Econ 399 Chapter6a

# Econ 399 Chapter6a - 6 Multiple Regression Analysis Further...

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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 Goodness-of-Fit and Selection of Regressors 6.4 Prediction and Residual Analysis 6.1 Data Scaling and OLS-Scaling 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 Coefficients-Due to scaling, the sizes of estimated coefficients can’t reflect the relative importance of a variable-ie: measuring in cents would create a smaller coefficient while measuring in thousands would create a larger coefficient-To avoid this, all variables can be STANDARDIZED (subtract mean and divide by standard deviation) and beta coefficients found 6.1 Beta Coefficients-to 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 ˆ ) ( ˆ ... ) ( ˆ ) ( ˆ ˆ ˆ ˆ ˆ ... ˆ ˆ ˆ ˆ 2 2 2 1 1 1 2 2 1 1 +- + +- +- + =- + + + + + = β β β β β β β β-adding sample standard deviations, σ hat, gives: y k k k k y k y y y u x x x x x x y y σ σ β σ σ σ β σ σ σ β σ σ σ ˆ ˆ ˆ ) ( ˆ ˆ ˆ ... ˆ ) ( ˆ ˆ ˆ ˆ ) ( ˆ ˆ ˆ ˆ ) ˆ ( 2 2 2 2 2 1 1 1 1 1 +- + +- +- =- 6.1 Beta Coefficients-Since standardizing a variable converts it to a z- score, we now have: error z b z b z b z k k y + + + + = ˆ ... ˆ ˆ ˆ 2 2 1 1-Where: 1,...k j ˆ ˆ ˆ ˆ = 2200 = j y j j b β σ σ 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 y’s standard deviation when x k ’s standard deviation changes-Magnitudes of coefficients can now be obtained-note that there is no intercept in this normalized equation 6.2 Functional Form - Logs-In 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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## This note was uploaded on 03/14/2009 for the course ECON ECON 399 taught by Professor Priemaza during the Spring '09 term at University of Alberta.

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Econ 399 Chapter6a - 6 Multiple Regression Analysis Further...

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