Slope Approximations
The fit below is simply Salest = + Advt + t. Beta represents slope. Alpha is the
intercept. Et is the error term. Pretty basic model. While contemporaneous advertising
is highly significant, the residual plot shows remaining structure
Alternate Variables
Dummy variables are used to separate the four different advertising regimes in the Lydia
Pinkham data set (annual data). There are several ways to define the dummies, and these
give different regression coefficient estimates, with diff
Annual Lydia Pinkham data, Part I
Here is another fit for the annual Lydia Pinkham data. It adds interaction terms to the
geometric decay model, the interaction between time periods and contemporaneous
advertising, and between time periods and lagged sale
Bivariate Settings
Consider the bivariate setting of sales and advertising. Is there one-way causation, and if
so, what is the direction? Or is there mutual causation? The questions are complicated
by the fact that time series of both sales and advertisin
Distributed Lag Models
Let y be a dependent variable and x an independent variable. The model
is called a distributed lag model. The effect of the independent variable on the dependent
variable is distributed over a number of lags (e.g., y is sales and x
K Factor Correlation
rk = Corr(eAdv, t, eSales, t-k) .
If rk is significantly different from zero for any k > 0, then we have evidence that sales
Granger-causes advertising. If a cross correlation is significant for k < 0, then there is
evidence that adve
P-Value Analysis
For annual data we find that advertising does not Granger-cause sales, unless we include
the contemporaneous advertising term and include it in the test (the p-value is then 0.01).
A study of sales and advertising by Ashley, Granger, and
Regression Coefficients
In this framework the base for comparison is the average of the four time periods, after
adjusting for the variables At and St1. The estimate of this base position is the intercept
estimate, 224.8863. The regression coefficients fo
Structural Time Analysis
This fit with interactions does give residuals which indicate decent reduction to white
noise. The fit does suggest some structural differences between the time periods.
However, the time periods are rather short, especially the f