Ch08nonlinear - Nonlinear Regression Functions (SW Chapter...

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1 Nonlinear Regression Functions (SW Chapter 8) The linear model assumes that ( ) | ii EY X is linear in X . But the linear approximation is not always a good one The multiple regression framework can be extended to handle regression functions that are nonlinear in one or more X . Outline 1. Nonlinear regression functions – general comments 2. Nonlinear functions of one variable 3. Nonlinear functions of two variables: interactions 2 The TestScore STR relation looks linear (maybe)… 3 But the TestScore Income relation looks nonlinear. .. 4 What is wrong with this regression? A. Forecasts will be inconsistent B. Heteroskedasticity C. Omitted variable bias. D. All of the above
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5 Nonlinear Regression Population Regression Functions – General Ideas (SW Section 8.1) If a relation between Y and X is nonlinear : The effect on Y of a change in X depends on the value of X that is, the marginal effect of X ( ( ) () | EYX X )is not constant A linear regression is mis-specified – the functional form is wrong (e.g. u i contains 2 i X or omitted variable bias) The estimator of the effect on Y of X is biased – it needn’t even be right on average. The solution to this is to estimate a regression function that is nonlinear in X 6 The general nonlinear population regression function Y i = f ( X 1 i , X 2 i ,…, X ki ) + u i , i = 1,…, n Assumptions 1. E ( u i | X 1 i , X 2 i ,…, X ki ) = 0 (same); implies that f is the conditional expectation of Y given the X ’s ( (| ) ). 2. ( X 1 i ,…, X ki , Y i ) are i.i.d. (same). 3. Big outliers are rare (same idea; the precise mathematical condition depends on the specific f ). 4. No perfect multicollinearity (same idea; the precise statement depends on the specific f ). 7 8 Nonlinear Functions of a Single Independent Variable (SW Section 8.2) We’ll look at two complementary approaches: 1. Polynomials in X The population regression function is approximated by a quadratic, cubic, or higher-degree polynomial 2. Logarithmic transformations Y and/or X is transformed by taking its logarithm this gives a “percentages” interpretation that makes sense in many applications
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9 1. Polynomials in X Approximate the population regression function by a polynomial: Y i = β 0 + 1 X i + 2 2 i X +…+ r r i X + u i This is just the linear multiple regression model – except that the regressors are powers of X ! Estimation, hypothesis testing, etc. proceeds as in the multiple regression model using OLS The coefficients are difficult to interpret, but the regression function itself is interpretable 10 Example : the TestScore Income relation Income i = average district income in the i th district (thousands of dollars per capita) Quadratic specification: TestScore i = 0 + 1 Income i + 2 ( Income i ) 2 + u i Cubic specification: TestScore i = 0 + 1 Income i + 2 ( Income i ) 2 + 3 ( Income i ) 3 + u i 11 Estimation of the quadratic specification Dependent Variable: TESTSCR Method: Least Squares Date: 04/12/09 Time: 17:50 Sample: 1 420 Included observations: 420 White Heteroskedasticity-Consistent Standard Errors & Covariance Variable Coefficient Std. Error
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Ch08nonlinear - Nonlinear Regression Functions (SW Chapter...

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