Lecture_7 - Nonlinear Regression Functions Everything so...

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1 Nonlinear Regression Functions Everything so far has been linear in the X ’s But CEF can be nonlinear 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

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2 The TestScore STR relation looks linear (maybe)…
3 But the TestScore Income relation looks nonlinear. ..

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4 Nonlinear Regression Population Regression Functions – General Ideas 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 average effect on Y of changing X is biased
5 Nonlinear Functions of a Single X Variable 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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6 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
7 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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8 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….
9 Interpreting the estimated regression function: (a) Plot the predicted values TestScore = 607.3 + 3.85 Income i – 0.0423( Income i ) 2 (2.9) (0.27) (0.0048)

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10 Interpreting the estimated regression function, ctd : (b) Compute “effects” for different values of X TestScore = 607.3 + 3.85 Income i – 0.0423( Income i ) 2 (2.9) (0.27) (0.0048) Predicted change in TestScore for a change in income from \$5,000 per capita to \$6,000 per capita: TestScore = 607.3 + 3.85 6 – 0.0423 6 2 – (607.3 + 3.85 5 – 0.0423 5 2 ) = 3.4
11 TestScore = 607.3 + 3.85 Income i – 0.0423( Income i ) 2 Predicted “effects” for different values of X : Change in Income (\$1000 per capita) from 5 to 6 3.4 from 25 to 26 1.7 from 45 to 46 0.0 The “effect” of a change in income is greater at low than high

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Lecture_7 - Nonlinear Regression Functions Everything so...

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