# Does dummy intervention variable (pulse or step) must be differenced when it is added to ARIMA model?

I have read some opinions from this forum and from other sources that when the dependent variable in any from of ARIMA model (whether ARIMA errors, ARIMAX or transfer function)is differenced, you should difference all covariates as well, including dummy variables. At the same time, by reading different ARIMA analyses/discussions etc, it seems that it is usual practice not to difference intervention variables. Unfortunately I have not found a good material that would tackle this issue directly and thoroughly. So, my question is, do you necessarily have to difference dummy variables in ARIMA model and how this affects the intrepretation of estimated coefficients?

Thanks for the comment. But being more specific with regard with coefficient interpretations, lets assume that we have ordinary regression model:

My point is that when we have Dt as specified above (i.e. linear trend starting at some point) in the original model, then after first differences we should get usual intervention dummy but now it is in the differenced model. Just like we get intercept in the differenced model from the trend variable. So, from this I derive that when I difference all the variables in regression model and then add usual intervention dummy without differencing it (i.e. 0 before intervention and 1 after intervention), I should interpret estimated coefficient of that dummy as the coefficient of Dt specified above in the original model. For instance, lets say the value of dummy coefficient in differenced model will be estimated as 0.5, then it means that intervention has initiated a positive linear trend, i.e. the value of dependent variable increases by 0.5 unit each period after the intervention.

On the other hand, if I add differenced intervention dummy (that in practice means the value of 1 only in the intervention period and 0 otherwise), I interpret it in usual way. Specifically, if the coefficient value will be estimated as 0.5 again, it means that the mean level of dependent variable has increased by 0.5 unit, i.e. there has been a level shift.

Is my logic correct? And if it is, it should apply to the model of regression with ARIMA errors as well.

EDIT:The simulations below are interesting experiments and helped me on little bit but I am still confused in regard with interpreting the coefficients under different dummy specifications. But couldn't the logic be as follows. Let`s take the famous and often cited Box and Tiao (1975) model for LA Oxidant Data. They specify the following model:

I am not completely sure but as I understand it, they apply seasonal differences on dependent variable and on the first intervention variable $\xi_{t1}$. They do not difference second and third intervention variables $\xi_{t2}$ and $\xi_{t3}$ because their model could be equivalently represented as follows:

In their article, they also classify alternative transfer functions and one of them, called the “ramp” response, seems to be applied to $\xi_{t2}$ and $\xi_{t3}$ variables. This “ramp” response, to my understanding, is exactly the case under which the step dummy (without differencing it) is applied to differenced data. I copy here its visual appearance from Box and Tiao (1975):

It actually looks something I described above, i.e. there is a linear trend starting from intervention period (or from other period, if necessary, specified by the backshift operator in the numerator). So, I am wondering if my logic is correct if I state that when the step dummy is applied to differenced data, it is the ramp response model and if differenced step dummy is applied to differenced data it should be interpreted as usual immediate step change? Or am I mixing something up...

• If the power of numerator $B^0$ then then linear trend would start at intervention period else would start at what ever the power is. Example, if $B^2$ in numerator then there would be a delay of 2 periods post intervention at which linear trend would start – forecaster Dec 30 '14 at 16:39
• If your data is non stationary, then you need to difference both dependent and independent variables otherwise it leads to spurious regression.source 1 source 2 – forecaster Dec 30 '14 at 17:06
• The ramp components e12 and e13 become permanent changes in the steady state constant (period to period differential ) > When you apply the seasonal differencing operator on all series this becomes clear. So as forecaster says (in this case since the noise term has a denominator differencing operator) you apply this to all series to clear fractions. In cases models with noise term denominator differencing need to difference all series in the model. Note that y(t)=b0+b1*t +a(t) has no differenxing operators present. – IrishStat Dec 30 '14 at 17:21
• @forecaster If Y and X are themselves upward trending a very possible TF model is Y=b0+B1*X+a(t) . Just because Y and X need to be differenced to IDENTIFY a suitable model does not imply the need to carry unwarranted luggage(i.e.differencing operators) into the TF. – IrishStat Dec 30 '14 at 17:24
• Thanks for these comments. I think the main source of my confusion arose from misunderstanding of how lag operators work in regression model. So, if I understand correctly, although differencing all series (that is reflected by the denominator of noise term) clear the fraction of second and third intervention variable, one still must apply the ramp TF (where the decay parameter delta = 1) to estimate this model. – IS_Stat Dec 30 '14 at 20:35