Chapter13 - Nonlinear Regression STAT 563 Spring 2007...

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Nonlinear Regression STAT 563 Spring 2007
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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 0 , 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
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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 0 γ γ
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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 0 =ln( θ 0 ) 1 0 1 0 ) , , ( θ θ θ θ x x f = ) ln( ) ln( ) , , ( ln( 1 0 1 0 x x f θ θ θ θ + = * ** ε θ α + + = ) ln( ) ln( 1 0 x y ***
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Example If ε ~N(0, σ 2 ), it follows that the model for y has the form
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