Only observing sign of the output of a linear model under Gaussian assumption

Question

Suppose the linear model is $y = \beta x + \epsilon$, where $X \sim \mathcal{N}(0, 1), \epsilon \sim \mathcal{N}(0, s^2)$. If we only observe the sign of the output $y_i$, and the number of observations is large, $i$ how can we estimate $\beta$?

Answer 1

Probit regression operates under the assumption that there is some latent linear model that has Gaussian errors, but we only see the signs. This is exactly the situation described in the question, so probit regression sounds like the way to go.

A simulation shows this to be pretty slick and to give estimated coefficients that are close to the OLS coefficients and close to the true values.

set.seed(2023)
N <- 10000
x <- rnorm(N, 0, 1)
b0 <- -1
b1 <- 1
y <- b0 +b1*x + rnorm(N)

# Set indicator for positivity of y
#
z <- y > 0

# Fit a linear model to the original x and y
#
L <- lm(y ~ x)

# Fit a probit model to x and the indicator for the sign of y
#
P <- glm(z ~ x, family = binomial(link = "probit"))

summary(L)$coef[, 1]
summary(P)$coef[, 1]

> summary(L)$coef[, 1]
(Intercept)           x 
 -0.9939896   1.0080528 
> summary(P)$coef[, 1]
(Intercept)           x 
 -0.9853826   0.9783042 

EDIT

Worth a mention is that the standard errors on the probit model are higher. Since there is less information available to the probit model (the z <- y > 0 line is lossy compression of the y variable), I suspect that this is not just a fluke. Consequently, if you have the original y, it seems that you can get tighter coefficient estimates by using the linear model, even though you can do something reasonable if you only have the lossy-compressed z <- y>0.