Nonlinear Regression Functions SW Chapter 8

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Nonlinear Regression Functions(SW Chapter 8)
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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 Population Regression
Functions General Ideas (SW Section 8.1)
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The general nonlinear population regression
function
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Nonlinear Functions of a Single Independent
Variable (SW Section 8.2)
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1. Polynomials in X
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Example the TestScore Income relation
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Estimation of the quadratic specification in
STATA
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Interpreting the estimated regression function
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Interpreting the estimated regression function,
ctd
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Estimation of a cubic specification in STATA
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Ramseys RESET Test REgression Specification
Error Test

• Consider the model (1)
• General test for misspecification of functional
  form
• If LSA 1 holds, then no non-linear function of
  the Xs should be significant when added to the
  model.
• Consider (2)
• Null hypothesis is that (1) is correctly
  specified
• How many powers of predicted values to include?
• Conduct F-test on powers of predicted values
• J.B. Ramsey (1969), Tests for Specification Error
  in Classical Linear Least Squares Regression
  Analysis. Journal of the Royal Statistical
  Society, Series B 31, 350371

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Ramseys RESET Test
. reg test str avginc, r Linear regression
Number of obs
420
F( 2, 417) 132.65
Prob gt F
0.0000
R-squared 0.5115

Root MSE 13.349 -------------------------
--------------------------------------------------
--- Robust
testscr Coef. Std. Err. t Pgtt
95 Conf. Interval -------------------------
--------------------------------------------------
-- str -.6487401 .3533403 -1.84
0.067 -1.34329 .04581 avginc
1.839112 .114733 16.03 0.000 1.613585
2.064639 _cons 638.7292 7.301234
87.48 0.000 624.3773
653.081 ------------------------------------------
------------------------------------
. estat ovtest (can just type . ovtest)
Ramsey RESET test using powers of the fitted
values of testscr Ho model has no
omitted variables F(3, 414)
18.36 Prob gt F 0.0000
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Ramseys RESET Test
. reg test str avginc avginc2, r Linear
regression
Number of obs 420
F( 3, 416)
286.55
Prob gt F 0.0000

R-squared 0.5638
Root MSE
12.629 ------------------------------------------
------------------------------------
Robust testscr Coef.
Std. Err. t Pgtt 95 Conf.
Interval ---------------------------------------
--------------------------------------
str -.9099512 .3545374 -2.57 0.011
-1.606859 -.2130432 avginc 3.881859
.2709564 14.33 0.000 3.349245
4.414474 avginc2 -.044157 .0049606
-8.90 0.000 -.053908 -.034406
_cons 625.2308 7.087793 88.21 0.000
611.2984 639.1631 ----------------------------
--------------------------------------------------
. estat ovtest Ramsey RESET test using powers
of the fitted values of testscr Ho model
has no omitted variables F(3,
413) 2.48 Prob gt F
0.0605
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Ramseys RESET Test
. reg test str avginc avginc2 avginc3, r Linear
regression
Number of obs 420
F( 4, 415)
207.23
Prob gt F 0.0000

R-squared 0.5663
Root MSE
12.608 ------------------------------------------
------------------------------------
Robust testscr Coef.
Std. Err. t Pgtt 95 Conf.
Interval ---------------------------------------
--------------------------------------
str -.9277523 .3562919 -2.60 0.010
-1.628114 -.2273905 avginc 5.124736
.7045403 7.27 0.000 3.739824
6.509649 avginc2 -.1011073 .0287052
-3.52 0.000 -.157533 -.0446815
avginc3 .0007293 .0003414 2.14 0.033
.0000582 .0014003 _cons 617.8974
7.926373 77.95 0.000 602.3165
633.4782 -----------------------------------------
------------------------------------- . estat
ovtest Ramsey RESET test using powers of the
fitted values of testscr Ho model has no
omitted variables F(3, 412)
1.79 Prob gt F 0.1490
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Ramseys RESET Test
. reg test str el_pct meal_pct , r Linear
regression
Number of obs 420
F( 3, 416)
453.48
Prob gt F 0.0000

