Econometrics-I-14 - Applied Econometrics William Greene...

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Applied Econometrics William Greene Department of Economics Stern School of Business
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Applied Econometrics 14. Nonlinear Regression and Nonlinear Least Squares
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Nonlinear Regression What makes a regression model “nonlinear?” Nonlinear functional form?     Regression model: y i   =  f(  x i  ,  β  )  +   ε i Not necessarily:          y i  = exp( α ) +  β 2 *x i  +  ε i                                 β  = exp( α )                                y i   = exp( α )x i β exp( ε i )     is “loglinear” Models can be nonlinear in the functional form of the  relationship between y and x, and not be nonlinear for  purposes here. We will redefine “nonlinear” shortly, as we proceed.
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Nonlinear Least Squares Least squares: Minimize wrt  β  ½  Σ i  [y i   - f( x i , β )] 2                                           = ½  Σ i  e i 2 First order conditions:     ½ Σ i [y i - f( x i , β )] 2  ]/ ∂β             = ½ Σ i (-2)[y i - f( x i , β )]  f( x i , β )/ ∂β           =  - Σ i  e i   x i 0   0 (familiar?) There is no explicit solution,  b  = f( data ) like LS. (Nonlinearity of the FOC defines nonlinear model)
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Example: NIST How to solve this kind of set of equations:  Example,            y i   =   β 0 +  β 1x i β 2  +  ε i . [ ½ Σ i  e i 2 ]/ ∂β 0  =  Σ i  (-1) (y i  -  β 0 -  β 1x i β 2 )  1             =  0 [ ½ Σ i  e i 2 ]/ ∂β 1  =  Σ i  (-1) (y i  -  β 0 -  β 1 x i β 2 ) x i β 2           =  0 [ ½ Σ i  e i 2 ]/ ∂β 2  =  Σ i  (-1) (y i  -  β 0 -  β 1 x i β 2 β 1 x i β 2 lnx i  =  0 Nonlinear equations require a nonlinear solution.  We’ll  return to that problem shortly. This defines a nonlinear regression model.  I.e., when the  first order conditions are not linear in  β . (!!!) Check your understanding.  What does this produce if  f(  x i  ,  β  ) =  x i ′ β ?  (I.e., a linear model)
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Linear Taylor series:    y  = f( x i , β )  +   ε .    Expand the regression around some point,  β 0 . f( x i , β )    f( x i , β 0 )  +  Σ k [ f( x i , β 0 )/ ∂β k 0 ](  β k   -   β k 0 )   = f( x i , β 0 )  +  Σ k  x i 0  (  β k   -   β k 0 )   = [f( x i , β 0 ) -   Σ k  x i 0 β k 0 ] +  Σ k  x i 0 β k           =  f 0   +  Σ k  x i 0 β k  which looks linear. The ‘pseudo-regressors’ are the derivative 
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This note was uploaded on 06/20/2011 for the course ECON 803 taught by Professor Pp during the Spring '11 term at Thammasat University.

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Econometrics-I-14 - Applied Econometrics William Greene...

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