Lecture 12 Prof. Arkonac's Slides (Ch 8) ECO 4000_1

# Lecture 12 Prof. Arkonac's Slides (Ch 8) ECO 4000_1 -...

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Nonlinear Regression Functions ECO 4000, Statistical Analysis for Economics and Finance Lecture 12 by Seyhan Erden Arkonac, PhD 1

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2 Nonlinear Regression Functions (SW Chapter 8) Everything so far has been linear in the X ’s 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
3 The TestScore STR relation looks linear (maybe)…

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4 But the TestScore Income relation looks nonlinear. ..
8-5

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6 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 is not constant A linear regression is mis-specified – the functional form is wrong 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
7 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 ).

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9 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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10 1. Polynomials in X Approximate the population regression function by a polynomial: Y i = 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
11 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

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12 Estimation of the quadratic specification in STATA generate avginc2 = avginc*avginc; Create a new regressor reg testscr avginc avginc2, r; Regression with robust standard errors Number of obs = 420 F( 2, 417) = 428.52 Prob > F = 0.0000 R-squared = 0.5562 Root MSE = 12.724 ------------------------------------------------------------------------------ | Robust testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- avginc | 3.850995 .2680941 14.36 0.000 3.32401 4.377979 avginc2 | -.0423085 .0047803 -8.85 0.000 -.051705 -.0329119 _cons | 607.3017 2.901754 209.29 0.000 601.5978 613.0056 ------------------------------------------------------------------------------ Test the null hypothesis of linearity against the alternative that the regression function is a quadratic….
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Lecture 12 Prof. Arkonac's Slides (Ch 8) ECO 4000_1 -...

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