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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


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)


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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