1 QUADRATIC EXPLANATORY VARIABLES We will now consider models with quadratic explanatory variables of the type shown. Such a model can be fitted using OLS with no modification.

2 However, the usual interpretation of a parameter, that it represents the effect of a unit change in its associated variable, holding all other variables constant, cannot be applied. It is not possible for X 2 to change without X 2 2 also changing. QUADRATIC EXPLANATORY VARIABLES

3 Differentiating the equation with respect to X 2, one obtains the change in Y per unit change in X 2. Thus, the impact of a unit change in X 2 on Y, (   3 X 2 ), is a function of X 2. QUADRATIC EXPLANATORY VARIABLES

4 This means that  2 has an interpretation that is different from that in the ordinary linear model where it is the unqualified effect of a unit change in X 2 on Y. QUADRATIC EXPLANATORY VARIABLES

5 In this model,  2 should be interpreted as the effect of a unit change in X 2 on Y for the special case where X 2 = 0. For nonzero values of X 2, the marginal effect will be different. QUADRATIC EXPLANATORY VARIABLES

6  3 also has a special interpretation. If we rewrite the model as shown,  3 can be interpreted as the rate of change of the coefficient of X 2, per unit change in X 2. QUADRATIC EXPLANATORY VARIABLES

7 Only  1 has a conventional interpretation. As usual, it is the value of Y (apart from the random component) when X 2 = 0. QUADRATIC EXPLANATORY VARIABLES

8 There is a further problem. We know that the estimate of the intercept may have no sensible meaning if X 2 = 0 is outside the data range. If X 2 = 0 lies outside the data range, the same type of distortion can happen with the estimate of  2. QUADRATIC EXPLANATORY VARIABLES

9 We will illustrate this with the earnings function. The table gives the output of a quadratic regression of earnings on schooling (SSQ is defined as the square of schooling). QUADRATIC EXPLANATORY VARIABLES. gen SSQ = S*S. reg EARNINGS S SSQ Source | SS df MS Number of obs = F( 2, 537) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | SSQ | _cons |

10 The coefficient of S implies that, for an individual with no schooling, the impact of a year of schooling is to decrease hourly earnings by $2.77. QUADRATIC EXPLANATORY VARIABLES. gen SSQ = S*S. reg EARNINGS S SSQ Source | SS df MS Number of obs = F( 2, 537) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | SSQ | _cons |

11 The intercept also has no sensible interpretation. Literally, it implies that an individual with no schooling would have hourly earnings of $22.25, which is implausibly high. QUADRATIC EXPLANATORY VARIABLES. gen SSQ = S*S. reg EARNINGS S SSQ Source | SS df MS Number of obs = F( 2, 537) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | SSQ | _cons |

12 The quadratic relationship is illustrated in the figure. Over the range of the actual data, it fits the observations tolerably well. The fit is not dramatically different from those of the linear and semilogarithmic specifications. QUADRATIC EXPLANATORY VARIABLES EARNINGS | Coef S | SSQ | _cons |

13 QUADRATIC EXPLANATORY VARIABLES However, when one extrapolates beyond the data range, the quadratic function increases as schooling decreases, giving rise to implausible estimates of both  1 and  2 for S = EARNINGS | Coef S | SSQ | _cons |

14 QUADRATIC EXPLANATORY VARIABLES In this example, we would prefer the semilogarithmic specification, as do all wage-equation studies EARNINGS | Coef S | SSQ | _cons |

15 QUADRATIC EXPLANATORY VARIABLES The slope coefficient of the semilogarithmic specification has a simple interpretation and the specification does not give rise to nonsensical predictions outside the data range EARNINGS | Coef S | SSQ | _cons |

16 The data on employment growth rate, e, and GDP growth rate, g, for 25 OECD countries in Exercise 1.4 provide a less problematic example of the use of a quadratic function. QUADRATIC EXPLANATORY VARIABLES Average annual percentage growth rates Employment GDP Employment GDP Australia Korea Austria Luxembourg Belgium Netherlands Canada New Zealand Denmark Norway Finland– Portugal France Spain Germany Sweden– Greece Switzerland Iceland– Turkey Ireland United Kingdom Italy– United States Japan

17 The output from a quadratic regression is shown. gsq has been defined as the square of g.. gen gsq = g*g. reg e g gsq Source | SS df MS Number of obs = F( 2, 22) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = e | Coef. Std. Err. t P>|t| [95% Conf. Interval] g | gsq | _cons | QUADRATIC EXPLANATORY VARIABLES

18 The quadratic specification appears to be an improvement on the hyperbolic function fitted in a previous slideshow. It is more satisfactory than the latter for low values of g, in that it does not yield implausibly large negative predicted values of e. QUADRATIC EXPLANATORY VARIABLES e | Coef g | gsq | _cons |

19 The only defect is that it predicts that the fitted value of e starts to fall when g exceeds 7. QUADRATIC EXPLANATORY VARIABLES e | Coef g | gsq | _cons |

20 Why stop at a quadratic? Why not consider a cubic, or quartic, or a polynomial of even higher order? There are usually several good reasons for not doing so. QUADRATIC EXPLANATORY VARIABLES

21 QUADRATIC EXPLANATORY VARIABLES Diminishing marginal effects are standard in economic theory, justifying quadratic specifications, at least as an approximation, but economic theory seldom suggests that a relationship might sensibly be represented by a cubic or higher-order polynomial.

22 QUADRATIC EXPLANATORY VARIABLES The second reason follows from the first. There will be an improvement in fit as higher- order terms are added, but because these terms are not theoretically justified, the improvement will be sample-specific.

23 QUADRATIC EXPLANATORY VARIABLES Third, unless the sample is very small, the fits of higher-order polynomials are unlikely to be very different from those of a quadratic over the main part of the data range.

24 QUADRATIC EXPLANATORY VARIABLES These points are illustrated by the figure, which shows cubic and quartic regressions with the quadratic regression. Over the main data range, from g = 1.5 to g = 4, the fits of the cubic and quartic are very similar to that of the quadratic.

25 QUADRATIC EXPLANATORY VARIABLES R 2 for the quadratic specification is For the cubic and quartic it is and 0.658, relatively small improvements.

26 QUADRATIC EXPLANATORY VARIABLES Further, the cubic and quartic curves both exhibit implausible characteristics. The cubic declines even more rapidly than the quadratic for high values of g, and the quartic has strange twists at its extremities.

Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 4.3 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course EC2020 Elements of Econometrics