# Transfer function in forecasting models - interpretation

I am occupied with ARIMA modelling augmented with exogenous variables for promotional modelling purposes and i have hard time explaining it to business users. In some cases software packages end up with a simple transfer function i.e. parameter * Exogenous Variable. In this case the interpretation is easy i.e. promotional activity X (represented by the exogenous binary variable) impacts the dependent variable (e.g. demand) by Y amount. So in business terms we could say that promotional activity X results in the increase of demand by Y units.

Some times the transfer function is more complicated e.g. division of polynomials* Exogenous Variable. What i could do is make the division of the polynomials so as to find all the dynamic regression coefficients and say that e.g. the promotional activity not only influences demand during the period that it takes place but also in future time periods. But since software packages output transfer functions as division of polynomials business users cannot make an intuitive interpretation. Is there anything that we could say about a complicated transfer function without making the division?

The parameters of a relevant model and the related transfer function are presented below:

Constant =4200, AR(1), Promotional Activity Coefficient 30, Num1=-15, Num2=1.62, Den1=0.25

So i guess that if we do a promotional activity this period the level of demand will increase by 30 units. Also since there exists a transfer function (division of polynomials), the promotional activity will have an impact not only to the current time period but also to subsequent periods. the question is how can we find how many periods in the future will be impacted by the promotional and what will be their impact per period in units of demand.

• This is a very good question, not many software/textbooks have addressed this issue but absolutely required in real world business forecasting. I know R and SAS have the ability to do this. There are specialists in this website who could answer this, I'll try to put something if I find time. Nov 18, 2014 at 19:53
• What about in the following case (two coefficients in the denominator): MA(1),MA(2), Promotional Activity Coefficient=0.12187, Num1=-0.2999, Num2=-0.19770, Den1=-1.35, Den2=-0.47. Thanks a lot! Jul 7, 2020 at 10:45

This answer is based on notation from Makridakis et. al textbook on forecasting. I would assume it is similar in any standard textbooks on transfer function modeling. I would also check out excellent text by Alan Pankratz on transfer function modeling as the following answer is motivated by excellent graphics in these two books. I'm using a notation called $r,s,b$ in transfer function equation you need to understand this from the reference text books for you to understand the material below. I have summarized them below:

1. $r$ is the number of denominator terms. (what is the decay pattern - rapid or slow?)
2. $s$ is the number of numerator terms. (when does the effect happens ?)
3. $b$ is the how much delay in taking effect.

A general transfer function takes the form:

$$Y_t = \mu + \frac{(\omega_0-\omega_1B^1- .....-\omega_sB^s)} {1-\delta_1B^1 - ...\delta_r B^r} X_{t-b}+e_t$$

It might help to put your coefficients in an equation format as shown below. Also consider $Y_t$ as Sales and $X_t$ as promotion/advertisement at time $t$ for easy understanding.

In your case $r$=1, $s$=2 and $b$ = 0

$$Y_t = \mu + \frac{(\omega_0-\omega_1B^1-\omega_2B^2)} {1-\delta B} X_t+e_t$$ where $e_t$ is an $AR(1)$ process. $\mu$ is the constant/level and $\omega$ is the numerator coefficients and $\delta$ is the denominator coefficient.

Applying your coefficients to the above equation translates to:

$$Y_t = 4200 + \frac{(30 + 15B^1- 1.62 B^2)} {1-0.25B} X_t+e_t$$

The numerator denotes the moving average (moving average) part and denominator denotes the auto regressive part of the transfer function. Think of numerator as when the effect starts and denominator will control the decay of the numerator factor. IT might further help to break down just the transfer function in an additive format using basic algebra to illustrate the effects.

$$\frac{30} {1-0.25B}X_t + \frac{15B^1} {1-0.25B}X_t - \frac{1.62B^2} {1-0.25B}X_t$$

I used SAS to do most of my calculations(see this website). Now performing recursive calculation on the first part of equation as noted in the website translates to the following figure. What this tells you is that Advertisement at time $t = 0$ causes 30 incremental units in Sales all things being equal. This advertisement also has an effect in subsequent periods example at $t = 1$ the effect is 7.5 incremental units, and so on caused by denominator coefficient $\delta = 0.25$.

The second part and third part of the transfer function, by applying recursive calculation translates to following chart. For the second part notice that sales at $t=0$ equates to 15 unit of sales lag 2 and decays further. For third part of numerator causes sales to decline by -1.62 units at lag 3 and decays further.

Combining all the 3 parts of transfer function additively using basic algebra translates to the final form as shown below:

What this tells you is that advertisement at $t=0$ causes 30 units of sales at $t=0$ and 22.5 units of sales at $t=1$ and rapidly decreases to 4 units of sales at $t=2$ and so on ....

Lets see what happens if you change the denominator coefficient from 0.25 to 0.70 and keeping the numerator as 30. By the way the following equation is a simple form of transfer function that works very well in practice is also called infinite distributed lag model or Koyck lag model.

$$\frac{\omega_0} {1-\delta B}X_t => \frac{30} {1-0.70B}X_t$$

This would be represented as the following figure, as you can see the decay is very slow due to the decay factor increased from 0.25 to 0.70.

