Polynomial Models

Using polynomials to capture nonlinearity in regression is quite easy and often effective. Stu­dents of economics are quite used to seeing U-shaped cost curves and S-Shaped production functions and these shapes are simply expressed using quadratic and cubic polynomials, respectively. Since the focus so far has been on simple regression, i. e., regression models with only one independent variable, the discussion in POE4 is simplified to include only a single squared or cubed value of the independent variable.

The general form of a quadratic equation y = a0 + a1x + a2x[17] [18] includes a constant, the level of x and its square. The latter two terms are multiplied times coefficients, ai and a2 that determine the actual shape of the parabola. A cubic equation adds a cubed term, y = a0 + a1x + a2x2 + a3x[19]. The simple regressions considered in this section include only the constant, ao and either the squared term in a quadratic model or the cubed term in the cubic model.

The simple quadratic regression has already been considered. The regression and its slope are

У = ві + ^x2 dy/dx = 2e2x

From this you can see that the function’s slope depends on the parameter в as well as the value of the variable x.

The cubic model and its slope are

У = ві + e2x3

dy/dx = 3в2 x2

Since x is squared in the slope, the algebraic sign of в2 determines whether the slope is positive or negative. Both of these models are considered using examples below.

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