model to examine concave relationship

[Issa Almaharmeh]

Dear all,
Is there any model to examine if there is a concave relationship between x and y?

[Carlo Lazzaro]

Issa:
yes, -regression- can do it, as you can see from the following toy-example:

sysuse auto.dta
. reg price c.mpg##c.mpg

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(2, 71)        =     18.28
       Model |   215835615         2   107917807   Prob > F        =    0.0000
    Residual |   419229781        71  5904644.81   R-squared       =    0.3399
-------------+----------------------------------   Adj R-squared   =    0.3213
       Total |   635065396        73  8699525.97   Root MSE        =    2429.9

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |  -1265.194   289.5443    -4.37   0.000    -1842.529   -687.8593
             |
 c.mpg#c.mpg |   21.36069   5.938885     3.60   0.001     9.518891    33.20249
             |
       _cons |   22716.48   3366.577     6.75   0.000     16003.71    29429.24
------------------------------------------------------------------------------

. di -(-1265.194)/(2* 21.36069)///there's a stationary point (max) at:
29.615008

[Issa Almaharmeh]

thank you, Carlo But how the second variable will measure the concave relationship?

[Carlo Lazzaro]

issa:
the second variable (which is the squared term of the first one) is need for quadratic equation.
In this case, you have an inverse U-shaped relationship between depvar (price) and indepvar (mpg); being the stationary point of this function a maximum, you have a concave function as the segment between two points of the graph of this function is below and within the graph itself.