Cubic Functions in Econometrics Are Good for Inflexion

With a cubic function, you allow the effect of the independent variable (X) on the dependent variable (Y) to change. As the value of X increases (or decreases), the impact of the dependent variable may increase or decrease. However, unlike a quadratic function, this relationship changes at some unique value of X.

In other words, at some specific point, a decreasing effect becomes increasing or an increasing effect becomes decreasing. The point at which this occurs is called the inflexion point.

The mathematical representation of an econometric model with a cubic function is

image0.jpg

If you estimate this type of regression, numerous outcomes are possible for your coefficients. However, the two most common results lead to either of the following curves:

  • A decreasing slope followed by an increasing slope, as shown in part (a)

  • An increasing slope followed by a decreasing slope, as shown in part (b)

    image1.jpg

Among many other possibilities, part (a) depicts the potential shape of a total variable cost (TVC) or total cost (TC) curve. Part (b) approximates a short-run total product (TP) curve if initially marginal productivity is increasing and then it diminishes.

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