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# lecture_7_slides - 8-1Nonlinear Regression Functions(SW...

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Unformatted text preview: 8-1Nonlinear 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. Outline1.Nonlinear regression functions – general comments 2.Nonlinear functions of one variable 3.Nonlinear functions of two variables: interactions 8-2The TestScore– STRrelation looks linear (maybe)… 8-3But theTestScore– Incomerelation looks nonlinear... 8-4Nonlinear Regression Population Regression Functions – General Ideas (SW Section 8.1) If a relation between Yand Xis nonlinear: •The effect on Yof a change in Xdepends on the value of X– that is, the marginal effect of Xis not constant •A linear regression is mis-specified – the functional form is wrong •The estimator of the effect on Yof Xis biased – it needn’t even be right on average. •The solution to this is to estimate a regression function that is nonlinear in X8-5The general nonlinear population regression function Yi= f(X1i, X2i,…, Xki) + ui, i= 1,…, nAssumptions 1.E(ui|X1i,X2i,…,Xki) = 0 (same); implies that fis the conditional expectation of Ygiven the X’s. 2.(X1i,…,Xki,Yi) 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). 8-68-7Nonlinear Functions of a Single Independent Variable (SW Section 8.2) We’ll look at two complementary approaches: 1. Polynomials in XThe population regression function is approximated by a quadratic, cubic, or higher-degree polynomial 2. Logarithmic transformations •Yand/or Xis transformed by taking its logarithm •this gives a “percentages” interpretation that makes sense in many applications 8-81. Polynomials in XApproximate the population regression function by a polynomial: Yi= β+ β1Xi+ β22iX+…+ βrriX+ ui•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 8-9Example: the TestScore– Incomerelation Incomei= average district income in the ithdistrict (thousands of dollars per capita) Quadratic specification: TestScorei= β+ β1Incomei+ β2(Incomei)2+ uiCubic specification: TestScorei= β+ β1Incomei+ β2(Incomei)2+ β3(Incomei)3+ ui8-10Estimation of the quadratic specification in STATA generate avginc2 = avginc*avginc; Create a new regressorreg 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] [95% Conf....
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