(Consistent) Feature Selection Over Time Let us assume the following simple scenario: Your goal is to forecast the sales figures of different companies. You assume that the sales figures are determined by company-specific characteristics (size, industry, etc.) and economic covariates (GDP growth, etc.).
Assume further that each company occurs in only one particular year of your database (cross-sectional data). When calibrating the model, you use a rolling-window approach. In particular, you calibrate your model with data from 2010-2014 and intend to make predictions for 2015. In a next step, you move the window to 2011-2015 for calibration and 2016 for prediction and so on. This approach ensures that you maintain the time dependence of the observations.
Since you can draw on a wide set of features, you use a selection method, such as Lasso. However, this may result in completely non-overlapping characteristics in each period, which does not make sense from a business perspective. Due to the time dependence, it is also not possible to identify the features in a first step across all observations. Ultimately, you want to identify a set of features that work well over time.
The question now is: Are there any proven methods or ways to deal with such a situation?

Instead of doing feature selection and modelling exclusively per-period that would generate different selection sets, one would devise a conditional selection to different time-windows or use feature selection techniques developed for temporal data.  See Feature selection for high-dimensional temporal data.
