Chapter 18 Econometrics

Slide 18.11Time Structured DataTime Structured Data

MathematicalMathematicalMarketingMarketing

Chapter 18 Econometrics

This series of slides will cover a subset of Chapter 18

Data and Operators

Autocorrelated

Lagged Variables

Partial Adjustment



Repeated Firm or Consumer Data

Time Structured Data - [y1, y2, …, yt, …, yT]

Error Structure - Not Gauss-Markov (2I)



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



Autocorrelation

Time

ResponseVar

Cov(yt, yt-1)?



Table for Autocorrelation

y1 y2 y3 y4 y5 y6 y7 y8

y1 y2 y3 y4 y5 y6 y7 y8



Autocorrelated Error

ttt eβxy

et = et-1 + t

~ N(0,2I)



Recursive Substitution in Time Series

et = et-1 + t

= (et-2 + t-1) + t

= [(et-3 + t-2) + t-1] + t



Now We Leverage the Pattern

et = [(et-3 + t-2) + t-1] + t

= t + t-1 + 2t-2 + 3t-3 + …

=

0iit

iερ



Time to Figure Out E(·)

0)E(ερ

ερE)E(e

0iit

i

0iit

it



And Now Of course V(·)

V(et) = E[et - E(et)]2

The previous slide showed that E(et) = 0

V(et) = E[et2]



Now We Use the Pattern (Squared)

.)1(

)(E)(E)(E)e(E

242

24222

22t

421t

22t

2t



A Big Mess, Right?

V(et) = E(et2) = (1 + 2 + 4 + …)2

Uh-oh… an infinite series…



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 - 2s = 1

.1

1s

2



Putting It Together

.1

1s

2

2

222

e2t ρ1

σsσσ)E(e

Since



Applying the Same Logic to the Covariances

2e1tt ρσ)e,E(e

2e

jjtt σρ)e,Cov(e

For any pair of errors one time unit apart we have

and in general



Instead of the Gauss-Markov Assumption (2I) we have

VV2

22e

ρ1

σσ

1ρρρ

ρ1ρρ

ρρ1ρ

ρρρ1

3n2n1n

3n2

2n

1n2

V

V(e) =

So how do we estimate now?



Lagged Independent Variables

yt = 0 + xt-11 + et

Consumer behavior and attitude do not immediately change:

yt = 0 + xt-11 + xt-22 + ··· + et

Or more generally:



Koyck’s Scheme

Koyck started with the infinite sequence

yt = xt0 + xt-11 + xt-22 + ··· + et

and assumed that the values are all of the same sign

.c0i

i



Lagged effects can take on many forms:

i

i

0 s

i

i

0 s

i

i

0 s

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



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.



We Rewrite the Model Slightly

t2t21t1t0

t22t11t0tt

e]xwxwxw[

exxxy

where wi 0 for i = 0, 1, 2, ···, and

0i

i 1w



Bring in the Backshift Operator and Assume a Geometric Pattern for the wi

.ex]BwBww[ tt2

210

t2t21t1t0t e]xwxwxβ[wy

Now we assume that

wi = (1 - )i

0 < < 1



Given Those Assumptions

λB1

1λ)(1

)BλλBλ)(1(1BwBww 222

210

Anyone care to say how we got to this fraction?



Substitute That into the Equation for yt

)ee(yx)1(y

Bee)x-(1Byy

e)B1()x-(1y)B1(

exB1

)1(y

1tt1ttt

ttttt

ttt

ttt



Adaptive Adjustment

Define

as the expected level of x (prices, availability, quality, outcome)… So consumer

behavior should look like

.ex~y t1t0t

tx~



Updating Process

.xx~)1(x~

x~)1(x

x~x~xx~

)x~x(x~x~

t1tt

1tt

1t1ttt

1tt1tt

Expectations are updated by a fraction of the discrepancy between the current observation and the previous expectation



Redefine in Terms of a New Parameter

Define

= 1 -

so that

t1tt δxx~λx~

t1tt δxx~δ)(1x~



More Algebra

.xB1

x~

xx~)B1(

xx~x~

tt

tt

t1tt



Back to the Model for yt

We end up at the same place as slide 25

tt1

0

tt1

0

t1t0t

exλB1

λ)(1ββ

exλB1

δββ

eβx~βy

Chapter 18 Econometrics

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