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 β i : Y i = β 1 + β 2 X 2 i + β 3 X 3 i + ··· + β k X ki + u i ∼ true model • If it’s not we can’t use OLS, at least not directly Y i = β 1 + β 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 = β 1 + β 2 Z i + u i look at Dougherty slideshow ‘linearity and nonlinearity’ here Y i = β 1 X β 2 i • The above equation is not linear in parameters (have dropped errors for now) • Neither is the below: Y i = β 1 e β 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- ˆ 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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This note was uploaded on 01/14/2012 for the course ECON 201 taught by Professor Witte during the Spring '08 term at Northwestern.

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

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