Chapter13

Chapter13 - Nonlinear Regression STAT 563 Spring 2007...

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Unformatted text preview: Nonlinear Regression STAT 563 Spring 2007 Regression Model • Recall that we can write the normal theory regression model as • Where x is a n-vector of input variables, θ is a k-vector of parameters, and the errors are independent N(0, σ 2 ) • The assumption that f(x , θ ) is a linear function of θ defines the linear regression model – For example, in equation (13.1), we can set θ = (β , β 1 ,..., β κ 29 • Linear regression is popular – Simplicity – Broad applicability – Elegant distributional and inferential results • What about when the expected value of the response is not a linear function of the parameters? – Results in non-linear regression models ε θ + = ) , ( x f y Example • In our discussion on Forbes data, we looked at a theoretical relationship between the pressure and the boiling point of water as – Expected value function is not linear in parameters – We took logarithm to make it linear and verified that the assumptions of additive normal errors was reasonable in the transformed model + = BP PR 7 . 459 exp 1 γ γ Example • Cobb-Douglass function – Often used to develop economic models – Takes the form – Taking log on both sides will linearize the form – To fit the data (y,x) to a model with mean function specified above (*), we can use the linear model in (**) with α =ln( θ ) 1 1 ) , , ( θ θ θ θ x x f = ) ln( ) ln( ) , , ( ln( 1 1 x x f θ θ θ θ + = * ** ε θ α + + = ) ln( ) ln( 1 x y *** Example •...
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This note was uploaded on 03/08/2009 for the course 960 563 taught by Professor Unknown during the Spring '07 term at Rutgers.

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Chapter13 - Nonlinear Regression STAT 563 Spring 2007...

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