DOC

Economics 1123

By Mario Hawkins,2014-04-30 07:32
15 views 0
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

Report this document

For any questions or suggestions please email
cust-service@docsford.com