I have a variable $y$ that I would like to model using a set of independent variables $X$. Both $y$ and $X$ are monthly measurements (~10 years in total) and appear to be subject to seasonal variability. Theory and initial analysis suggest that multicollinearity and interaction effects are also present in my $X$.
Right now all I want to do is to explore the relationship between $y$ and the variables in $X$, but in the future, I probably will need to forecast future values of $y$. Initially, I just used the data as is in an OLS fit, only standardising them to z-scores because my theory suggested that I didn't need an intercept.
My colleagues said that this approach was naive and that there would be no way of differentiating genuine relationships from the seasonal cycle. To that end, I performed a seasonal adjustment on each variable by fitting and removing 12 dummy variables. However, I now find that my seasonally adjusted $X$ and $y$ fit very poorly compared to my previous model ($R^2$ with holdout data drops from 0.7 to 0.3).
Is there any way to effectively deal with seasonal variation without directly estimating & removing the seasonality in this way? Because of the seasonal cycle, I think that this could be described as an autoregressive process, so I guess I could try adding lagged variables but I am unsure of what to do next. What I don't understand is the following:
- If all my variables have a similar seasonal cycle (i.e. peak in summer, the trough in winter, though exact maxima & minima vary within 1-3 months), do I need to explicitly account for seasonality at all?
- If my fits with seasonally adjusted data are so poor, does this mean that my data is unusable and/or my theory is incorrect?
- As an autoregressive process, should I include both lagged $y$ and $X$ terms in my fits, as they all have seasonal variability? Most examples I've read online seem to be concerned with lags in $y$ only.
- For this problem should I be using $ARIMA$, $ARIMAX$, or something else entirely?