Time Series Machine Learning Feature Selection Problem
I will give you some other idea of it, but without mathematical. 

Like horaceT's mentioned, it's all about feature construction and feature selection, then you have many machine learning method which can do the regression. 

So, just look it as regression model, not just the time series model. Then you will get the following model:

y(Dependent variable 'Spending') = X1(moving average term)...X2(lastquarter moving average term) + X3(country id) + X4(Fourier term) ... + Error term

The moving average can catch the dependent variable's trend, Fourier term can catch the dependent variable's cycle. Maybe you can construct other feature.

Finally, you can selection the features using tree-based model, such as xgboost.