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# Economics 1123

By Mario Hawkins,2014-04-30 07:32
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Economics 1123Econ

Nonlinear Regression Functions

(SW Ch. 6)

( Everything so far has been linear in the X’s

( The approximation that the regression function is

linear might be good for some variables, but not for

others.

( The multiple regression framework can be extended to

handle regression functions that are nonlinear in one

or more X.

6-1

The TestScore STR relation looks approximately

linear…

6-2

But the TestScore average district income relation

looks like it is nonlinear.

6-3

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

6-4

The General Nonlinear Population Regression Function

Y = f(X,X,…,X) + u, i = 1,…, n i1i2ikii

Assumptions

1. E(u| X,X,…,X) = 0 (same); implies that f is the i1i2iki

conditional expectation of Y given the X’s.

2. (X,…,X,Y) are i.i.d. (same). 1ikii

3. “enough” moments exist (same idea; the precise

statement depends on specific f).

4. No perfect multicollinearity (same idea; the precise

statement depends on the specific f).

6-5

6-6

Nonlinear Functions of a Single Independent Variable

(SW Section 6.2)

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

6-7

1. Polynomials in X

Approximate the population regression function by a polynomial:

2r

Y = + X + +…+ + u XXi01i2riii

( 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

6-8

Example: the TestScore Income relation

th

Income = average district income in the i district i

(thousdand dollars per capita)

Quadratic specification:

2

TestScore = + Income + (Income) + u i01i2ii

Cubic specification:

2

TestScore = + Income + (Income) i01i2i

3

+ (Income) + u 3ii

6-9

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 ------------------------------------------------------------------------------

2

The t-statistic on Income is -8.85, so the hypothesis of

linearity is rejected against the quadratic alternative at the

1% significance level.

6-10

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