Introduction to econometrics II eco 356 faculty of social sciences course guide course Developers: Dr. Adesina-Uthman
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- 2.4.2.0 OBJECTIVE
- INTRODUCTION TO ECONOMETRICS II ECO 306 NOUN
| INTRODUCTION TO ECONOMETRICS II ECO 306 NOUN 88 In regression, however, a transformation to achieve linearity is a special kind of nonlinear transformation. It is a nonlinear transformation that increases the linear relationship between two variables. 2.4.2.0 OBJECTIVE The main objective of this unit is to show that regression analysis can be extended to fit nonlinear models through transformation of nonlinear model that can be made linear. 2.4.3.0 MAIN CONTENT A limitation out of other limitations of linear regression analysis is that it is contained in its very name, in that it can be used to fit only linear equations where every explanatory term, except the constant, is written in the form of a coefficient multiplied by variable …[2.74] Y equations such as the two below are nonlinear 1 1 2 X Y …[2.75] And …[2.76] Nevertheless, both [2.75] and [2.76] have been suggested as suitable forms for Engel curves, (the relationship between the demand fora particular commodity, Y and income, X). As an illustration, given data on Y and X, how could one estimate the parameters in these equations Actually, in both cases, with a little preparation one can actually use linear regression analysis. Here, first, note that [2.74] is linear in two ways. The right side is linear invariables because the variables are included exactly as defined, rather than as functions. It, therefore, consists of a weighted sum of the variables, the parameters being the weights. The right side is also linear in the parameters since it consists of a weighted sum of these as well, the X variables being the weights in this respect. INTRODUCTION TO ECONOMETRICS II ECO 306 NOUN 89 For the purpose of linear regression analysis, only the second type of linearity is important. Nonlinearity in the variables can always be sidestepped by using appropriate definitions. For example, suppose that the relationship was of the form √ …[2.77] By defining Z 2 = , Z 3 = √ , Z 4 = etc, the relationship can be rewritten …[2.78] and it is now linear invariables as well as in parameters. This type of transformation is only beautifying, and you will usually seethe regression equation presented with the variables written in their nonlinear form. This avoids the need for explanation and extra notation. But [2.76] is nonlinear in both parameters and variables and cannot be handled by a mere redefinition. That is, even if attempted, the equation cannot be made linear by defining Z = and replacing with Z; since you do not know , you have noway of calculating sample data for Z. However, you could define 1 Z X , the equation now becomes …[2.79] and this is linear, which is the regress of Y onZ. The constant term in the regression will bean estimate of and the coefficient of Z will bean estimate of Download 1.75 Mb. Share with your friends:
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