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Unformatted text preview: 81Nonlinear 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 82The TestScore– STRrelation looks linear (maybe)… 83But theTestScore– Incomerelation looks nonlinear... 84Nonlinear 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 misspecified – 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 X85The general nonlinear population regression function Yi= f(X1i, X2i,…, Xki) + ui, i= 1,…, nAssumptions 1.E(uiX1i,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). 8687Nonlinear 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 higherdegree polynomial 2. Logarithmic transformations •Yand/or Xis transformed by taking its logarithm •this gives a “percentages” interpretation that makes sense in many applications 881. 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 89Example: 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+ ui810Estimation 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 Rsquared = 0.5562 Root MSE = 12.724   Robust testscr  Coef. Std. Err. t P>t [95% Conf. Interval] [95% Conf....
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 Spring '11
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 Linear Regression, Regression Analysis, Yi, regression function

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