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ch06 - Multiple Regression Analysis y = 0 1x1 2x2 kxk u 4...

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Economics 20 - Prof. Anderson 1 Multiple Regression Analysis y = β 0 + β 1 x 1 + β 2 x 2 + . . . β k x k + u 4. Further Issues
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Economics 20 - Prof. Anderson 2 Redefining Variables Changing the scale of the y variable will lead to a corresponding change in the scale of the coefficients and standard errors, so no change in the significance or interpretation Changing the scale of one x variable will lead to a change in the scale of that coefficient and standard error, so no change in the significance or interpretation
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Economics 20 - Prof. Anderson 3 Beta Coefficients Occasional you’ll see reference to a “standardized coefficient” or “beta coefficient” which has a specific meaning Idea is to replace y and each x variable with a standardized version – i.e. subtract mean and divide by standard deviation Coefficient reflects standard deviation of y for a one standard deviation change in x
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Economics 20 - Prof. Anderson 4 Functional Form OLS can be used for relationships that are not strictly linear in x and y by using nonlinear functions of x and y – will still be linear in the parameters Can take the natural log of x, y or both Can use quadratic forms of x Can use interactions of x variables
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Economics 20 - Prof. Anderson 5 Interpretation of Log Models If the model is ln( y ) = β 0 + β 1 ln( x ) + u β 1 is the elasticity of y with respect to x If the model is ln( y ) = β 0 + β 1 x + u β 1 is approximately the percentage change in y given a 1 unit change in x If the model is y = β 0 + β 1 ln( x ) + u β 1 is approximately the change in y for a 100 percent change in x
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Economics 20 - Prof. Anderson 6 Why use log models? Log models are invariant to the scale of the variables since measuring percent changes They give a direct estimate of elasticity For models with y > 0, the conditional distribution is often heteroskedastic or skewed, while ln( y ) is much less so The distribution of ln( y ) is more narrow, limiting the effect of outliers
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Economics 20 - Prof. Anderson
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