# Intervention analysis with multi-dimensional time-series

I would like to do an intervention analysis to quantify the results of a policy decision on the sales of alcohol over time. I am fairly new to time series analysis, however, so I have some beginners questions.

An examination of the literature reveals that other researchers have used ARIMA to model the time-series sales of alcohol, with a dummy variables as regressor to model the effect of the intervention. While this seems like a reasonable approach, my data set is slightly richer than those I have encoutnered in the literature. Firstly, my data set is disaggregated by beverage type (i.e. beer, wine, spirits), and then further disaggregated by geographical zone.

While I could create separate ARIMA analyses for each disagregated group and then compare the results, I suspect there is a better approach here. Could anyone more familiar with multi-dimensional time-series data provide some poitners or suggestions?

## 3 Answers

The ARIMA model with a dummy variable for an intervention is a special case of a linear model with ARIMA errors.

You can do the same here but with a richer linear model including factors for the beverage type and geographical zones.

In R, the model can be estimated using arima() with the regression variables included via the xreg argument. Unfortunately, you will have to code the factors using dummy variables, but otherwise it is relatively straightforward.

If you wanted to model the sales of drinks types as a vector [sales of wine at t, sales of beer at t, sales of spirits at t], you might want to look at Vector Autoregression (VAR) models. You probably want the VARX variety that have a vector of exogenous variables like region and the policy intervention dummy, alongside the wine, beer and spirits sequences. They are fairly straightforward to fit and you'd get impulse response functions to express the impact of exogenous shocks, which might also be of interest. There's comprehensive discussion in Lütkepohl's book on multivariate time series.

Finally, I'm certainly no economist but it seems to me that you might also think about ratios of these drinks types as well as levels. People probably operate under a booze budget constraint - I know I do - which would couple the levels and (anti-)correlate the errors.

Each time series should be evaluated separately with the ultimate idea of collecting i.e. grouping similar series into groups or sections as having similar/common structure. Since time series data can be intervened with by unknown deterministic structure at unspecified poins in time,one is advised to do Intervention Detection to find where the intervention actually had an effect. If you know a law went into effect at a particular point of (de jure) this may in fact (de facto) not the date when the intervention actually happened. Systems can respond in advance of a known effect date or even after the date due to non-compliance or non-response. Specifying the date of the intervention can lead to Model Specification Bias. I suggest that you google "Intervention Detection" or "Outlier Detection". A good book on this would be by Prof. Wei of Temple University published by Addison-Wessley. I believe the title is "Time Series Analysis". One further comment an Intervention Variable might appear as a Pulse or Level/Step Shift or a Seasonal Pulse or a Local Time Trend.

In response to expanding the discussion about Local Time Trends:

If you have a series that exhibits 1,2,3,4,5,7,9,11,13,15,16,17,18,19... there has been a change in trend at period 5 and at 10. For me a main question in time series is the detection of level shifts e.g. 1,2,3,4,5,8,9,10,..or another example of a level shift 1,1,1,1,2,2,2,2, AND/OR or the detection of time trend breaks. Just as a Pulse is a difference of a Step, a Step is a difference of a Trend. We have extended the theory of Intervention Detection to the 4th dimension i,e, Trend Point Change. In terms of openess, I have been able to implement such Intervention Detection schemes in conjuction with both ARIMA and Transfer Function Models. I am one of the senior time-series statisticians who have collaborated in the development of AUTOBOX which incorporates these features. I am unaware of anyone else who has programmed this exciting innovation. Perhaps someone else can comment on an R package that might do that but I don’t think so.

• Could you elaborate a little more on what a Local Time Trend intervention variable looks like? I am familiar with the other three. Commented Apr 1, 2011 at 0:05
• Also, can you point me towards an R package that might be able to do intervention detection? Commented Apr 1, 2011 at 0:12
• If you have a series that exhibits 1,2,3,4,5,7,9,11,... there has been a change in trend at period 5. The main question in time series is the detection of level shifts e.g. 1,2,3,4,5,8,9,10,..or another example of a level shift 1,1,1,1,2,2,2,2, and or the detection of time trend breaks. Commented Apr 1, 2011 at 0:44
• How can one find interventions in a time series with external regressors? How would one know that the regressor does not explain the intervention? Commented Jul 5, 2019 at 22:43
• if an intervention in Y is found AFTER the impact of X and it's history an Y's history THEN it is declared an anomaly/pulse with external regressors. Commented Jul 6, 2019 at 6:45