FIGURE 7.1 The U-shaped marginal cost curve.



The stochastic version of (7.10.1) may be written as

which is called a second-degree polynomial regression.

The general kth degree polynomial regression may be written as

Yi = ¡0 + ¡1 Xi + ¡2 X2 + ••• + 3kXk + ui (7.10.3)

Notice that in these types of polynomial regressions there is only one explanatory variable on the right-hand side but it appears with various powers, thus making them multiple regression models. Incidentally, note that if Xi is assumed to be fixed or nonstochastic, the powered terms of Xi also become fixed or nonstochastic.

Do these models present any special estimation problems? Since the second-degree polynomial (7.10.2) or the kth degree polynomial (7.10.13) is linear in the parameters, the ft's, they can be estimated by the usual OLS or ML methodology. But what about the collinearity problem? Aren't the various Xs highly correlated since they are all powers of X? Yes, but remember that terms like X2, X3, X4, etc., are all nonlinear functions of X and hence, strictly speaking, do not violate the no multicollinearity assumption. In short, polynomial regression models can be estimated by the techniques presented in this chapter and present no new estimation problems.

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