Economics 712 Steven N. Durlauf
Fall 2007
Lecture Notes 4. Univariate Forecasting
1. The Wiener-Kolmogorov Prediction Formulas
From previous notes, he solution to the problem
2
min E
t
t k H x
x
ξ
ξ
+ ∈
− 4.1
is equal to the projection of
t k
x
+
onto
t
H x
, denoted as
t k t
x
+
. Once again letting denote the fundamental MA representation of
j t k
j j
α ε
∞ + −
=
∑
t
x , from the equivalence of
and
t
H x
H
t
ε , it is immediate that
t k t
x
+
may be written as
j k t j
t k t j
x
α ε
∞ +
− +
=
=
∑
4.2
The reason for this is simple: at time t, one can condition forecasts on
t
ε ,
1 t
ε
−
, i.e.
t
H
ε . The component of
t k
x
+
that is determined by
1
,...,
t k t
ε ε
+ +
, i.e. is
orthogonal to
k j
t k j
j
α ε
+ − =
∑
t
H
ε . To achieve a more parsimonious expression of expressions such as 4.2, define
the annihilation operator for lag polynomials.
Definition 4.1 Annihilation operator
1
Let be a polynomial lag operator of the form
The annihilation operator, denoted as
L
π
2 1
2 2
1 1
2
L L
L L
π π
π π
π
− −
− −
+ +
+ +
+ +
… …
+
⋅
eliminates all lag terms with negative exponents, i.e.
2 1
2
L L
L π
π π
π
+
+ +
+… 4.3
Using the annihilation operator, one can re-express the optimal predictor as
t t k t
k
L x
L α
ε
+ +
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
4.4
When the MA polynomial is invertible,
1 t
L x
t
α ε
−
= , so that one can explicitly express the predictor as a weighted average of current and lagged
t
x ’s, 4.4 implies
1 t
t k t k
L x
L x
L α
α
− +
+
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
4.5
Equations 4.4 and 4.5 are known as the Wiener-Kolmogorov prediction formulas.
Example 4.1. MA1 process.
Let
1 t
t t
x ε ρε
−
= − . The AR Wiener-Kolmogorov formula imply that optimal linear
predictions of the process is
1
1 1
t t k t
k
L x
L x
L
ρ ρ
− +
+
− ⎛
⎞ =
− ⎜
⎟ ⎝
⎠
4.6
2
which equals 0 for
1 j
, as one would expect since the process is unaffected by shocks more than one period in the past.
Example 4.2. AR1 process.
If
1 t
t
x x
t
ρ ε
−
= + , then the AR Wiener-Kolmogorov formula is
1
i k i
k t
t k t i
t
x L
L x x
ρ ρ
ρ
∞ +
+ =
⎛ ⎞
= −
⎜ ⎟
⎝ ⎠
∑
= 4.7
The AR1 thus has a very simple structure:
1 t k
t t k t
x x
ρ
+ + +
=
.
2. The Hansen-Sargent formula for geometrically discounted distributed leads
The Wiener-Kolmogorov formula has played a critical role in the development of modern empirical macroeconomics. The reason for this is many dynamic aggregate
models imply that one time series represents a combination of forecasts of others. In a pair of seminal papers both of which are on the reading list, Lars Hansen and Thomas
Sargent developed the machinery to understand how such expectations-based relationships lead to testable implications of macroeconomic theories. This machinery
has been of immense importance in the development of modern empirical macroeconomics. I develop the basic Hansen-Sargent formula in the context of the
constant discount dividend stock price model. This is an example of a “geometric distributed leads” model in that one variable is a geometrically weighted sum of expected
values of future levels of another variable Defining the variables
= dividends measured in real terms, = stock price
measured in real terms, and
t
D
t
P β = fixed discount rate, the constant discount dividend
stock price asserts that the stock price series obeys
3
4.8 E
E
i t
t i i
P D
β
∞ +
=
= =
∑
t
P
t i
D where
i t
i
P β
∞ +
=
=
∑
4.9
What is the economic interpretation of this model? The model can be thought of as follows. Suppose agents are risk neutral i.e. utility from consumption at t is linear in the
level and discount utility geometrically at rate β . One can think of ownership of a unit
of stock as providing a stream of real consumption via the dividends if held forever. The opportunity cost of ownership is the consumption foregone today by buying a unit of
stock. The price formula expresses the price at which an expected utility maximizer is indifferent between buying a unit or not, which is what the price must do in equilibrium.
How can one test whether stock prices are consistent with this theory? Suppose that dividends have the fundamental moving average representation.
.
t
D L
t
α ε
=
4.10
This representation is recoverable fromt the data. One may use this representation to test the theory by computing the projection of
onto
t
P
t
H D
that is implied by the theory, and comparing this projection to the direct projection
onto
t
P
t
H D
. Put differently, the stock price model that has been described implies that
j t
t t
j t
j
proj P H D proj D
H D
β
∞ +
=
=
∑
4.11
4
If one knows the process for the dividend series, one can derive the right hand side expression and compare it to the left hand side formula.
In order to derive the projection on the right hand side of this expression, one proceeds in two steps. First one projects
onto
t
P
t
H
ε to generate a time series
t
L
π ε and second, one inverts the
t
ε ’s to generate ’s. The derivation relies on the implication of the model that
may also be expressed as:
t
D
t
P
1 t
t t
t
E P H D E P
H D
D β
+ t
= +
4.12
which implies that
t t
L L
L π
t
L π
ε β
ε α ε
+
⎛ ⎞
= +
⎜ ⎟
⎝ ⎠
4.13
or
t t
L L
L L
π π
t
L
π ε
β ε α
ε
⎛ ⎞
= −
+ ⎜
⎟ ⎝
⎠
4.14
Algebraic manipulation yields
1
1 L
L L
L β
π α
βπ
1 −
−
− =
− 4.15
To manipulate this expression in order to express
L
π in terms of
, notice that
L
α L
β = , then
π α β
=
. Substituting into this expression and rearranging yields
1 1
1 L
L L
L α
βα β π
β
− −
− =
− 4.16
5
which leads to
1 1
1 1
1 E
1 1
t t
t
L L
L L
P H D
D L
L
α βα β
βα β α ε
β β
− −
− −
− −
= =
− −
1 t
−
4.17
This relationship between
and L
L
π α
is complicated. For example, if one writes
E
i t
t i
i
L P H D
L α
t
β ε
∞ =
+
⎛ ⎞
= ⎜
⎟ ⎝
⎠
∑
4.18
then it is apparent that
i k k
i k i
π β α
∞ −
=
=
∑
4.19
3. Some properties of forecasts