R-squared 0.7745
Root MSE
9.0801 ------------------------------------------
------------------------------------
Robust testscr Coef.
Std. Err. t Pgtt 95 Conf.
Interval ---------------------------------------
--------------------------------------
str -.9983092 .2700799 -3.70 0.000
-1.529201 -.4674178 el_pct -.1215733
.0328317 -3.70 0.000 -.18611
-.0570366 meal_pct -.5473456 .0241072
-22.70 0.000 -.5947328 -.4999583
_cons 700.15 5.56845 125.74 0.000
689.2042 711.0958 ----------------------------
--------------------------------------------------
. estat ovtest Ramsey RESET test using powers
of the fitted values of testscr Ho model
has no omitted variables F(3,
413) 6.29 Prob gt F
0.0004
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Ramseys RESET Test
. reg test str el_pct meal_pct avginc ,
r Linear regression
Number of obs 420
F( 4,
415) 467.42
Prob gt F 0.0000

R-squared 0.8053
Root MSE
8.4477 ------------------------------------------
------------------------------------
Robust testscr Coef.
Std. Err. t Pgtt 95 Conf.
Interval ---------------------------------------
--------------------------------------
str -.5603892 .2550641 -2.20 0.029
-1.061768 -.0590105 el_pct -.1943282
.0332445 -5.85 0.000 -.2596768
-.1289795 meal_pct -.3963661 .0302302
-13.11 0.000 -.4557895 -.3369427
avginc .674984 .0837161 8.06 0.000
.5104236 .8395444 _cons 675.6082
6.201865 108.94 0.000 663.4172
687.7992 -----------------------------------------
------------------------------------- . estat
ovtest Ramsey RESET test using powers of the
fitted values of testscr Ho model has no
omitted variables F(3, 412)
0.47 Prob gt F 0.7014
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Ramseys RESET Test replicated
. predict yh (option xb assumed fitted
values) . sum yh Variable Obs
Mean Std. Dev. Min
Max ---------------------------------------------
------------------------ yh 420
654.1565 17.09817 614.9183 702.8387 .
gen yhz (yh-r(mean))/r(sd) . sum yh
Variable Obs Mean Std. Dev.
Min Max --------------------------------
-------------------------------------
yh 420 654.1565 17.09817 614.9183
702.8387 yhz 420 1.22e-09
1 -2.294882 2.847214 . gen
yhz2yhzyhz . gen yhz3yhz3 . gen yhz4yhz4
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Ramseys RESET Test replicated
. reg test str el meal avginc yhz2 yhz3 yhz4
Source SS df MS
Number of obs 420 ------------------------
------------------- F( 7, 412)
244.48 Model 122595.145 7
17513.5921 Prob gt F 0.0000
Residual 29514.4488 412 71.6370116
R-squared 0.8060 ------------------------
------------------- Adj R-squared
0.8027 Total 152109.594 419
363.030056 Root MSE
8.4639 ------------------------------------------
------------------------------------ testscr
Coef. Std. Err. t Pgtt 95
Conf. Interval ---------------------------------
--------------------------------------------
str -.5500585 .2336368 -2.35 0.019
-1.009327 -.0907896 el_pct -.2170374
.0407058 -5.33 0.000 -.2970544
-.1370204 meal_pct -.400967 .0289303
-13.86 0.000 -.4578364 -.3440976
avginc .6476592 .1505253 4.30 0.000
.3517657 .9435527 yhz2 .7652051
.915534 0.84 0.404 -1.034495
2.564906 yhz3 -.0822669 .3243362
-0.25 0.800 -.7198272 .5552933
yhz4 -.0650369 .1767693 -0.37 0.713
-.412519 .2824453 _cons 675.8077
5.443279 124.15 0.000 665.1076
686.5077 -----------------------------------------
------------------------------------- . test
yhz2 yhz3 yhz4 ( 1) yhz2 0 ( 2) yhz3 0
( 3) yhz4 0 F( 3, 412) 0.47
Prob gt F 0.7014
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Summary polynomial regression functions
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2. Logarithmic functions of Y and/or X
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The three log regression specifications
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I. Linear-log population regression function
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Linear-log case, continued
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Example TestScore vs. ln(Income)
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The linear-log and cubic regression functions
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II. Log-linear population regression function
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Log-linear case, continued
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III. Log-log population regression function
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Log-log case, continued
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Example ln( TestScore) vs. ln( Income)
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Example ln( TestScore) vs. ln( Income), ctd.
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The log-linear and log-log specifications
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Summary Logarithmic transformations
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Other nonlinear functions (and nonlinear least
squares) (SW App. 8.1)
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Negative exponential growth
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Nonlinear Least Squares
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