When conducting an intervention analysis with time series data (aka Interrupted Time series) as discussed here for example one requirement I have is to estimate the total gain (or loss) due to the intervention - i.e. number of units gained or lost (the Y variable).

Not entirely understanding how to estimate the intervention function using a filter function within R, I went about it in a brute force manner, hoping this is general enough to work in any situation.

Lets say that given the data

 cds<- structure(c(2580L, 2263L, 3679L, 3461L, 3645L, 3716L, 3955L, 
    3362L, 2637L, 2524L, 2084L, 2031L, 2256L, 2401L, 3253L, 2881L, 
    2555L, 2585L, 3015L, 2608L, 3676L, 5763L, 4626L, 3848L, 4523L, 
    4186L, 4070L, 4000L, 3498L), .Dim = c(29L, 1L), .Dimnames = list(
        NULL, "CD"), .Tsp = c(2012, 2014.33333333333, 12), class = "ts")

we decide that the best fitting model is as follows, with the intervention function as

$m_t= \frac{\omega_0}{(1-\delta B)}X_t$ where $X_t$ is a pulse at October 2013.

fit4 <- arimax(log(cds), order = c(1,1,0),include.mean=FALSE, 
               xtransf = data.frame(Oct13 = 1*(seq_along(cds)==22)),
               transfer = list(c(1,0))
               ,xreg=1*(seq_along(cds)==3))
fit4

#    ARIMA(1,1,0)                    
    
#    Coefficients:
#              ar1    xreg  Oct13-AR1  Oct13-MA0
#          -0.0184  0.2718     0.4295     0.4392
#    s.e.   0.2124  0.1072     0.3589     0.1485
    
#    sigma^2 estimated as 0.02176:  log likelihood=13.85
#    AIC=-19.71   AICc=-16.98   BIC=-13.05

I have two questions:

1. Even though we have differenced the ARIMA errors , to assess the intervention function which was then technically fit using the differenced series $\bigtriangledown X_t $ is there anything we need to do in order to "change back" the estimate of $\omega_0$ or $\delta$ from using $\bigtriangledown X_t $ to $ X_t $?

2. Is this correct: In order to determine the gain of the intervention, I constructed the intervention $m_t$ from the parameters. Once I have $m_t$ then I compare the fitted values from the model fit4 (exp() to reverse the log) to exp( fitted values minus $m_t$ ) and determine that over the observed period, the intervention resulted in 3342.37 extra units.

Is this process the correct one to determine the gain generally from an intervention analysis?

    int_vect1<-1*(seq_along(cds)==22)
    wo<- 0.4392
    delta<-0.4295
    
    
    mt<-rep(0,length(int_vect1))
    
    for (i in 1:length(int_vect1))
    {
      
      if (i>1)
      {
        mt[i]<-wo*int_vect1[i]+delta*mt[i-1]
      }
      
    }
    
    
    mt

sum(exp(fitted(fit4)) - (exp(fitted(fit4) - mt)))