# Test which functional form that best explains data

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

• Hi @Zebraboard. There are potentially other methods, but have a look at the Ramsey RESET. Nov 15, 2021 at 11:44

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.

• I have a hard time seeing how the first model appears as a result of a parameter restriction of the second. The page on wiki you refer to explicitly has $\Theta_0$ being a subset of $\Theta$. It could be just me but I can only see $\beta_1$ being common. You could set up the more general model including both $x_1$, age terms, and dummies and test whether the dummies can be excluded. If that is the case, however, you could probably also exclude the age terms from the full model (since the dummy model is more flexible) and the choice of the model with age terms would then depend on simplicity. Nov 16, 2021 at 0:48
• From my point of view this would suggest a comparison based on information criteria. But maybe I am missing something? Nov 16, 2021 at 0:48
• Sorry, why is the one model more restrictive than the other? I thought more variables means more restrictions? Nov 16, 2021 at 8:27
• @Jesper Hybel: I tried to give more info for why the first model is a special case of the second one. In general if you have a discrete number of values for the covariates (like age) then creating a dummy for each value gives a fully non-parametric specification.
– tdm
Nov 17, 2021 at 7:11
• @tdm That makes sense. I completely overlooked that there was a dummy for each year. Nov 17, 2021 at 11:45