Hope this is helpful. I have learnt thru experience that visualization is the only way you can explain transfer function to a non technical audience including me.A practical suggestion, I would recommend conducting experiments on data due to the fact that this could be just illusions as noted by Armstrong. If possible, I would do experimentation of your "causal" variable to establish the "cause and effect". Also I don't know why your numerator 3 is -1.62, it could be just spurious.

Please provide feed back if you find this post useful as it took some effort to respond to this answer.I learnt the visualization of transfer function in this website thanks to @javlacalle.

• Hello. Thnak you very nuch for your answer. It is very detailed and helps a lot. I guess we cannot avoid the polynomial division to explain in detail the transfer of the effect of the independent variabes to the dependent. From what i have seen, software packages report the polynomials of the numerator and the denominator and not the result of their division. Finally how have you concluded to the values of the e.g. first graph (30, 7.5, 1.9 0.5 etc)? Nov 25, 2014 at 11:00
• Glad to see that the answer was helpful, to perform calculations, I used SAS. There is a function called ratio in proc iml that I used to calculate the outputs for transfer function. Nov 25, 2014 at 16:18
• Absolutely amazing way of presenting the transfer function's actual meaning. Apr 3, 2020 at 6:51

In many circumstances that I have consulted on, there often is exceptional activity before the promotion reflecting lead effects. Automatically/routinely detecting this phenomenon is critical to good model development. Additionally Pulses, Level Shifts, Local time trends need to be considered otherwise they thwart/distort the analysis. We have also found that although differences might be necessary to identify the Transfer Function, they are not necessarily part of the final model. This and other points were not addressed in the seminal work of Box and Jenkins but are now routinely addressed . If you wanted to post your data I and others might be able to help elucidate on that while also investigating any necessary transformations such as power transformations or weighted least squares. I have used software which restates the Transfer Function as an ordinary regression (Polynomial Distributed Lag / Auto-regressive Distributed Lag ) model. This is very useful in explaining the model to customers/clients and also useful in subsequent utilization of the equation .

• could you elaborate on "restates the Transfer Function as an ordinary regression" please -- how to do this and/or software ? Apr 27, 2015 at 10:22
• Thanks, but others could benefit from your explanation too; I'll ask a new question if I can. Apr 28, 2015 at 17:11
• @denis I have programmed AUTOBOX to restate the Transfer Function as a PDL or ADL . The file is called RHSIDE.TXT Apr 28, 2015 at 17:24

In terms of expressing the TF model as pure right-hand side

MODELS ARE PRESENTED:
1.PURE MODEL IN TERMS OF THE INPUTS
Y=K1+[W(B)/D(B)]*X+[THETA(B)/PHI(B)]*A
2.AS A MIXED MODEL INCLUDING LAGS OF Y
D(B)*PHI(B)*Y= K2
=+PHI(B)*W(B)*X
=+D(B)*THETA(B)*A
=+PHI(B)*W(B)*X =+D(B)*THETA(B)*A

    WHERE K2 = K1*[D(B)*PHI(B)]
OR   K1 = K2*/[D(B)*PHI(B)]


ESTIMATION IS ACTUALLY DONE AS A (2)
WHILE THE TABLE PRESENTS IT AS A (1).
IN THE TABLE THE CONSTANT IS K2 WHILE
PRESENTED IN FORM (1) THE CONSTANT IS K1
WE PRESENT IT HERE IN FORM (2).

MODEL EXPRESSED AS AN XARMAX
Y[t] = a1Y[t-1] + ... + a[p]Y[t-p]
+ w[0]X[t-0] + ... + w[r]X[t-r]
+ b1a[t-1] + ... + b[q]a[t-q]
+ constant

The model automatically built for the sales data from the Bpx-Jenkins text was

. Expressing it as a "regression model" we get

• This looks like nothing more than a dump of computer output. Could you be more explicit about how it answers the question "how can we find how many periods in the future will be impacted by the promotion and what will be their impact per period in units of demand?" Where in all this stuff are those answers and what technique(s) are you recommending?
– whuber
Apr 28, 2015 at 18:20
• @whuber I was politely responding to the OP'S request that I be more specific. I can't be more responsive or specific without giving him actual proprietary code. "could you elaborate on "restates the Transfer Function as an ordinary regression" please -- how to do this and/or software ? – denis yesterday" In terms of his question .... in general one needs to do polynomial operations including division to express a TF as a PDL/ADL The right-hand side gives coefficients to answer the question that you specified here. Apr 28, 2015 at 20:35
• Because this site focuses on methods and principles rather than software, a software-only demonstration of "how to do this" at best has marginal value. A description using English and mathematical notation will be much more generally accessible to and appreciated by your readers. As far as the mechanics go, it is better to edit your previous answer, compared to posting a new one that is intended to be a continuation or amplification of that answer. The disconnection between the two posts is confusing and makes this one even less understandable when first encountered.
– whuber
Apr 28, 2015 at 20:39
• @whuver I thought that a separate response was appropriate as the OP was considering posting a separate question.. Apr 28, 2015 at 20:40