That's basically the question - What to do when the continuous dependent variable is zero more than 50% of the time? Are there alternatives to ordinary least squares that are recommended? I know Tobit exists but don't believe that applies in this situation. Thanks
What kind of analysis are you doing? What problems is this causing with the analysis? It doesn't directly violate any assumptions (at least without telling us what type of analysis you're doing) so you'll need to let us know what problems it's giving you.
Here's just an idea, if you're looking for stuff to try (not sure if that's your aim). Categorize the values into quantiles (eg quartiles). Since you have so many 0's, you could make the lowest (quartile) all the 0's. Then set the rest of the quartiles up to be evenly distributed.
eg n=100, with 50 zeroes.
Q1 will have 50 subjects, Q2 will have 16, Q3 will have 17, Q4 will have 17.
With the variable now "categorized", you could do polytomous logistic regression instead of OLS.