Conditional variance of y

Problem

True or false?

In an LRM, \(y_i=β_1+β_2x_i+ε_i\) , if the error terms are homoscedastic then \(Var\left( y_i|x_i \right)\) is constant independent of \(i\) while if the error terms are heteroscedastic then \(Var\left( y_i|x_i \right)\) depends on \(i\) .

Solution

True.

\[Var\left( y_i|x_i \right)=Var\left( β_1+β_2x_i+ε_i|x_i \right)=Var\left( ε_i|x_i \right)\]

since, conditionally on \(x_i\) , \(β_1+β_2x_i\) are constants. We can use this to detect heteroscedasticity by creating a scatter plot of \(y\) against \(x\) . If there is more variation in \(y\) when \(x\) is large, we have signs of heteroscedasticity.