# What does it mean when the coefficient of the squared term is negative in regression?

I am reading a research paper which models a regression model where the returns are regressed on the number of ad exposures.

the equation looks something like this:

$Returns = beta_1*nExp + beta_2*nExp^2$

nExp: The number of times the user looks at the ad.

Returns: The revenue from the user


The result is that: $beta_1$ is positive (and statistically significant), which means that the ad exposures has a $+ve$ relation with the revenue.

However, $beta_2$ is negative. And the research paper says:

term for the square of the number of exposures, which has a negative coefficient suggesting diminishing returns to ad exposure.

What does it really mean? What does it mean when the term's coefficient is $+ve$ but the coefficient of it's squared term in the equation is $-ve$?

In this context probably the plain English way to put it is that 'returns increase with additional exposure to the ad, but there is a tapering effect at the upper end of exposures. Looking at this picture of a parabola here (i.e a graph of y= -ax^2+bx+c): • +1, spot on. The fitted response to Exp is parabola-shaped (because you have a squared term, but no higher powers), and the parabola opens downwards (because the estimated leading coefficient $\beta_2$ is negative). Depending on where you are on the $x$ = Exp axis, you see an increasing response with a diminishing positive slope, or a decreasing response with a negative slope that increases in absolute value. I recommend plotting the estimated equation over the range of Exp you have actually observed. – Stephan Kolassa May 27 '16 at 9:04