Instead of finding the K nearest neighbors and averaging their y values, which is effectively fitting a 0th order (constant) mode, I wonder why fitting a multip

Instead of finding the K nearest neighbors and averaging their y values, which is effectively fitting a 0th order (constant) mode, I wonder why fitting a multiple linear regression using those neighbors is not done. Since a linear model can approximate a function over a wider range than constant can, one can use a larger value of K. I know that local linear regression is often used in one dimension, but I wonder if it should be used more often when there are multiple predictors.