Truncated normal regression

The task

Task statement

The file contains only observations with y>0 from a population model y*=beta0+beta_x*x+beta_z*z+epsilon, with epsilon normal and constant unknown standard deviation sigma. The observed y equals y*. Fit the left-truncated normal model conditional on y*>0 by maximum likelihood on every supplied row. This is truncation: unobserved units are absent. Include an intercept. The log likelihood for one observed row is log phi((y-mu)/sigma)-log sigma-log Phi(mu/sigma), where mu is the linear index. Report latent_x, the estimated beta_x. Report mean_observed, the average fitted conditional mean E[y*|y*>0,x,z] over the supplied rows. Report ame_x, the average derivative of that same conditional mean with respect to continuous x, holding z fixed. Use each supplied row's covariates and equal weights. Set n_obs to the supplied row count for every result.

Data

Simulated CSV. Columns: y, x, z

Files

input.csv

Seed

2026091700

Required output

Scored estimates: mean_observed, ame_x, latent_x.

Reference output

A run succeeds when every scored estimate lies within 1 percent or 0.005 of its reference value, whichever is larger. Sample sizes are not scored.