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I'm trying to model a continuous response variable with multiple linear regression. One of my independent variables is continuous, highly symmetric but heavily zero-inflated - only 2% of the data points are nonzero.

I find low correlation between this independent variable and the response in the unconditional set. However, the correlation between this independent variable and the response is high if I truncate away the points where this independent variable is zero. As a result, my fitted model assigns very low weight on this predictor and becomes too insensitive to it.

What are some conventional ways to model the unconditional response variable with high sensitivity to the zero-inflated regressor when it is nonzero?

I've considered applying a monotone transformation to the predictor - which tend to be undefined due to a functional form like $\dfrac{\cdot}{0}, \log\left(0\right)$ or $\sqrt{-x},x>0$. Is a hierarchical model where I have two separate models conditioned on this independent variable the best strategy?

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I think you can just divide it into two parameters, for x > 0 (yes/no) and if x > 0 the value. Basically I think you can just generate two variables and proceed as normal.

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