Time Series Machine Learning Feature Selection Problem I have to solve a time series model that can take one of two shapes. It can probably take more but here are the two I'm going to ask about. If you have other ideas they are of course welcome. 

First Possible Model - 


  X(Dependent variable 'Spending') = X(lag1)...X(lagN) + X(Dummy
  variable when US) + X(Dummy Variable when Mexico) ... + Error term


Or Make a separate model for each Country like -


  X(Total Spending in the US only) = X(lag1)...X(lagN) + Error term
  
  X(Total Spending in the Mexico only) = X(lag1)...X(lagN) + Error term
  
  … for Each country


Mathematically I can't decide what approach is better. I will use an F-Statistic, dickey fuller statistic to check the auto regression for stationarity and then compare the two models but I wanted to see what others thought of the theory and if you should ever include the dummy variables. 

I'm looking more for an answer that includes mathematical reasoning.
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.