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Unformatted text preview: 6. Multiple Regression Analysis: Further Issues 6.1 Effects of Data Scaling on OLS Statistics 6.2 More on Functional Form 6.3 More on GoodnessofFit and Selection of Regressors 6.4 Prediction and Residual Analysis 6.1 Data Scaling and OLSScaling data will have NO effect on the conclusions (tests and predictions) that we obtain through OLS 1) If a dependent variable is scaled by dividing by C:estimated coefficients and standard errors will also be divided by C (thus t stats and tests are unaffected)R 2 will be unaffected, but SSR will be divided by C 2 and SER by C as they are unbounded 6.1 Data Scaling and OLS 2) If an independent variable is scaled by dividing by C:the coefficient and standard error of that variable are multiplied by C (thus t statistics and tests are constant) 3) If a dependent OR independent variable in log form is scaled by C:only the intercept is affected, due to the fact that logs in regressions deal with PERCENTAGE changes 6.1 Beta CoefficientsDue to scaling, the sizes of estimated coefficients cant reflect the relative importance of a variableie: measuring in cents would create a smaller coefficient while measuring in thousands would create a larger coefficientTo avoid this, all variables can be STANDARDIZED (subtract mean and divide by standard deviation) and beta coefficients found 6.1 Beta Coefficientsto obtain beta coefficients, begin with the normal OLS regression and subtract means (note that residuals have zero sample average): u x x x x x x y y u x x x y k k k k k ) ( ... ) ( ) ( ... 2 2 2 1 1 1 2 2 1 1 + + + + + = + + + + + = adding sample standard deviations, hat, gives: y k k k k y k y y y u x x x x x x y y ) ( ... ) ( ) ( ) ( 2 2 2 2 2 1 1 1 1 1 + + + + = 6.1 Beta CoefficientsSince standardizing a variable converts it to a z score, we now have: error z b z b z b z k k y + + + + = ... 2 2 1 1Where: 1,...k j = 2200 = j y j j b 6.1 Beta Coefficients These new coefficients are called STANDARDIZED COEFFICIENTS or BETA COEFFICIENTS (which is confusing as the typical OLS regression uses Betas).This regression estimates the change in ys standard deviation when x k s standard deviation changesMagnitudes of coefficients can now be obtainednote that there is no intercept in this normalized equation 6.2 Functional Form  LogsIn this course (and most economics in general) log always refers to the NATURAL LOG (ln)a typical regression including logs is of the form: u ) log( ) log( 2 2 1 1 +...
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 Spring '09
 Priemaza
 Econometrics

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