[Music] hi in this quick video we want to see how to model non-linear relationships in a regression framework we expect the relationship between x and y to follow a nonlinear shape perhaps to look like a u what is a function that we can use to model such a relationship how about a simple quadratic form y is equal to x plus x to the power of 2. when we add parameters beta1 and beta0 we can have this function be perfectly linear or look like a u or look like an inverted u using a quadratic functional form
is quite powerful we can not only model u's and hums but also decreasing returns the first part of an inverted u and even increasing returns the second part of a u whether two variables follow a hum shape relationship or whether there are simply decreasing returns makes a big difference of course the good news is that we can test that with simple econometrics we want to figure out whether there is a u or not we estimate the parameters of the quadratic function with a regression then we compute the extreme point maximum or minimum given the estimates
of beta1 and beta2 and see whether the extreme point is in a reasonable range of the values of the x variable here's an example of a non-linear relationship that is not exactly hum shaped but rather has diminishing returns how do we figure out the extreme point well we take the derivative with respect to x set it to 0 and solve for x it is as simple as that for instance if x is defined on a range from 0 to 100 and the implied extreme point is way outside of this interval then you know that the
curve is most likely similar to the blue one with diminishing returns you can also compute the confidence interval around the extreme point because the extreme point itself is of course just an estimate one last hint say you expect a hum shape then you may want to test whether the curve has a positive slope in the beginning and the negative slope towards the end of a reasonable scale of the x variable if you're interested in more details and the actual implementation of the u-test procedure in your statistical package check out the link in the description of
this video
