Question

Suppose you estimate the following regression model using OLS: Y_i = β₀ + β₁X_i + β₂X_i² + β₃X_i³+ u_i. You estimate that the p-value of the F-test that β₂= β₃ =0 is 0.01. This implies:

options:

	You can reject the null hypothesis that the regression function is linear.
	You cannot reject the null hypothesis that the regression function is either quadratic or cubic.
	The alternate hypothesis is that the regression function is either quadratic or cubic.
	Both (a) and (c).

Answer 1

Answer: Both (a) and (c)

The null hypothesis here is . This is essentially the hypothesis that the regression function is linear. (that is, the only valid term is the slope of the term, and not the slope of the or terms).

A p-value of 0.01 means that the results are significant at 1% level of significance. Now, the question doesn't mention what level of significance is required, but usually 1% is sufficient to reject the null hypothesis (in social science atleast).

So you can reject the null hypothesis that the regression function is linear, given the p-value and the null hypothesis.

Next, if the null hypothesis is that the regression function is linear, then the alternative hypothesis must be of the following form:

This hypothesis says that either the slope of the quadratic term is different from 0 (in which case the equation is quadratic) or the slope of the cubic term is different from 0 (in which case the equation is cubic) or both are different from 0 (in which case also the equation is cubic). So the alternative hypothesis says that the equation is either quadratic or cubic.