chapter 4 [article]

chapter 4 [article] - Contents 1 Nonlinear models 1 1.1...

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Unformatted text preview: Contents 1 Nonlinear models 1 1.1 Linearity in parameters . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Linearisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Logarithmic and semilogarithmic models 3 2.1 Logarithmic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Semilogarithmic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 Specification 5 3.1 Where to start . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 How to choose . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1 Nonlinear models Contents 1.1 Linearity in parameters Contents In order to use OLS, we need our true model to be linear in parameters That is, RHS must be a linear function of the i : Y i = 1 + 2 X 2 i + 3 X 3 i + + k X ki + u i true model If its not we cant use OLS, at least not directly Y i = 1 + 2 1 X i + u i true model This model is linear in parameters We can just redefine Z i = 1 X i and run OLS on: Y i = 1 + 2 Z i + u i look at Dougherty slideshow linearity and nonlinearity here Y i = 1 X 2 i The above equation is not linear in parameters (have dropped errors for now) Neither is the below: Y i = 1 e 2 X i Our simple regression OLS estimators b 1 and b 2 would not minimise the RSS for these models Our OLS estimators for simple regression were derived assuming the true model were linear in parameters Recall that b 1 and b 2 were chosen to minimise the RSS : RSS = n X i =1 e 2 i = n X i =1 Y i- Y i 2 = n X i =1 ( Y i- b 1- b 2 X i ) 2 Linearity of parameters implicit in second line above OLS just not designed to deal with nonlinear models 1.2 Linearisation Contents Basic approach when confronted with nonlinear model: try and linearise it...
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chapter 4 [article] - Contents 1 Nonlinear models 1 1.1...

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