analysis, Variance, Probability theory, Trigraph, Yt

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