Best Predictor

[Lad]

Consider the following nonlinear regression equation y = aX^2 + u where a, B (beta) are constants, E(u|X) = 0, and X has a standard normal distrib.

a) derive the best predictor of y given X
b) derive E(y)
c) derive Cov(X^2,u)

 

[BGM]

Sorry is there any \( \beta \) in your model?

By the way it would be best for you to show some effort, but not just to put down a question. Or try to ask a more specific question, which part you have difficulties?

 

[bryangoodrich]

The equation,

\(Y = \alpha X^2 + \epsilon\)​

is still a linear regression since it follows the general linear form,

\(Y = \alpha X^* + \epsilon.\)​

Therefore, the best predictor would be to use your standard OLS regression methods and interpret \(X^*\) appropriately (i.e., that it is the square of the \(X\) values). More appropriately, this model would be considered curvilinear because the functional shape of it would be curvy instead of a straight line. A regression model is nonlinear when it is nonlinear in the coefficients. For instance, the following model is nonlinear:

\(Y = \alpha e^{\beta X}\)​