Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 13J / Solution

This is attenuation bias from classical measurement error, also called regression dilution. The latent predictor is $Z$, but $X_1$ and $X_2$ are noisy proxies. The noisier proxy produces a slope closer to zero.

For a regression with an intercept, the population slope isHere all variables are centred and the noises are independent. Thereforeso regressing $Y$ on $Z$ has slope $\beta =2$. Forwe haveThusWith the stated values, the slopes arematching the simulation.

Under this independent additive measurement-error model, the magnitude of the $Y$-on-$X_1$ slope is generally smaller than the magnitude of the $Y$-on-$Z$ slope whenever ${\sigma }_{x_1}^2>0$. Increasing ${\sigma }_y$ does not change the population slope, because response noise contributes neither to $\mathrm{Cov}(X_1,Y)$ nor to $\mathrm{Var}(X_1)$. It increases residual variance and the sampling variability of the estimate; with one million observations, doubling it should leave the displayed slope close to $1.6$ while increasing its standard error.