# Applying regression to a time series where each season has a a linear relationship with a different independent variable

I am trying to apply regression to a problem where I have a time series that I want to predict - say y. Y is seasonal and let's say has 3 seasons summer, winter and other.

I have 3 predictors X1, X2 and X3. Y has a linear relationship with X1 during Summers but not during the other seasons. Similarly it is linearly related to X2 in winters and X3 in other months. ie Y(summer) ~ X1. Y(winter) ~ X2. Y(other) ~ X3

My current approach is to run 3 separate regressions after splitting the dataset into 3 parts (summer, winter and shoulder) We get the final time series predictions by combining predictions from each season. In the current approach the final residuals seem to be expressing some trend and are autocorrelated. So we are thinking of modelling the residuals separately

Is this a correct approach to such a problem or is there a better time series way to approach this? Any pointers to books/articles that solve similar problems will be really helpful

• This sounds to me like your indicators of season are perfectly separate wrt your three predictors. In that case you can just use a factor indicator of season as your predictor variable and begin building the model from there. Apr 11 at 0:52