Slide 18.1 Time Structured Data MathematicalMarketing Chapter 18 Econometrics This series of slides will cover a subset of Chapter 18  Data and Operators  Autocorrelated  Lagged Variables  Partial Adjustment

Slide 18.2 Time Structured Data MathematicalMarketing Repeated Firm or Consumer Data  Time Structured Data - [y 1, y 2, …, y t, …, y T ]  Error Structure - Not Gauss-Markov (  2 I)

Slide 18.3 Time Structured Data MathematicalMarketing Backshift Operator The backshift operator, B, by definition produces x t-1 from x t Bx t = x t-1 Of course, one can also say BBx t = B 2 x t = x t-2 In general, B j x t = x t-j

Slide 18.4 Time Structured Data MathematicalMarketing Autocorrelation Time Response Var Cov(y t, y t-1 )?

Slide 18.5 Time Structured Data MathematicalMarketing Table for Autocorrelation y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8

Slide 18.6 Time Structured Data MathematicalMarketing Table for Autocorrelation y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8

Slide 18.7 Time Structured Data MathematicalMarketing Autocorrelated Error e t =  e t-1 +  t  ~ N(0,  2 I)

Slide 18.8 Time Structured Data MathematicalMarketing Recursive Substitution in Time Series e t =  e t-1 +  t =  (  e t-2 +  t-1 ) +  t =  [  (  e t-3 +  t-2 ) +  t-1 ] +  t

Slide 18.9 Time Structured Data MathematicalMarketing Now We Leverage the Pattern e t =  [  (  e t-3 +  t-2 ) +  t-1 ] +  t =  t +  t-1 +  2  t-2 +  3  t-3 + … =

Slide Time Structured Data MathematicalMarketing Time to Figure Out E(·)

Slide Time Structured Data MathematicalMarketing And Now Of course V(·) V(e t ) = E[e t - E(e t )] 2 The previous slide showed that E(e t ) = 0 V(e t ) = E[e t 2 ]

Slide Time Structured Data MathematicalMarketing Now We Use the Pattern (Squared)

Slide Time Structured Data MathematicalMarketing A Big Mess, Right? V(e t ) = E(e t 2 ) = (1 +  2 +  4 + …)  2 Uh-oh… an infinite series…

Slide Time Structured Data MathematicalMarketing Let’s Define the Infinite Series “s” s = 1 +  2 +  4 +  8 + …  2 s =  2 +  4 +  8 +  16 + … What is the difference between the first and second lines? s -  2 s = 1

Slide Time Structured Data MathematicalMarketing Putting It Together Since

Slide Time Structured Data MathematicalMarketing Applying the Same Logic to the Covariances For any pair of errors one time unit apart we have and in general

Slide Time Structured Data MathematicalMarketing Instead of the Gauss-Markov Assumption (  2 I) we have V(e) = So how do we estimate  now?

Slide Time Structured Data MathematicalMarketing Lagged Independent Variables y t =  0 + x t-1  1 + e t Consumer behavior and attitude do not immediately change: y t =  0 + x t-1  1 + x t-2  2 + ··· + e t Or more generally:

Slide Time Structured Data MathematicalMarketing Koyck’s Scheme Koyck started with the infinite sequence y t = x t  0 + x t-1  1 + x t-2  2 + ··· + e t and assumed that the  values are all of the same sign

Slide Time Structured Data MathematicalMarketing Lagged effects can take on many forms: ii i 0 s ii i 0 s i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

Slide Time Structured Data MathematicalMarketing 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.

Slide Time Structured Data MathematicalMarketing We Rewrite the Model Slightly where w i  0 for i = 0, 1, 2, ···,  and

Slide Time Structured Data MathematicalMarketing Bring in the Backshift Operator and Assume a Geometric Pattern for the w i Now we assume that w i = (1 - ) i 0 < < 1

Slide Time Structured Data MathematicalMarketing Given Those Assumptions Anyone care to say how we got to this fraction?

Slide Time Structured Data MathematicalMarketing Substitute That into the Equation for y t

Slide Time Structured Data MathematicalMarketing Adaptive Adjustment Define as the expected level of x (prices, availability, quality, outcome)… So consumer behavior should look like

Slide Time Structured Data MathematicalMarketing Updating Process Expectations are updated by a fraction of the discrepancy between the current observation and the previous expectation

Slide Time Structured Data MathematicalMarketing Redefine  in Terms of a New Parameter Define = 1 -  so that

Slide Time Structured Data MathematicalMarketing More Algebra

Slide Time Structured Data MathematicalMarketing Back to the Model for y t We end up at the same place as slide 25