# transfer function-noise modelling in R

I recently started to learn transfer-function model, which, more specifically, is transfer-function-noise model(TFN). I have also attempted modeling it in R.

I found one thing that is baffling.

according to Tsay(2009) Lecture notes, Montgomery et al(2015), and Bisgaard et al(2011), in practice, the transfer function Model comes with innovations (the noise term). However if I understand correctly, the arima/arimax function from R package TSA does not provide an argument to account for the ARIMA noise term. e.g. they provide xtransf and transfer to help formulate the transfer function itself, but nothing for modeling the noise term which is normally taken to follow ARIMA.

Do they just not model the noise term like I said or I missed a critical part?

Or, they are taking the original data as an ARIMA noise series and model it with arimax and use the transfer function arguments to model the change in mean function (Cryer, 2008)? Cryer(2008) has presented the intervention analysis model, which is suitably derived from transfer funciton-noise model, as:

$$Y_{t} = m_{t} + N_{t}\space\space\space (1)$$

a more TFN-alike representation of $$(1)$$ is from Montgomery(2015):

$$y_{t} = \frac{w(B)}{\delta(B)}\xi_{t}^T + \frac{\theta(B)}{\phi(B)}\epsilon_{t} = v(B)\xi_{t}^T + N_{t}$$

In intervention analysis $$\xi_{t}^T$$ is just a dummy/indicator variable.

• actually I think I was wrong; armiax has taken into account the noise model by asking for order to start with. transfer argument is to model the transfer function. I think I didn't understand the function properly earlier on. But it also means that I did not fully understand your explanation above.. – stucash Apr 30 '19 at 3:03