# 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?

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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.

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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.

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Could you elaborate a little more on what a Local Time Trend intervention variable looks like? I am familiar with the other three. –  fmark Apr 1 '11 at 0:05