Nonlinear Regression Functions

Nonlinear Regression Functions

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The TestScore ? STR relation looks linear (maybe)...

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But the TestScore ? Income relation looks nonlinear...

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Nonlinear Regression ? 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 effect on Y of X is biased: in general it isn't even right on average.

The solution is to estimate a regression function that is nonlinear in X

SW Ch 8

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The general nonlinear population regression function

Yi = f(X1i, X2i,..., Xki) + ui, i = 1,..., n

Assumptions 1. E(ui| X1i, X2i,..., Xki) = 0 (same) 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)

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Outline 1. Nonlinear (polynomial) functions of one variable 2. Polynomial functions of multiple variables: Interactions 3. Application to the California Test Score data set 4. Addendum: Fun with logarithms

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Nonlinear (Polynomial) Functions of a One RHS Variable

Approximate the population regression function by a polynomial:

Yi

=

0

+

1Xi

+

2

X

2 i

+...+

r

X

r i

+

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

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Example: the TestScore ? Income relation Incomei = average district income in the ith district

(thousands of dollars per capita)

Quadratic specification:

TestScorei = 0 + 1Incomei + 2(Incomei)2 + ui

Cubic specification:

TestScorei = 0 + 1Incomei + 2(Incomei)2 + 3(Incomei)3 + ui

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