What are some ways to identify that a quadratic term is need

What are some ways to identify that a quadratic term is needed in a multiple regression model?

Is the model valid for all values of x? Try to give an example if you can.

I\'ve tried understanding this and it just won\'t click.

Solution

Residual plot, especially, if the square of residuals against the covariate values is linear, is an indication of a quadratic term in multiple linear regression model. A regression model is always to be guarded against extrapolation beyond the x values.