Solved – Intervention analysis with multi-dimensional time-series

arimaintervention-analysismultivariate analysistime 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?

Best Answer

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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Solved – How to calculate proportion of variance of an aggregate time-series accounted for by the variance of a sub-series

It turns out that, as always, there are multiple, competing ways of figuring this out. The most common involves estimating the finding the change in $R^2$ when the independent variable is added to the model, as in a stepwise regression. In order to eliminate ordering effects, all possible orderings are tried and the resulting change in $R^2$ is averaged. This seems to have been invested multiple times - the best explanation I have come across is Kruskal (1987), "Relative Importance by Averaging Over Orderings", The American Statistician 41(1):6-10. This method is elaborated on as hierarchical partitioning in Chevan and Sutherland (1991), The American Statistician 45(2):90-96.

The R packages hier.part implement this algorithm (and the relaimpo package includes many more). Indeed, I'd recommend anyone interested in this to check out the relaimpo package which can also estimate confidence intervals using bootstrapping, either assuming simple random sampling or using the survey package.

Using the example given above with the hier.part package gives the following:

> require(hier.part)
> p <- hier.part(y, data.frame(x1, x2, x3, x4), family="gaussian")
> p$I.perc
          I
x1 10.96334
x2 18.18595
x3 12.67995
x4 58.17075
>

bargraph of relative importance of regressors, showing x4 to have 58%

I should probably add that this method isn't appropriate for time-series data given the for all the usual reasons that regression of time-series data should be undertaken with caution.

Solved – Interrupted time-series analysis for panel data

I would identify the true date of the intervention i.e. when it was fully realized for each state separately using Intervention Detection schemes. The true date (de facto) can often be different from the "known date" (de jure) because of either a pre or a post effect (delay). With a common ARIMA model and one composite intervention variable constructed from the de facto dates, I would then globally estimate all parameters ie. the ARIMA coefficients and the national impact. The coefficient for the composite Intervention Variable would then reflect the national effect. When conducting this Global estimation you need to ensure that the prediction/fitted value for the first reading in the second state is not affected by the latest readings for the first state. For example if the ARIMA model is an AR(12) the first 12 expectations for each state would be the 12 observed values thus generating 12 zero errors. Commercial software with this built-in feature is rare but available.

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