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Chapter 18 Econometrics
This series of slides will cover a subset of Chapter 18
 Data and Operators
 Autocorrelated
 Lagged Variables
 Partial Adjustment
Mathematical
Marketing
Slide 18.1
Time Structured Data
Repeated Firm or Consumer Data
 Time Structured Data - [y1, y2, …, yt, …, yT]
 Error Structure - Not Gauss-Markov (2I)
Mathematical
Marketing
Slide 18.2
Time Structured Data
Backshift Operator
The backshift operator, B, by definition produces xt-1 from xt
Bxt = xt-1
Of course, one can also say
BBxt = B2xt = xt-2
In general,
Bjxt = xt-j
Mathematical
Marketing
Slide 18.3
Time Structured Data
Autocorrelation
Response
Var
Time
Cov(yt, yt-1)?
Mathematical
Marketing
Slide 18.4
Time Structured Data
Table for Autocorrelation
y1 y2 y 3 y4 y5 y6 y7 y 8
y1 y2 y 3 y4 y5 y6 y7 y 8
Mathematical
Marketing
Slide 18.5
Time Structured Data
Table for Autocorrelation
y1 y2 y 3 y4 y5 y6 y7 y 8
y1 y 2 y3 y4 y5 y6 y 7 y8
Mathematical
Marketing
Slide 18.6
Time Structured Data
Autocorrelated Error
y t  xtβ  e t
et = et-1 + t
 ~ N(0,2I)
Mathematical
Marketing
Slide 18.7
Time Structured Data
Recursive Substitution in Time Series
et = et-1 + t
= (et-2 + t-1) + t
= [(et-3 + t-2) + t-1] + t
Mathematical
Marketing
Slide 18.8
Time Structured Data
Now We Leverage the Pattern
et = [(et-3 + t-2) + t-1] + t
= t + t-1 + 2t-2 + 3t-3 + …

i
ρ
=  ε t i
i 0
Mathematical
Marketing
Slide 18.9
Time Structured Data
Time to Figure Out E(·)
 i

E(et )  E  ρ ε t i 
 i 0


  ρi E(ε t i )  0
i 0
Mathematical
Marketing
Slide 18.10
Time Structured Data
And Now Of course V(·)
V(et) = E[et - E(et)]2
V(et) = E[et2]
The previous slide showed that E(et) = 0
Mathematical
Marketing
Slide 18.11
Time Structured Data
Now We Use the Pattern (Squared)
E (e 2t )  E ( 2t )   2 E ( 2t 1 )   4 E ( 2t  2 )  
 2  22  42  
 (1   2   4  ) 2 .
Mathematical
Marketing
Slide 18.12
Time Structured Data
A Big Mess, Right?
V(et) = E(et2) = (1 + 2 + 4 + …)2
Uh-oh… an infinite series…
Mathematical
Marketing
Slide 18.13
Time Structured Data
Let’s Define the Infinite Series “s”
s = 1 + 2 + 4 + 8 + …
2s = 2 + 4 + 8 + 16 + …
What is the difference between the first and second lines?
s - 2 s = 1
1
s
.
2
1 
Mathematical
Marketing
Slide 18.14
Time Structured Data
Putting It Together
Since
1
s
.
2
1 
E(e 2t )  σ e2  sσ 2
Mathematical
Marketing
σ2
1 ρ
2
Slide 18.15
Time Structured Data
Applying the Same Logic to the Covariances
For any pair of errors one time unit apart we have
E(e t , e t 1 )  ρσ
2
e
and in general
j 2
Cov(e t , e t  j )  ρ σ e
Mathematical
Marketing
Slide 18.16
Time Structured Data
Instead of the Gauss-Markov Assumption (2I) we have
2
V(e) = σ e V 
 1
ρ

1
 ρ
V   ρ2
ρ



ρ n 1 ρ n 2
σ2
1 ρ
ρ2
ρ
1

ρ n 3
2
V
 ρ n 1 
n 2 
 ρ 
 ρ n 3 

  

