# 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.

## 1 Answer

I have not and would not use the software you mention BUT if I had no choice .......

As you intimate they are not allowing for explicit incorporation of the model for the error term . In sympathy for the authors they are assuming the you the ma/ar structure and multiply both the Y and the X and voila you have a white noise error term as the model for the original noise term is now non-parsimoniously implicit in the expanded Y and X coefficients.

You asked if I knew of a piece of software that was VERY general in this area. I have helped develop this it is called AUTOBOX http://www.autobox.com and follows How to predict the next number in a series while having additional series of data that might affect it? . It also has an automatic model detection feature which includes intervention detection . https://autobox.com/capable.pdf is a useful introduction. It is available in R .

• thanks for your insight :) I think in R, it is almost the only practical package/function that does some level of Transfer Function Modeling; I can't find other packages that provide this, apart from Tsay's padegogical MTS package. i.e. MTS::tfm, MTS::tfm1, MTS::tfm2, all of which explicitly ask for noise term model. Any chance you know a package that models TFN correctly? thanks a lot! – stucash Apr 7 '19 at 18:20
• I take back what I said in the above comment; this package also was done for the author's book, therefore I at least shouldn't say "practical" whereby I meant robust and heavily tested. – stucash Apr 7 '19 at 20:05
• 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
• at this juncture I think the best path is for us to have a SKYPE session and I will walk you through the interface culminating in the monte-carlo bootstrap to generate confidence intervals for the forecasts.including identified pulses. – IrishStat Apr 30 '19 at 7:54
• let me know when you want to do that . Thanks again for a good question.. – IrishStat May 1 '19 at 5:41