multivariate analysisoutliersrtime seriesvector-autoregression
Data: Multivariate Time Series, Series
Approach: I am trying to estimate the effect of rainfall on demand of the product using VAR( Vector Autoregression) model. Demand data has some outliers, like a month of sudden high demand and followed by zero values.
Question: How to treat these outliers (I am working in R), since I already have few data and deleting them is not an option for me.
I work with time series rather than being an expert in it and I have never seen this raised. I don't think, however, that you can go from data aggregated at the quarterly level to the monthly level. Why is it important to your company to answer the question at the monthly level if there are no patterns at the monthly level and thus no predictability?
We predict at the monthly level as well, but due to unpredictable changes over time (my organization has no knowledge and limited interest in our spending process since we have little capacity to refuse services) our results are rarely accurate at less than the yearly level which is what I tell the people I report the data to.
In response to the IP response. We get asked for monthly data even though it is doubtful we can predict such accurately given the level of variation in our poorly documented process. When you have to do something you do, the key is to point out that what is being asked is not reasonable given the nature of the process. That way decision makers know the limits on what is being provided.
Have you considered using exponential smoothing to do predictions (Holt simple, damp trend etc)? Its not a perfect solution (its what we use) but its easy to do, has a good reputation, and can be accurate as long as the basic patterns don't change too much I think.
I took your 1380 monthly values and introduced them to AUTOBOX and the following useful model ( in 3 parts ) was automatically developed/identified 
and 
and 
. The residual plot is here 
with acf here 
. A significant error variance change reduction) was identified here 
. Forecasts are here 
which were generated using monte-carlo / bootstrapping procedures.
As it turned out no expected value forecast was negative , but if it had one should/could simply convert it to zero as no constraint is available .. just a logical constraint.
In terms of your forecast function based upon a model you didn't share ... I would suggest better analytics might be helpful ... including remedying unusual values and non-constant error variance. The ARIMA model that was developed was (0,0,0)(0,1,1)12 . The ARIMA model should always be identified using data adjusted for deterministic structure.
You might want to look at How to improve this time series model? for a similar case study.
Best Answer
Why would you do VAR on product demand and rainfall? VAR assumes that the impact goes both ways, and I find it unusual to assume that demand for the product causes rainfall. It's not entire impossible, of course. After all our farming impacts weather according to global warming alarmists. However, me thinks in your case it's not what you are trying to say.
That's why I'd start with ARIMAX models, here's MATLAB example. In R there's astsa package with similar functionality. In ARIMAX X stands for the exogenous time series, in your example it would rainfall. Your dependent variable would be the demand. This is a univariate set up, much simpler and makes more sense, in my opinion.
The things you have to be aware of is the causality issues. It's often very difficult to establish causality. What if your product demand is not driven by rainfall, but by something else, which is linked to the annual cyclicality, which in turn is correlated with a rainfall? So, the fact that betas are significant doesn't automatically mean that rainfall causes or impacts the demand.
There are statistical methods for testing causality in the sense of Granger, for instance.
However, I would rely less on stats and more on your underlying theory or a domain knowledge. Let's say we're talking about umbrellas. Clearly, one would expect demand to depend on rainfall.