College enrollment probability model

Question

Im working on college enrollment linear probability model where enrollment is regressed on income.

I have the following regressions

1. $$y=\beta_0+\beta_1x_1+u$$
2. $$y=\beta_0+\beta_1x_1^2+u$$
3. $$y=\beta_0+\beta_1x_1+\beta_2x_1^2+u$$

where $x_1$ is Income and $y$ is a binary variable which indicates

In cases (1) and (2) I have statistical significance at the 1% level ( *** ), but in regression (3)I have no significance at all in these variables.

How do I remedy/ interpret this?

Answer 1

Model (1) is a regular linear probability model. Your results say that $x_1$ is significantly correlated with $y$.

Model (2) is strange. It means that the probability is quadratic in $x_1$ and the turning point is exactly $x_1=0$. You will have difficulty justifying the restriction.

The coefficients in Model (3) being insignificant may be a symptom of multicollinearity, i.e., $x_1$ and $x_1^2$ are strongly correlated. To see if that's the case, you can test $H_0: \beta_1 = \beta_2 = 0$. I am quite confident they are jointly significant considering the significant results in (1) and (2). If they are jointly significant but individually insignificant, it's a typical symptom of multicollinearity.

Plotting the fitted probabilities for (1) and (3) often helps. Also see if the range of $x_1$ contains the turning point ($-\frac{1}{2} \beta_1/\beta_2$) in Model (3). You usually have insignificant quadratic terms if the turning point is not in data range, because $y$ is already explained well by the linear model, and then the linear term also becomes insignificant due to multicollinearity.