How to determine if the mean of 1 time series is significantly greater than that of a group of other time series?

Question

I have 20 different time series data over the past 60 days. Each time series is collected from 20 different geographic zones. A test was run in zone A but not in the remaining 19 zones. Is there a way to determine whether the time series from zone A (the experimental one) is significantly different from the time series belonging to the control zones (the remaining 19 time series)?

To be clear, there are a couple ways that times series could be similar. For example, they may be correlated (they go up and down together) or they may have a similar mean. I am specifically interested in the later type of similarity. How can I determine if the mean of the time series from zone A is significantly higher than the mean of the mean of the time series from the control zones?

Answer 1

It is crucial what you assume about the time dependence of your data series.

If you do not assume any particular functional form of how your populations develop over time, each time point will have a different underlying population/data generating process. At each time point $t=1,...,60$ your situation is equivalent to the following. You have drawn $20$ random numbers, each from a different population, and you are asking whether the mean of population $P_{it}$ (where $i=1,...,20$ indexes the twenty populations at a given time $t$) is different from the means of the other populations $P_{jt}$ where $j \neq i$. You cannot answer this question with any certainty because you only have one observation from each population. Note that this is in contrast to the cross-sectional setting where you assume that $P_{it} \equiv P_i$ so that you have $60$ observations from each population. Meanwhile, in this time series setting you have $1$ observation for each of $20 \times 60$ populations.

If you do assume some particular functional form of how your populations develop over time, then you have some parameters and hyperparameters that you may be interested in. For example, you may assume that $P_{it}$ is distributed as $N(\mu_{it},\sigma^2)$ where $\mu_{it}=\alpha_{i}+\beta t$. Then you could formulate a hypothesis about the parameters $\mu_{it}$, e.g. that $\mu_{1t}=...=\mu_{20,t}$ for all $t$. Given the assumptions above, it would amount to estimating the hyperparameters $\alpha_i$ and testing whether $\alpha_1=...=\alpha_{20}$. Of course, you may want to test the assumptions as well; otherwise rejecting the null hypothesis may be due to a failure of the assumptions. This is just one example, but I hope it conveys the message.

Answer 2

This is a problem in pooled cross-sectional time series analysis.Use time series methods to identify an appropriate ARIMA model never assume a structure. Identify an ARIMA model that roughly characterizes each of the 20 series. Estimate this model globally and locally therefore 21 individual models/sets of parameters . Form an F test based upon the respective error sum of squares to test the hypothesis of a common set of parameters. Upon rejecting the hypothesis of a common set of parameters one can then find which subset of the 20 is different from the others. Good advanced time series software should have this option routinely available.