A simple model of measurement errors in a LRM

Summary

• We have a random sample \(\left( y_1,x_1 \right),…,\left( y_n,x_n \right)\) where \(x_i\) is a measurement of \(z_i\) which is not directly observed.
• Suppose that

\[E\left( z_i,x_i \right)=β_1+β_2z_i\]

• such that

\[y_i=β_1+β_2z_i+ε_i , i=1,…,n\]

• Suppose that

\[x_i=z_i+u_i , i=1,…,n\]

• where \(u_i\) is the measurement error.
• Substituting \(z_i\) into the linear regression model

\[y_i=β_1+β_2x_i+ε_i-β_2u_i , i=1,…,n\]

• Since \(x_i\) is correlated with the measurement error \(u_i\) , it is correlated with the error term \(ε_i-β_2u_i\) and therefore not exogenous.
• If \(Var\left( x_i \right)=σ_x^2\) and \(Var\left( u_i \right)=σ_u^2\) for all \(i\) and if \(z_i\) is independent of \(u_i\) then we define the reliability ratio as

\[λ=1- \frac{σ_u^2}{σ_x^2}\]

• It is sometimes possible to find consistent estimates of the parameters in a LRM even if the explanatory variables suffer from measurement errors if the reliability ratios are known.