Multivariate linear regression accounting for threshold / data cleaning I am trying to make a linear regression model for the sale price of a house based on many variables (based on the data from this Kaggle challenge https://www.kaggle.com/c/house-prices-advanced-regression-techniques) 

The distribution above is the 2nd-floor size in square feet and the y-axis is the sale price. It shows a clear linearity, except for the fact that homes without a 2nd floor clearly are not for sale for 0 dollars. 

I have many variables like this that have some threshold either upper or lower that has a large distribution of response for a single value. If I simply exclude these values then the intercept for this curve will be through the origin. Should I let that be the case and assume that the $0 price tag will be corrected by the other variables in my regression? 

What is the best way to treat/fit data such as this? 

Thanks!
You could try including "Does not have a 2nd floor" as a separate categorical variable, which you could encode as 1 or 0 in your linear model.  Something like:

price ~ no_second_floor + area_of_second_floor + [Other Variables]


Event without the other variables, it's already going to be able to fit the data better than price ~ area_of_second_floor because, instead of forcing the price of the single-floor houses to $0, it would be able to fit it to the average price of all the single-floor houses in your data set - which is the best you can do for those houses until you add other variables besides area_of_second_floor.

(Actually, it wouldn't be forced to $0 if you were to include a constant term in your model, but the constant term that best lets you fit the linear portion of the data still probably isn't what you want for the single-floor houses, some of which - as you see - are quite expensive).