dougherty3e_ch04 - Dougherty: Introduction to Econometrics...

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Dougherty: Introduction to Econometrics 3e Study Guide Chapter 4 Transformations of variables Overview This chapter shows how least squares regression analysis can be extended to fit nonlinear models. Sometimes an apparently nonlinear model can be linearized by taking logarithms. and are examples. Because they can be fitted using linear regression analysis, they have proved very popular in the literature, there usually being little to be gained from using more sophisticated specifications. If you plot earnings on schooling, using the EAEF data set, or expenditure on a given category of expenditure on total household expenditure, using the CES data set, you will see that there is so much randomness in the data that one nonlinear specification is likely to be just as good as another, and indeed a linear specification may not be obviously inferior. Often the real reason for preferring a nonlinear specification to a linear one is that it makes more sense theoretically. The chapter shows how the least squares principle can be applied when the model cannot be linearized, and it concludes with a transformation for comparing the fits of linear and logarithmic specifications. 2 1 β X Y = X e Y 2 1 = Further material This section has three unrelated topics: (1) the use of interactive explanatory variables, and the interpretation of their coefficients, (2) Ramsey’s RESET test for functional form, and (3) Box–Cox tests of functional specification. Interactive explanatory variables Consider the model u X X X X Y + + + + = 3 2 4 3 3 2 2 1 This is linear in parameters and it may be fitted using straightforward OLS, provided that the regression model assumptions are satisfied. However, the fact that it is nonlinear in variables has implications for the interpretation of the parameters. When multiple regression was introduced at the beginning of the previous chapter, it was stated that the slope coefficients represented the separate, individual marginal effects of the variables on Y , holding the other variables constant. In this model, such an interpretation is not possible. In particular, it is not possible to interpret 2 as the effect of X 2 on Y , holding X 3 and X 2 X 3 constant, because it is not possible to hold both X 3 and X 2 X 3 constant if X 2 changes. To give a proper interpretation to the coefficients, we can rewrite the model as ( ) u X X X Y + + + + = 3 3 2 3 4 2 1 The coefficient of X 2 , ( 2 + 4 X 3 ), can now be interpreted as the marginal effect of X 2 on Y , holding X 3 constant. This representation makes explicit the fact that the marginal effect of X 2 depends on the value of X 3 . The interpretation of 2 now becomes the marginal effect of X 2 on Y , when X 3 is equal to zero . One may equally well rewrite the model as ( ) u X X X Y + + + + = 3 2 4 3 2 2 1 From this it may be seen that the marginal effect of X 3 on Y , holding X 2 constant, is ( 3 + 4 X 2 ) and that 3 may be interpreted as the marginal effect of X 3 on Y , when X 2 is equal to zero.
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dougherty3e_ch04 - Dougherty: Introduction to Econometrics...

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