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l61 - Nonlinear relationships Sources Berry Feldmans...

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Nonlinear Relationships Page 1 Nonlinear relationships Sources : Berry & Feldman’s Multiple Regression in Practice 1985; Pindyck and Rubinfeld ’s Econometric Models and Economic Forecasts 1991 edition; McClendon ’s Multiple Regression and Causal Analysis, 1994 ; SPSS’s Curvefit documentati on. Also see Hamilton’s Statistics with Stata, Updated for Version 9, for more on how Stata can handle nonlinear relationships. Linearity versus additivity. Remember again that the general linear model is Y X X X X E Y X j j j k kj j i ij j i k j j 1 1 2 2 1 ... ( | ) The assumptions of linearity and additivity are both implicit in this specification. Additivity = assumption that for each IV X, the amount of change in E(Y) associated with a unit increase in X (holding all other variables constant) is the same regardless of the values of the other IVs in the model. That is, the effect of X1 does not depend on X2; increasing X1 from 10 to 11 will have the same effect regardless of whether X2 = 0 or X2 = 1. With non-additivity, the effect of X on Y depends on the value of a third variable, e.g. gender. As we’ve just discussed, we u se models with multiplicative interaction effects when relationships are non-additive. Linearity = assumption that for each IV, the amount of change in the mean value of Y associated with a unit increase in the IV, holding all other variables constant, is the same regardless of the level of X, e.g. increasing X from 10 to 11 will produce the same amount of increase in E(Y) as increasing X from 20 to 21. Put another way, the effect of a 1 unit increase in X does not depend on the value of X. With nonlinearity, the effect of X on Y depends on the value of X; in effect, X somehow interacts with itself. The interaction may be multiplicative but it can take on other forms as well, e.g. you may need to take logs of variables. Examples:
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Nonlinear Relationships Page 2 Y0N0 X1 4 3 2 1 0 -1 -2 -3 -4 2 0 -2 -4 -6 -8 -10 Observ ed Linear Quadratic YEXP X1 4 3 2 1 0 -1 -2 -3 -4 30 20 10 0 -10 Observ ed Linear Exponential Dealing with Nonlinearity in variables. We will see that many nonlinear specifications can be converted to linear form by performing transformations on the variables in the model. For example, if Y is related to X by the equation E Y X i i ( ) 2 and the relationship between the variables is therefore nonlinear, we can define a new variable Z = X 2 . The new variable Z is then linearly related to Y, and OLS regression can be used to estimate the coefficients of the model. There are numerous other cases where, given appropriate transformations of the variables, nonlinear relationships can be converted into models for which coefficients can be estimated using OLS. We’ll cover a few of the most important and common ones here, but there are many others.
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