1 
So how do we estimate  now?
Mathematical
Marketing
Slide 18.17
Time Structured Data
Lagged Independent Variables
Consumer behavior and attitude do not immediately change:
yt = 0 + xt-11 + et
Or more generally:
yt = 0 + xt-11 + xt-22 + ··· + et
Mathematical
Marketing
Slide 18.18
Time Structured Data
Koyck’s Scheme
Koyck started with the infinite sequence
yt = xt0 + xt-11 + xt-22 + ··· + et
and assumed that the  values are all of the same sign


i
 c  .
i 0
Mathematical
Marketing
Slide 18.19
Time Structured Data
Lagged effects can take on many forms:
i
i
s
0
i
i
s
0
i
s
0
i
Koyck (and others) have come up with ways of estimating different shaped
impacts (1) assuming that only s lag positions really matter, and that (2) the
impact of x on y takes on some sort of curved pattern as above
Mathematical
Marketing
Slide 18.20
Time Structured Data
Further Assumptions
1. How many lags matter? In other words, how far back do we
really need to go? Call that s.
2. Can we express the impact of those s lags with an even fewer
number of unknowns. Any pattern can be approximated with
a polynomial of degree r  s (Almon’s Scheme). In Koyck’s
Scheme, we will use a geometric rather than polynomial
pattern.
Mathematical
Marketing
Slide 18.21
Time Structured Data
We Rewrite the Model Slightly
y t  x t  0  x t 11  x t 2 2    e t
 [ w 0 x t  w 1 x t 1  w 2 x t 2  ]  e t
where wi  0 for i = 0, 1, 2, ···,  and

w
i 0
Mathematical
Marketing
i
1
Slide 18.22
Time Structured Data
Bring in the Backshift Operator and Assume a
Geometric Pattern for the wi
y t  β[w 0 x t  w1x t 1  w 2 x t 2  ]  e t
 [w 0  w1B  w 2 B2  ]x t  e t .
Now we assume that
wi = (1 - )i
0<<1
Mathematical
Marketing
Slide 18.23
Time Structured Data
Given Those Assumptions
w 0  w 1B  w 2 B    (1  λ)(1  λB  λ B  )
2
2
2
1
 (1  λ)
1  λB
Anyone care to say how we got to this fraction?
Mathematical
Marketing
Slide 18.24
Time Structured Data
Substitute That into the Equation for yt
(1  )
yt 
x t  et
1  B
(1  B) y t  (1 - )x t  (1  B)e t
y t  By t  (1 - )x t  e t  Be t
y t  (1  ) x t  y t 1  (e t  e t 1 )
Mathematical
Marketing
Slide 18.25
Time Structured Data
Adaptive Adjustment
Define
~
xt
as the expected level of x (prices, availability, quality, outcome)… So consumer
behavior should look like
~
y t  0  x t 1 e t .
Mathematical
Marketing
Slide 18.26
Time Structured Data
Updating Process
Expectations are updated by a fraction of the discrepancy between the
current observation and the previous expectation
~
xt  ~
x t 1  ( x t  ~
x t 1 )
~
x t  x t  ~
x t 1  ~
x t 1
 x t  (1  )~
x t 1
~
x t  (1  )~
x t 1  x t .
Mathematical
Marketing
Slide 18.27
Time Structured Data
Redefine  in Terms of a New Parameter 
~
~
x t  (1  δ)x t 1  δx t
Define
=1-
so that
~
~
x t  λx t 1  δx t
Mathematical
Marketing
Slide 18.28
Time Structured Data
More Algebra
~
x t  ~
x t 1  x t
(1  B)~
x t  x t
~
xt 
Mathematical
Marketing

xt.
1  B
Slide 18.29
Time Structured Data
Back to the Model for yt
yt  β0  ~
x t β1  e t
δβ1
 β0 
x t  et
1  λB
β1 (1  λ)
 β0 
x t  et
1  λB
We end up at the same place as slide 25
Mathematical
Marketing
Slide 18.30
Time Structured Data