# On dichotomizing a continuous variable and how it affects MSE of OLS

Recently I have learned about the practice of dichotomizing a continuous independent variable (or maybe even discretize it into more than 2 categories), and then run a predictive model (e.g. multiple linear regression, random forest). I am interested in its theoretical properties/justifications. I have tried to study the following simple situation, but couldn't work out the algebra.

Assume the true mode is $$Y = a + 1_{\{X\ge c\}}b + \epsilon$$ where $$a, b,c, X$$ are non-stochastic, and $$\epsilon \sim \mathcal{N}\left(0,\sigma^2\right)$$. Suppose we observe i.i.d. samples $$\left(X_1,Y_1\right),\cdots,\left(X_n, Y_n\right)$$. We mistakenly thought that the true model is $$Y = a + Xb + \epsilon$$ and calculated OLS estimator $$\beta_{wrong}$$. Let $$\beta_{right}$$ be the OLS estimator we would have calculated according to the correct model, and let $$\beta = \begin{bmatrix}a & b \end{bmatrix}^T$$. Is it true that $$\mathbb{E}\left[ \left\|\beta_{right} - \beta\right\|_2^2\right] \le \mathbb{E}\left[ \left\|\beta_{wrong} - \beta\right\|_2^2\right]\,?$$

I can find what LHS is. What would be the best way to approach the RHS? Should I look for an explicit expression of $$\mathbb{E}\left[ \left\|\beta_{wrong} - \beta\right\|_2^2\right]$$?