How to analyse my time series experiment I have a question about how I can analyze the data from a time series experiment. I'm still a student and therefore am not really familiar with all the procedures and analyzes in SPSS, so i hope someone with the proper knowledge can help me with this problem. 
The goal is to show a relation between advertising en sales. To show this we have conducted a experiment with an experimental and control group. The experimental group is a province in which the people gets to see more advertising. In the control group the advertising stays the same as usual.
The dataset contains weekly sales data from several stores throughout the country from the control area and experimental area. And how much advertising is used in all the periods. The data goes back for 1.5 years so there are 70 measures before the experiment, and 10 measures in/after the experiment.
So my question is, how can I prove a relation between those two factors. I know I can do a simple t-test to test the difference between the two groups. But this doesn’t  take care of any trends of long/short term effects. 
 A: In order to test for the effect of a change in a known variable (advertising spend) one needs to control for both known and unknown impacts that have affected sales. Weekly indicators are certainly known as is the history of the advertising variable. If the series has had a level shift or a change in trend in the history of the data prior to the advertising change then this needs to be empirically identified along with any one time unusual activity and incorporated into the model providing robust estimates of the model parameters. Furthermore there may be autoregressive structure in the tentative model residuals that needs to be addressed in order to render an error process to be Gaussian and any subsequent F or T test to be meaningful. With this in place one can then form a test of the importance of the change in the advertising variable as a Level Shift at the point of the change in advertising would be detected. Unfortunately simple solution tools like SPSS sometimes fall far short of providing this kind of analysis. The literature of the Interrupted Time Series may help you. Search for McCleary and Hay as possible sources to get you started. Unfortunately their work was done in the 80's and is not fully up to date with the modern concepts of Intervention Detection.
EDIT:
Figure 1 from Charles's reference showing a clear level shift(intercept change) and a possible (but probably dubious/insignificant) trend change. Again Charles a thank you for showing the clear need for Intervention Detection procedures. 

A: I think that you could consider using a changepoint analysis. Some other methods that you might consider include:


*

*Split-Plot ANOVA

*MANOVA

*Mixed Models


In the split-plot ANOVA, we model the data as follows: $$Y_{ijk} = \mu + \alpha_i + \beta_{j(i)}+ \tau_{k}+(\alpha \tau)_{ik}+ \epsilon_{ijk}$$
where $Y_{ijk}$ denotes the sales for treatment $i$ from store $j$ at time $k$. Also, $\alpha_i$ is a treatment effect and $\beta_{j(i)}$ is a random effect of store $j$ receiving treatment $i$. Likewise, $\tau_k$ is a time effect, $(\alpha \tau)_{ik}$ is a treatment-time interaction term, and $\epsilon_{ijk}$ is an error term.
In the MANOVA approach, we would collect the sales data for each store over time and put it in a vector. So let $$\textbf{Y}_{ij} = \begin{bmatrix} Y_{ij1} \\ \  \vdots \\ Y_{ijt} \end{bmatrix}$$
denote the sales data for store $j$ receiving treatment $i$. Doing a MANOVA would test if there is an overall difference between the mean vectors of the two treatments.
The mixed models approach is very similar to the split-plot ANOVA approach except that assumptions are made about the error structure within a single unit.
