Polynomials
One way to allow for nonlinear relationships between independent and dependent variables is to introduce polynomials of the regressors into the model. In this example the marginal effect of an additional dollar of advertising is expected to diminish as more advertising is used. The model becomes:
salesi = ві + e2pricei + faadverti + e4advert[21] [22] + e^ i = 1, 2,…, N (5.6)
To estimate the parameters of this model, one creates the new variable, advert2, adds it to the model, and uses least squares.
OLS, using observations 175
Dependent variable: sales
Coefficient 
Std. Error 
tratio 
pvalue 

const 
109.719 
6.79905 
16.1374 
0.0000 
price 
7.64000 
1.04594 
7.3044 
0.0000 
advert 
12.1512 
3.55616 
3.4170 
0.0011 
a2 
2.76796 
0.940624 
2.9427 
0.0044 
Mean dependent var 77.37467 S. D. dependent var 6.488537
Sum squared resid 1532.084 S. E. of regression 4.645283
R2 0.508235 Adjusted R2 0.487456
F(3,71) 24.45932 Pvalue(F) 5.60e11
Loglikelihood 219.5540 Akaike criterion 447.1080
Schwarz criterion 456.3780 HannanQuinn 450.8094
The variable a2, which is created by squaring advert, is a simple example of what is sometimes referred to as an interaction variable. The simplest way to think about an interaction variable is that you believe that its effect on the dependent variable depends on another variablethe two variables interact to determine the average value of the dependent variable. In this example, the effect of advertising on average sales depends on the level of advertising itself.
Another way to square variables is to use the square command
square advert
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