# Proving that the variance observed in time series A is due to the variance observed in time series B

I'm testing the hypothesis that the the variance observed in the number of medical consultations felt from 2009 through 2013 is due to the variance in user fees price, and not due to something else. I have 3 time series (from Jan. 2009 to Dec. 2013, measured quarterly) corresponding to the following variables (same population):

1. User fees price
2. Ratio of exempt consultations by exempt population
3. Ratio of not exempt consultations by not exempt population

In theory, with the exception of the user fee price, the variables that influence 2 and 3 are the same. I've already established that there is a strong correlation between 1 and 3 (Pearson r = -.83; df = 18; p < 0.01). Also, between 1 and 2 I've found a weak, insignificant correlation (r = -.4).

Question: Is this enough to statistically prove my theory? If not, how should I approach this?

Notes:

Healthcare demand is influenced by several variables, some of them easily measurable (such as population ageing, number of doctors, mean price of consultation, etc.), but others not so much (such as technology advances in healthcare), which makes it difficult to create a good regression model.

In this context, a user fee is a small fixed contribution payed by users for the medical consultation. Some people are exempt from this fee (the poor, children, patients with chronic diseases, pregnant women, ...).

@Tivie, you are into the realm of time series analysis. One of the pieces of baggage that comes with these kinds of data is a question about stationarity which is a property that time series may have, or may not have. To be a bit fast and loose with the concepts, stationarity implies that the (population) distributions underlying your time series data have (1) a defined mean (or a defined mean about some defined trend), and (2) a finite variance. If these two conditions obtain, then you can proceed with basic, possibly familiar, statistical approaches, such as a regression model of medical consultations as explained by time with user fee prices as an additional predictor.

Unfortunately, if your data are not stationary, the basic statistical models break (usually with some form of spurious correlation). You have not demonstrated that your data are stationary (using, for example a test for stationarity, such as the Dickey-Fuller test. There are many such, which one you need to use depends on the particulars of your data.) So the answer to your question is a provisional: No. You have not proved your theory.

What happens if you data are not stationary? Such a condition is often referred to as 'non-stationary,' 'integrated' or even 'unit-root' (or 'nearly integrated' or 'nearly unit root') time series data. As mentioned above statistical approaches appropriate to stationary data don't work so well with non-stationary data. You will need to read up on appropriate time series models. There are many different approaches to modeling time series. I like De Boef and Keele's article because they show how many such time series models relate, and offer a decent general model to start with (the single equation generalized error correction model).

There are also a number of time-series analysis textbooks, I have included a reference to one below. These tend to be a bit heavy on the level of required statistical chops. If you aren't quite comfortable, it might be time to call up an ally with time series analysis skills.

References

Banerjee, A., Dolado, J. J., Galbraith, J. W., and Hendry, D. F. (1993). Co-integration, error correction, and the econometric analysis of non-stationary data. Oxford University Press, USA.

De Boef, S. and Keele, L. (2008). Taking time seriously. American Journal of Political Science, 52(1):184–200.

• Thanks for the tips. I've followed your suggestion regarding reading more about time series analysis and stationarity. I have a question though: If I treat the data as data-pairs with no specific order as opposed to a time-series, do I still need to test for stationarity? – Tivie May 21 '14 at 14:45
• @Tivie, if your data are non-stationary, ignoring their time series data generating process will not change the fact that they are not i.i.d.. Of course, you could test for stationarity and if you find it proceed with some form of non-time series analysis. Also: if my answer works feel free to upvote and/or click the check-mark. – Alexis May 21 '14 at 14:50