Test which functional form that best explains data

Question

Tried asking this on Math Stack Exchange. Got no answer after a week, so trying here.

I had this question in an exam lately and I was not sure how to answer it. Now the exam is done, and I can't go back, but it's been in my head ever since and I'm really curious about the answer.

Suppose you have a data set, with variables:

$age$: A persons age

$age^2$: age to the power of 2

And dummy variables: $D45 = (age=45)$, $D46 = (age=46)$ ... $D55 = (age=55)$ etc.

And suppose you have two models, where

$$ y = \beta_1 x_1 + \beta_2 age + \beta_3 age^2 $$ and $$ y = \beta_1 x_1 + \beta_2 D_{45} + \beta_3 D_{46} ... \beta_{22} D_{65} $$

How would you guys test which of the functional forms in the two models best explains the data?

I suppose we are to test $\beta_i = 0$ for both models. But I am not sure.

What would you guys have done in this situation?

Kind regards

Answer 1

Notice that: $$ y = \beta_1 x_1 + \beta_2 age + \beta_3 age^2 $$ is a more restrictive model than: $$ y = \delta_1 x_1 + \delta_2 D_{45} + \delta_3 D_{46} + \ldots $$ where you have a dummy for every age level.

To see this, take for example the case where age is $a$. Then the first regression gives: $$ y = \beta_1 x_1 + \beta_2 \times a + \beta_3 \times (a)^2 $$ The second regression gives: $$ y = \delta_1 x_1 + \delta_a. $$ where $\delta_a$ is the coefficient of the dummy $D_a$.

So for $\delta_1 = \beta_1$ and $\delta_a = \beta_2 \times a + \beta_3 \times (a)^2$ the two are the same.

This means that the first regression is a special case of the second one where we specify $\delta_1 = \beta_1$ and $\delta_a = \beta_2 a + \beta_3 a^2$.

Given that the first regression is a restrictive version of the second one, in principle, you could test for the fit of the first model versus the second model using something like a likelihood ratio test wiki.