# How to identify functional form of relationship between response & input series in dynamic regression/arimax?

Problem statement

A US insurance company advertises on national television in an attempt to increase the number of insurance quotations provided (and consequently the number of new policies).

Objective

I want to identify correct functional form of relation between Quotes & the below variables

1. Lag1 of Quotes
3. Lag1 of TV advertising expenditure
4. Lag2 of TV advertising expenditure
5. Lag3 of TV advertising expenditure

and this functional form should to suitable for forecasting Quotes as well.

Since we are dealing with time series data I thought that dynamic regression/Arimax can be used to regression reponse series and input series as shown below to get the coefficients of the input series.

library(fpp)
# Lagged predictors. Test 0, 1, 2 , 3 lags.
insurance[,2],
c(NA,insurance[1:39,2]),
c(NA,NA,insurance[1:38,2]),
c(NA,NA,NA,insurance[1:37,2]))

Dynamic regression output

> summary(fit)
Series: insurance[4:40, 1]
ARIMA(1,1,1)

Coefficients:
-0.0069  0.6003     -0.0563  1.2837  0.1808  -0.1444  -0.0784
s.e.   0.3393  0.2785      0.4082  0.0931  0.5685   0.1078   0.0615

sigma^2 estimated as 0.2142:  log likelihood=-22.41
AIC=60.83   AICc=66.16   BIC=73.5

Training set error measures:
ME      RMSE       MAE        MPE   MAPE      MASE        ACF1
Training set -0.006497616 0.4500943 0.3529388 0.03671748 2.7024 0.2247559 0.001845781

I understand that the ARIMA(1,1,1) denotes the arima structure of errors/residuals & not response series.

My problem is that I want to use the functional form of this relationship in an optimization model as a constraint, so can i ignore the ARIMA(1,1,1) in the functional form because I cant think of a way to inlcude ARIMA(1,1,1) of residual in the optimization model.

Questions

1. When I write the functional form relating response series and input series can I omit the ARIMA(1,1,1) part of error? Or is the functional form wrong if I omit the ARIMA(1,1,1)? i.e. can i just say the below ignoring the arima(1,1,1)?