I did a multiple variable regression using R, for $Grade_i = β_0 + β_1(Hours_i) + β_2(Hours_i^2) + μ_i$, this is what I got: .

But I'm having trouble interpreting the estimated value for $β_2$, Since $ \frac{6.8472}{0.3699} = 18.5109489$, can I say, if students’ study hours are over 18.5109489 hours, then if students increase study time by 1 hour, the grade will decrease by 0.3699 points? Is it right?

## Best Answer

The addition of a squared term is often used when you suspect that the marginal payoff in terms of grade of an extra study hour is dependent on how many hours you study.

In your case the first order term is positive while the second order term is negative. To get an impression of the effect you can plot the parable

To find the maximizer you differentiate

$$6.84 hours - 0.37 hours^2$$

with respect to hours and set equal to 0, this gives you

$$6.84 - 2 \times 0.37 hours = 0$$

such that maximizer is $hours^\star = 9.24$.

Beyond that more hours of study have a negative effect on grade.

However there are two things you should ask yourself now:

(1) Am I doing extrapolation?

To see if this is the case you need to check that you actually observe some students using more than 9.24 hours of study, otherwise that part of the curve is simply irrelevant for estimation and you should not conclude anything from it.

If such observations exists then the model tells you that an extra hour of study has a negative impact on grade when student has already used 9.24 hours. However you need to consider whether you really believe this. If I estimated such a model and got those results I would be very suspicious that something else was wrong. If you do not believe in this effect you have to consider the standard problems of regression equation specification hence perhaps most importantly

(2) You should ask yourself whether you really have the correct functional form and are not forgetting to control for some important variables etc.