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My apologies if this has been addressed before, however my experience with statistics is a bit limited, particularly when it comes to time series analyses.

My current question relates to the analysis of a trend in monthly sales before and after an event for a single product. Suppose there is a new product with increasing monthly sales, however the company decides to initiate a promotion 12 months after it launches to further increase sales. A year later, the company wants to examine if the sales trend after the promotion is significantly different from the trend prior to the promotion (ideally the unit sales would be increasing at a faster rate after initiation of the promotion). So essentially, I want to determine if I had forecasted 12 months into the future just prior to the promotion, are the sales significantly different from what they actually turned out to be.

What statistical test is the most appropriate to use to determine if the monthly sales trend is significantly different before vs. after the promotion? I only have one data point for each time point (monthly sales) before and after, and since the sales are increasing I have come across problems due to the "stationarity" requirement for many tests, as well as the fact that I would only have 12 historical data points on which to base the initial forecast.

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Would a time series intervention analysis suit your needs? It estimates how much an intervention has changed a time series, if at all.

how to in R: http://www.r-bloggers.com/time-series-intervention-analysis-wih-r-and-sas/

example use case: What test should I use to determine if a policy change had a statistically significant impact on website registrations?

online course notes: https://onlinecourses.science.psu.edu/stat510/?q=node/76

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I realise this answer is a bit late for the poster of this question, but I thought it may help others reviewing this question.

There is an excellent tutorial paper on interrupted time series analysis by Bernal et al:

Bernal, J. L., Cummins, S., Gasparrini; A. (2017). Interrupted time series regression for the evaluation of public health interventions: a tutorial, International Journal of Epidemiology, 46(1): 348–355.

You can download it free at: https://www.researchgate.net/publication/303883790_Interrupted_time_series_regression_for_the_evaluation_of_public_health_interventions_A_tutorial

And best of all the paper's supplementary material includes R code for all of the examples discussed in the paper.

It is particularly relevant to the poster's question since it uses segmented linear regression which is more appropriate for smaller data sets (i.e. less than 100 time points) than ARIMA.

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Without knowing more about the data, especially the independent variables (the regressors), it will be impossible to give a one-fits-all solution for anyone here.

However the issue you are adressing falls into the area of Econometrics. As such, there are a couple of good texts which will bring you further.

They easiest book which will give you the tools to tackle this issue is Introduction to Econometrics by Stock&Watson. It is an undergraduate textbook which offers a very modern approach. Your small sample might give you issues.

The standard books one would recommend in this instance are either "Econometrics" by Fumio Hayashi or Econometric Analysis by Greene.

Finally, if you really want to dig deep into this problem, the go to guide is the landmark "Time Series Analysis" by Hamilton. It is, however, a challenging book.

Be aware that there are other approaches in Statistics, often with their own terminology and goals. However in this case Econometrics is the best fit as it is designed exactly for problems such as this.

Which tests you'd have to run specifically depend very much on data and model and approach. It will be some kind of structural break test, if you want to look into what's available there. But, as you already realized, with 12 observations the sample size and therefore the model selection will be an issue before you even get to these tests.

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    $\begingroup$ The Wikipedia article on structural breaks provides a good place to start, and cites a variety of approaches. en.wikipedia.org/wiki/Structural_break $\endgroup$ – JBK Jun 10 '13 at 22:08

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