Chapter 8 Nonlinear Regression Functions Nonlinear Regression Functions

Chapter 8 Nonlinear Regression Functions

Nonlinear Regression Functions (SW Chapter 8) 2

The Test. Score – STR relation looks linear (maybe)… 3

But the Test. Score – Income relation looks nonlinear. . . 4

Nonlinear Regression Population Regression Functions – General Ideas (SW Section 8. 1) 5

The general nonlinear population regression function 6

7

Nonlinear Functions of a Single Independent Variable (SW Section 8. 2) 8

1. Polynomials in X 9

Example: the Test. Score – Income relation 10

Estimation of the quadratic specification in STATA 11

Interpreting the estimated regression function: 12

Interpreting the estimated regression function, ctd: 13

14

Estimation of a cubic specification in STATA 15

16

Summary: polynomial regression functions 17

2. Logarithmic functions of Y and/or X 18

The three log regression specifications: 19

I. Linear-log population regression function 20

Linear-log case, continued 21

Example: Test. Score vs. ln(Income) 22

The linear-log and cubic regression functions 23

II. Log-linear population regression function 24

Log-linear case, continued 25

III. Log-log population regression function 26

Log-log case, continued 27

Example: ln( Test. Score) vs. ln( Income) 28

Example: ln( Test. Score) vs. ln( Income), ctd. 29

The log-linear and log-log specifications: 30

Summary: Logarithmic transformations 31

Other nonlinear functions (and nonlinear least squares) (SW App. 8. 1) 32

Negative exponential growth 33

Nonlinear Least Squares 34

35

36

Interactions Between Independent Variables (SW Section 8. 3) 37

(a) Interactions between two binary variables 38

Interpreting the coefficients 39

Example: Test. Score, STR, English learners 40

(b) Interactions between continuous and binary variables 41

Binary-continuous interactions: the two regression lines 42

Binary-continuous interactions, ctd. 43

Interpreting the coefficients 44

Example: Test. Score, STR, Hi. EL (=1 if Pct. EL 10) 45

Example, ctd: Testing hypotheses 46

(c) Interactions between two continuous variables 47

Interpreting the coefficients: 48

Example: Test. Score, STR, Pct. EL 49

Example, ctd: hypothesis tests 50

Application: Nonlinear Effects on Test Scores of the Student-Teacher Ratio (SW Section 8. 4) 51

Strategy for Question #1 (different effects for different STR? ) 52

Strategy for Question #2 (interactions between Pct. EL and STR? ) 53

What is a good “base” specification? 54

55

Tests of joint hypotheses: 56

Interpreting the regression functions via plots: 57

Next, compare the regressions with interactions: 58

Summary: Nonlinear Regression Functions 59