# Decomposing Historical Data - Arimax versus linear regression

In the creation of a "marketing mix model", past sales data, is regressed against various marketing spend (TV, radio, billboards etc) along with other aspects influencing a companies sales such as pricing changes, competitors, seasonality, etc.

The goals are to understand the impacts of marketing on sales and how one might optimize these investments.

What is typically output from the model are both a retrospective and a forecast.

1) Forecast : If next year we assume pricing, competitors and given levels of marketing spend in these channels, what will sales be?

2) Retrospective : A decomposition of the prior years sales into baseline (the intercept) and the other elements of our model.

Often linear regression is used for these models, but it seems Arimax would be more suitable. Can this type of model still be used to decompose historical or future periods (we have the regression component but I am wondering if the arima errors portion keep us from making this inference)?

Here is an example :

dat<-read.table(dat.csv)  #The data
sales_ts <- ts(dat$$SalesVol, start=c(1988, 1), end=c(1991, 12), frequency=12) arima_mod<-Arima(sales_ts,order = c(1,1,1),xreg = as.matrix(subset(dat,,c(TVAds,RadioAds)))) summary(arima_mod) arima_mod$$fitted[3]


The summary of the model is:

Regression with ARIMA(1,1,1) errors

Coefficients:
0.4298  -0.3219  7.6774   81.2355
s.e.  0.4032   0.3995  9.2620  108.2095


If I look at the fitted value for the 3rd observation is it 38169.22. The prediction from the linear model is

arima_mod$$coef['TVAds'] * dat$$TVAds[3] + arima_mod$$coef['RadioAds']*dat$$RadioAds[3]

#4069.048


So in this case the "baseline" would be 38169- 4069.

Would this work for the other observations and for predictions?