Intervention modeling allows you to deal with simple types of structural breaks while using an ARIMA model. It does this by adding a deterministic component, such as a dummy variable, to the basic ARIMA model. Its scope is somewhat limited because it requires that the date and general form of the change be known. It also requires the rather strong assumption that the time series properties of the series stay the same after the break. For instance, if a country chooses to devalue its currency and then attempts to keep its value in line with (say) the US dollar, we would know the date, and the form of the break, but the time series structure would likely be quite different before and after, making intervention modeling impractical.

 

Tools

Intervention models are estimated using BOXJENK. However, before running BOXJENK, you must create the deterministic components which will create the effect that you want. These are added to the model using the INPUTS option. Usually there is just one input, so we’ll demonstrate that:

 

boxjenk(inputs=1,other options) series  start end

# input p d q

 

For intervention modeling, it is usually simplest to keep the time series in its undifferenced form. Then, by using the APPLYDIFFERENCES option you will be able to figure out the form of the input based upon what change it makes to the level of the series. If you work with the differenced series, you will have to work out the input as a shift in the differenced series, which is usually much harder.

 

Here are some examples of intervention inputs:

 

For a permanent change in level beginning at T0

 

set change = (t>=t0)

 

For a short term change in level from T0 to T1

 

set change = (t>=t0.and.t<=t1)

 

For a permanent change in growth rate beginning at T0

 

set change = %max(0.0,t-(t0+1))

 

The instruction DUMMY can also be used to create these dummies (with the LS (level shift) option for the first two and the RAMP option for the third), and others. However, DUMMY uses equivalent definitions which are zero at the end of the sample; for instance, DUMMY(LS=T0) will be -1 through T0-1 and 0 afterwards. This makes it easier to extrapolate the effects beyond the end of the sample.

 

Notes

Even if you know the date of the intervention and are satisfied that you understand its form, and you are willing to assume that the time series properties are the same except for the structural break, you still have one problem: figuring out the proper form for the base ARIMA model. You can’t apply the standard identification techniques to the original series, as the autocorrelations will be thrown off by the structural break. There are a couple of ways to approach this:

1.You can apply the identification procedures to whichever stretch of “stable” data is longest; that is, restrict your attention only to the part of the sample before the break, or the part after it.

2.You can apply the identification procedure to the time series after accounting for the type of change that you are planning to permit. This works best for a level change, as you can just take the residuals from the regression on CONSTANT and your dummy variable. If you have a growth rate change, you’re almost certainly looking at data which, at minimum, will be differenced once. The equivalent procedure in that case would be to difference the data, and then take the residuals from the regression on CONSTANT and a level change dummy starting at T0.

 

The “denominator lags” parameter on the input line can be used if there is a “phase-in” before the permanent change takes full effect. When applied to a temporary effect, the denominator lags create a “phase-out.” A single denominator lag allows for a geometric convergence to the new level. If you use this, you have to be careful interpreting the coefficients. In all the examples above, if you use a simple input without lags, the coefficients on the CHANGE variable will be easy to interpret: in the first two, they will show the size of the level shift, and in the last the difference in the growth rate. However, in the two permanent effects, if the coefficient on the input is 3.0 with a denominator lag coefficient of .6, the long-run effect will be 3.0/(1–.6) or 7.5.

 

Example

The U.S. stock market took a very large tumble in October, 1987. The example INTERVENTION.RPF applies intervention techniques to the S&P 500. Because the series we are using consists of monthly averages (not the ideal choice for a serious analysis of this event—daily closing prices would be better), the October, 1987 value does not fully show the decline, since over half the values averaged into the October value were from before the large drop on October 19. This means that we’re going to need a lag in the numerator as well, to give the effect two months to take full effect. We’ll also look at two possible ways to model the change: a permanent drop, or a temporary drop with a phase out (see DeLurgio (1998) for another treatment of this same basic example).

 

We won’t go through the process of choosing the base ARIMA model for this series. It proves to be a relatively easy choice of an ARIMA(0,1,1) with a CONSTANT on the logs price (The ARIMA(0,1,1) is the expected model for the time-averages of a series which is likely to be fairly close to a random walk in continuous time).

 

The following are the regression tables from the more straightforward of the models. The results show a roughly 25% "permanent" drop in the index through the two month period in October and November.

 

1.  CONSTANT                  0.010310788  0.002907481      3.54630  0.00050947

2.  MA{1}                     0.200127855  0.077450051      2.58396  0.01063889

3.  N_PERM{0}                -0.128493661  0.031743366     -4.04789  0.00007949

4.  N_PERM{1}                -0.137958367  0.031372235     -4.39747  0.00001963

 

Copyright © 2025 Thomas A. Doan