Doubts calculating the minimum point of the parabola suggested by an OLS with quadratic independent variable

[Rui Agostinho]

Hello everyone,

I am having issues in calculating the inflection point of the parabola that is suggested by the coefficients obtained when running an OLS which includes an independent variable that has a quadratic term.


The regression I'm running is the following

reg Percentage_changes_per_industry c.age_median##c.age_median gender log_vn_median

Where Percentage_changes_per_industry is the percentage of new entrepreneurs, per industry, per year;

age_median is the median age of the workers, per industry, per year;

gender is the mean gender of each industry, per year (0 for male, 1 for female=

log_vn_median is the log of the volume of sales (in hundreds of thousands of euros), per industry, per year;


The results I am obtaining are

      Source |       SS           df       MS      Number of obs   =       892
-------------+----------------------------------   F(4, 887)       =     99.72
       Model |  52.6487941         4  13.1621985   Prob > F        =    0.0000
    Residual |  117.075867       887   .13199083   R-squared       =    0.3102
-------------+----------------------------------   Adj R-squared   =    0.3071
       Total |  169.724661       891  .190487835   Root MSE        =    .36331

-------------------------------------------------------------------------------------------
Percentage_changes_per_~y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
--------------------------+----------------------------------------------------------------
               age_median |   -.109139   .0468088    -2.33   0.020    -.2010079     -.01727
                          |
c.age_median#c.age_median |   .0010078     .00059     1.71   0.088      -.00015    .0021657
                          |
                   gender |  -.0019343   .0006177    -3.13   0.002    -.0031466   -.0007221
            log_vn_median |   -.221265   .0132558   -16.69   0.000    -.2472814   -.1952486
                    _cons |   5.951701   .9600941     6.20   0.000      4.06738    7.836022
--------------------

The issue I have is that, while the coefficients for median age do show statistically significant results for both the linear and the quadratic term, I want to know where the inflection point of the parabola is, but am not sure how to calculate it. I know from my calculus classes that the inflection point of a parabola given by y= ax² + bx + c is (-b/2a) - in my case, if this is the way to calculate it for the results I obtained, it would be at (0.109139 / (2* 0.0010078)) = 54.147.


The questions I have are twofold then:
- is this the correct way of knowing the inflection point for the results obtained in such a regression? If not, how should I calculate it?

- If the result obtained for the inflection point is not within the range of my data, what does that exactly mean? For my sample, the maximum value of age_median is of 49.5 - if the inflection point is, in fact, of 54.147, which is outside this range, does this mean that, for my sample, Percentage_changes_per_industry and age_median have in fact just a linear relationship?

Thank you very much for any help you can provide!

Rui

[Rich Goldstein]

you have the formula correct but the name wrong - an "inflection point" is where a curve changes from concave to convex (or vice versa) and this does not happen for a quadratic; the max (min) of a quadratic is generically known as the "vertex"

[Nick Cox]

Or turning point.

[Rui Agostinho]

Thank you very much for both your replies.

Well, since I was correct in how to calculate the turning point, my second question still remains. Given its value falls outside the range of values in my sample, does this mean that, in this case, the relationship between age_median and Percentage_changes_per_industry is linear and negative, and not quadratic in nature?

Thank you

[Rich Goldstein]

no, it does not; to get a better idea maybe try graphing it using "two function"; see

help twoway_function

[Nick Cox]

twoway function -0.109139 * x + 0.0010078 * x^2, ra(35 60) xli(49.5)