lecture7 - ECON 5350 Class Notes Nonlinear Regression...

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Unformatted text preview: ECON 5350 Class Notes Nonlinear Regression Models 1 Introduction In this chapter, we examine regression models that are nonlinear in the parameters and give a brief overview of methods to estimate such models. 2 Nonlinear Regression Models The general form of the nonlinear regression model is y i = h ( x i ;&; i ) ; (1) which is more commonly written in a form with an additive error term y i = h ( x i ;& ) + i : (2) Below are two examples 1. h ( x i ;&; i ) = & x & 1 1 i x & 2 2 i exp( i ) . This is an intrinsically linear model because by taking natural loga- rithms, we get a model that is linear in the parameters, ln( y i ) = & + & 1 ln( x 1 i ) + & 2 ln( x 2 i ) + i . This can be estimated with standard linear procedures such as OLS. 2. h ( x i ;& ) = & x & 1 1 i x & 2 2 i . Since the error term in (2) is additive, there is no transformation that will produce a linear model. This is an intrinsically nonlinear model (i.e., the relevant &rst-order conditions are nonlinear in the parameters). Below we consider two methods for estimating such a model linearizing the underlying regression model and nonlinear optimization of the objective function. 2.1 Linearized Regression Model and the Gauss-Newton Algorithm Consider a &rst-order Taylor series approximation of the regression model around & y i = h ( x i ;& ) + i h ( x i ;& ) + g ( x i ;& )( & & & ) + i where g ( x i ;& ) = ( @h=@& 1 j & = & ;:::;@h=@& k j & = & ) : 1 Collecting terms and rearranging gives Y = X & + where Y & Y h ( X;& ) + g ( X;& ) & X & g ( X;& ) : The matrix X is called the pseudoregressor matrix. Note also that will include higher-order approxima- tion errors. 2.1.1 Gauss-Newton Algorithm Given an initial value for & , we can estimate & with the following iterative LS procedure b t +1 = [ X ( b t ) X ( b t )] & 1 [ X ( b t ) Y ( b t )] = [ X ( b t ) X ( b t )] & 1 [ X ( b t ) ( X ( b t ) b t + e t )] = b t + [ X ( b t ) X ( b t )] & 1 X ( b t ) e t = b t + W t t g t where W t = [2 X ( b t ) X ( b t )] & 1 , t = 1 and g t = 2 X ( b t ) e t . The iterations continue until the di/erence between b t +1 and b t is su ciently small. This is called the Gauss-Newton algorithm. Interpretations for W t , t and g t will be given below. A consistent estimator of 2 is s 2 = 1 n k X n i =1 ( y i h ( x i ;b )) 2 : 2.1.2 Properties of the NLS Estimator Only asymptotic results are available for this estimator. Assuming that the pseudoregessors are well-behaved (i.e., plim 1 n X X = Q , a &nite positive de&nite matrix), then we can apply the CLT to show that b asy N [ &; 2 n ( Q ) & 1 ] , where the estimate of & 2 n ( Q ) & 1 is s 2 ( X X ) & 1 : 2 2.1.3 Notes 1. Depending on the initial value, b , the Gauss-Newton algorithm can lead to a local (as opposed to global) minimum or head o/ on a divergent path.global) minimum or head o/ on a divergent path....
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This note was uploaded on 11/27/2011 for the course ECON 101 taught by Professor Robert during the Fall '08 term at Montgomery College.

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lecture7 - ECON 5350 Class Notes Nonlinear Regression...

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