12.4 Functional Form
To model non-linear relationships, an independent variable can be transformed by squaring it. For example, consider the relationship between income and food expenditure. The regular OLS assumes a linear relationship in the sense that an increase in income always leads to a proportional increase in food expenditure. In realty, there is likely a flattening out of food expenditure for high incomes because only so much money can be spent on food.
##
## Call:
## lm(formula = log(total) ~ yards + att + exp + exp2 + draft1 +
## veteran + changeteam + pbowlever, data = nfl)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.4477 -0.5010 -0.0807 0.4452 3.1638
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.0270821 0.1228294 -8.362 4.93e-16 ***
## yards 0.0003353 0.0001426 2.351 0.019087 *
## att 0.0005137 0.0009728 0.528 0.597691
## exp 0.0702541 0.0601679 1.168 0.243452
## exp2 -0.0037576 0.0036704 -1.024 0.306398
## draft1 0.8910570 0.1132812 7.866 1.90e-14 ***
## veteran 0.5752784 0.1493617 3.852 0.000131 ***
## changeteam -0.3221519 0.0901474 -3.574 0.000383 ***
## pbowlever 0.4158535 0.0938203 4.432 1.12e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7909 on 559 degrees of freedom
## (441 observations deleted due to missingness)
## Multiple R-squared: 0.5509, Adjusted R-squared: 0.5445
## F-statistic: 85.71 on 8 and 559 DF, p-value: < 2.2e-16