To Test Whether or Not the Population Regression Function Is $R ^ { 2 }$

Short Answer

To test whether or not the population regression function is linear rather than a polynomial of order r, a. check whether the regression $R ^ { 2 }$ for the polynomial regression is higher than that of the linear regression.
b. compare the TSS from both regressions.
c. look at the pattern of the coefficients: if they change from positive to negative to positive, etc., then the polynomial regression should be used.
d. use the test of $( r - 1 )$ restrictions using the $F$ -statistic.