Dealing with population instability
There could be significant problems from using one time chunk in a time series to predict the next chunk. For example, if you were using data from September and October to predict retail shopping expenditure in November and December, you would be very wrong. Instead, you need to note the annual trend of increased shopping between Thanksgiving and Christmas to make correct predictions.

So, while this sort of skirts the questions you asked, I think that if you are having difficulty using one month to predict the next month, you need to start looking at longer-term trends, especially annual ones. I would investigate this first before looking into population instability.