Inflection point in qudrating relationship

[Faisal Abdullah]

Dear experts
in my study, I am using fixed effect regression in order to test a quadratic relationship between firms' cash and profitability

the dependent variable is cash and the main independent variables are profitability and profitability^2 and other independent variables

Profitability	-1.273***
Profitability2	2.225***

the results illustrate that the relationship between cash and profitability is quadratic. since the coefficient of profitability is positive, the curve is U shape.
I derived the inflection point = -0.286

my concern is about how how to interpret it regarding the effect of profitability on firms' cash. how it is related to the distribution of the data?

your support is very appreciated
Faisal

[Clyde Schechter]

Your calculation is incorrect; you got the sign wrong. Also, quadratic relationships don't have an inflection point: they have a vertex. The vertex in your model is at profitability = +0.286. (Location of vertex = -linear coefficient/(2*quadratic coefficient)) So for values of profitability < 0.286, increasing profitability is associated with decreasing cash. For values of profitability > 0.286, further increases in profitability are associated with increasing cash. Cash achieves its minimum value when profitability = 0.286.

Now, an important point in interpreting quadratic models is to ask where this number 0.286 falls with respect to the range of observed, or even possible, values of profitability. Not being in finance, and not seeing your data, I can't advise you. But the principle is this: if that 0.286 is outside the range of values of profitability, then you don't really have a U-shaped relationship. What you have in that case is a relationship that is non-linear in that the increase in cash associated with a given increase in profitability increases at higher levels of profitability but the nadir and turnaround never actually happen! If the 0.286 value is within the range of values of profitability but is near one end or the other, then you have a barely U-shaped relationship (more like a J or backwards J), and given the uncertainties in the coefficient estimates due to measurement error, sampling error, etc., you have to consider whether the apparent nadir is real. On the other hand, if 0.286 lies well within the range of data, then you have a genuine U-shaped relationship.

[Faisal Abdullah]

Dear Clyde, I must say thank you so much for your endless support.