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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?

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  • $\begingroup$ Have you graphed the data? This would normally be modeled as logistic regression rather than polynomial regression. Ignore the statistical significance, that just tells you the result is not due to chance. Lots of things are not due to chance but also are not useful. $\endgroup$ – Dave Harris Jun 17 '17 at 3:12
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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.

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