Time Series Models: Lecture XXXIV
Charles B. Moss
December 2, 2010
I. Nonspherical Errors  Autocorrelation in Structural Models
A. Time Series versus Structural Models
1. Theoretical models of supply and demand.
a) Most students have been introduced to the utility model
for the derivation of consumer demand. Taking for exam
ple the CobbDouglas utility function
max
x
1
,x
2
x
α
1
x
β
2
s
.
t
.p
1
x
1
+
p
2
x
2
≤
Y
(1)
yields two demand equations
x
1
(
p
1
,p
2
,Y
) =
Y
p
1
±
α
α
+
β
²
x
2
(
p
1
,p
2
,Y
) =
Y
p
2
±
β
α
+
β
²
.
(2)
Taking the natural logarithms of the demand equations
in Equation 2 yields structural models of demand
ln (
x
1
) = ln
±
α
α
+
β
²

ln (
p
1
) + ln (
Y
)
ln (
x
2
) = ln
±
β
α
+
β
²

ln (
p
2
) + ln (
Y
)
⇒
³
ln (
x
1
) =
α
01
+
α
11
ln (
p
1
) +
α
21
ln (
p
2
) +
α
31
ln (
Y
)
ln (
x
2
) =
α
02
+
α
12
ln (
p
1
) +
α
22
ln (
p
2
) +
α
32
ln (
Y
)
´
(3)
1
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View Full DocumentAEB 6571 Econometric Methods I
Professor Charles B. Moss
Lecture XXXIV
Fall 2010
So the estimated model is directly linked to the theoretic
model.
b) The structural models can also be derived analytical ex
pansions of theoretic relationships. Speciﬁcally, in pro
duction economics we derive the existence of the cost
function for a ﬁrm (and the expenditure function for the
consumer)
±
min
x
x
0
w
s
.
t
.f
(
x,y
) = 0
²
⇒
C
(
w,y
)
(4)
often this relationship is estimated a second order Taylor
series expansion of an unknown function
C
(
w,y
) =
α
0
+
α
0
x
+
1
2
x
0
Ax
+
β
0
y
+
1
2
y
0
By
+
x
0
Γ
y.
(5)
The cost function and input demand functions derived us
ing Shephard’s lemma are derived from economic theory.
c) These models are typically referred to as structural models
where economic theory can be used to directly justify the
speciﬁcation.
2. Time series models model the current value of an economic
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 Spring '10
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 Taylor Series, The Land, Natural logarithm, Autoregressive moving average model, Time series analysis, Lecture XXXIV

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