VAR model: combining multiple short samples generated by similar DGPs I have a problem which only has 35 observations. I like to apply around 6 variables in a VAR model to predict 25 observations into the future. Clearly, the data set is way too small. However, I do have the same sets of data for over 5000 markets. I wonder if I can leverage the large number of markets (each with 35 observations) by treating it like a sampling problem, where the coefficients derived from single market data are collected and merged to arrive at a meaningful mean with a small variance. Does this make sense?
 A: It depends on the application. I will give you some ideas without going much into details.
For example (a very contrived example, I must admit!), if the 5000 markets are all independent and behave in exactly the same way (the 35-observation samples have been generated independently by the same data generating process), then you effectively have a sample of $35 \times 5000$ observations. You could then create the response vectors and the design matrices (containing lagged variables) for each market separately and then stack them and feed into a VAR model. You would end up with $(35-p)\times 5000$ effective observations, where $p$ is the lag order of the VAR model.
Since the above example is too ideal to be interesting, you may consider softening the assumption that the data generating process is exactly the same for all the markets. But if the data generating processes are similar, then you could estimate 5000 VAR models jointly imposing some shrinkage of parameters towards common values, something like doing a fused lasso, so that you would incorporate the information of similarity between the markets.
For some related literature on regularized VAR models, see the Google Scholar profile of Christophe Croux and the website on Ines Wilms, especially the "software" section. E.g. Gelper et al. "Identifying Demand Effects in a Large Network of Product Categories" (2016). 
Further, if the data is gathered at the same time in 5000 different markets, some of the markets may be hit by common shocks, and thus the information from each different market will likely not be entirely "original". In an extreme case, you would be observing the same 35-observations-long process 5000 times, contaminated by some measurement error. Something should be done to account for such dependence, which effectively reduces the sample size you have. (At the moment I do not have a good idea of what could be done, though, but this should be a known problem and solutions should exist.) Maybe take a look at the panel data literature?
