I am doing a Multiple Linear Regression with 19 explanatory variables and around 500,000 data points. When I plot the Residuals vs Fitted plot, I see weird shapes where there appears to be multiple regions in it. One centered around 0, one vertical band below 0 and a funnel like band above 0.
Generally the funnel shape may indicate heteroskedasticity, but what would these multiple shapes in my plot indicate?
There are clearly latent clusters in your data. Do you have any other variables, especially categorical variables, that might account for the different bands? If so, there is an interaction between that (those) categorical variable(s) and something else.
In general, you should look at your data before you fit a model. You don't want to be surprised by this. What are your variables? What do they mean? Try looking at scatterplot matrices, etc.
I saw something like this in a data set once. The graph was of charitable contributions (Y) vs income (X). There was a distinct linear pattern with a slope of 0.10 that looked different from the rest of the data scatter. These were (I strongly believe) the tithers. The variable "tither", a 0/1 variable, is a type of latent variable mentioned by gung. It is latent because you don't know who the tithers are.
Bottom line, it looks like there are different mechanisms at work in your data, and you do not necessarily know which observations are responding to which mechanism (like tithers vs non-tithers). To tease out these latent structures, you could consider using switching regressions.One reference is here.
