Ch. 8 Nonlinear Regression Functions

Ch. 8 Nonlinear Regression Functions - NONLINEAR REGRESSION...

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1 | Nonlinear NONLINEAR REGRESSION FUNCTIONS (Ch. 8) The recommended exercise questions from the textbook: Chapter 8: All except (8.9), for the 2 nd ed; All except (8.9), (8.11) – (8.12), for the 3 rd ed.

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2 | Nonlinear (1) Motivation
3 | Nonlinear [Example] Return to our TestScore example. Wish to look at the relation between students’ performances and district income: T e s t S c o r e . AVGINC (district average income, \$1,000).

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4 | Nonlinear 600 620 640 660 680 700 720 0 10 20 30 40 50 60 AVGINC TESTSCR Set up this model: 2 01 2 TestScore AVGINC AVGINC u   .
5 | Nonlinear [Quadratic regression model] Dependent Variable: TESTSCR Method: Least Squares Included observations: 420 White Heteroskedasticity-Consistent SE & Covariance Variable Coefficient Std. Error t-Statistic Prob. C 607.30 2.902 209.3 0.0000 AVGINC 3.851 0.268 14.36 0.0000 AVGINC^2 -0.042 0.005 -8.851 0.0000 R-squared 0.556 Mean dependent var 654.2 Adjusted R-squared 0.554 S.D. dependent var 19.05 Log likelihood -1662.7 F-statistic 261.28 Durbin-Watson stat 0.951 Prob(F-statistic) 0.0000

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6 | Nonlinear 600 620 640 660 680 700 720 740 0 10 20 30 40 50 60 AVGINC TESTSCR TESTSCRF_1 TESTSCRF_2 How can we test H o : The correct model is a linear regression model?
7 | Nonlinear (2) General model 1) General model: Y = f ( X 1 , . .. , X k ) + u . w h e r e f ( ) is called “nonlinear regression function.” 2) How to estimate the effect on Y of a change in X . • Expected change in Y ( Y ) associated with the change in X 1 ( 1 X ) holding X 2 ,..., X k constant: 11 2 1 2 ( , ,..., ) ( , ,..., ) kk Y f XX f X  . • Estimated change: 1 ˆˆ ( ,..., ) ( ,..., ) Y f XXX f .

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8 | Nonlinear [Example] Wish to estimate the change in test scores associated with a change in district income of \$1,000, based on the estimated quadratic regression function. 22 01 2 2 12 ˆˆ ˆ ˆ (1 1 1 1 ) 0 1 0 ) 21 . Testscr    How can we estimate () SE Testscr ? • Using eviews, test H 0 : β 1 + 21 β 2 = 0. 21 2.963 Testscr . 1 1 2 2 ˆ ˆ ˆ ˆ var( 21 ) var( ) 42cov( , ) 441var( )  . ( ) 0.17 SE Testscr .
9 | Nonlinear Wald Test: Equation: Untitled Test Statistic Value df Prob F-statistic 299.94 (1, 417) 0.000 Chi-square 299.94 1 0.000 Null Hypothesis Summary: Normalized Restric. (=0) Value Std. Err. C(2) + 21*C(3) 2.963 0.171

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10 | Nonlinear (3) Nonlinear functions A. Polynomials: Y = β 0 + β 1 X + β 2 X 2 + . .. + β r X r + u . 1) Testing linearity: T e s t H o : β 2 = . .. = β r = 0. 2) Determining maximum r . (General to specific) • Pick a maximum value of r and estimate the model for that r . • Check the significance of β r . If significant, then keep using that r . • If β r is not significant, estimate the model with maximum ( r -1). If β r -1 is significant, then keep using ( r -1). • If β r -1 is not significant, try ( r -2). Repeat this procedure until the last β becomes significant.
11 | Nonlinear [Example] (Specific to general approach) 23 01 2 3 TestScore AVGINC AVGINC AVGINC u   .

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Ch. 8 Nonlinear Regression Functions - NONLINEAR REGRESSION...

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