1. Economics 105: Statistics GH 24 due Wednesday

2. Hypothesis Tests on Several Regression Coefficients Consider the model (expanding on GH 22) Is “race” as a group significant?

3. Hypothesis Tests on Several Regression Coefficients obsln(hrwage)educexpexp2bluegreenredpurplecheckrace 12.1484341614196010012 22.42036812749010012 31.60943816441936000114 41.6094388502500100011 52.3025851224576100011 62.01490312416100011 71.6094381011100011 82.9575111230900010012 93.1322281612144001013 102.99573211431849010012 112.6741491430900001013 123.58490712361296001013 132.3025851329841001013 142.5745191023529100011 152.7880931626676001013

4. Hypothesis Tests on Several Regression Coefficients To test Use F statistic Impose the restrictions to get “restricted” terms m is the number of restrictions Reject H 0 if Intuition?

5. Hypothesis Tests on Several Regression Coefficients

8. Multiple Regression: Example where Sign Switches Correlations Rating Age Income Rating 1.000 0.587 0.885 Age 0.587 1.000 0.829 Income 0.885 0.829 1.000 Survey of 75 consumers Rating = rating of likelihood of purchase of a PDA (e.g., palm pilot) on a scale of 1-10, 10 indicating highest likelihood. Age = age in years Income = income in thousands of dollars

9. Multiple Regression: Example where Sign Switches Regression of Rating on Age Estimate Std Error t Ratio Prob>|t| Intercept 2.067 0.487 4.24 <.0001 Age 0.059 0.009 6.19 <.0001 Regression of Rating on Income Term Estimate Std Error t Ratio Prob>|t| Intercept -0.596 0.352 -1.69 0.0951 Income 0.070 0.004 16.20 <.0001

10. Multiple Regression: Example where Sign Switches Multiple Regression Estimates Term Estimate Std Err t Ratio Prob>|t| Intercept -0.736 0.295 -2.50 0.0149 Age -0.047 0.008 -5.74 <.0001 Income 0.101 0.006 15.63 <.0001 Conclusions?

11. Variable Selection (or Model Building) OLS Assumption #1 (and #2 and #5) Use theory and prior research Use your hypotheses But what if you don’t have much theoretical guidance? –Parsimony=f(simplicity, fit) –Using adj R 2 … fit, controlling for complexity

12. Empirical Indicators in Model Building When adding a variable, check for: –Improved prediction (increase in adj R 2 ) –Statistically and substantively significant estimated coefficients –Stability of model coefficients Do other coefficients change when adding the new one? Particularly look for sign changes

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14. Risks in Model Building Including irrelevant X’s –Increases complexity –Reduces adjusted R 2 –Increases model variability across samples Omitting relevant X’s –Fails to capture fit –Can bias other estimated coefficients Where omitted X is related to both other X’s and to the dependent variable (Y)

15. More Risks: Samples Can Mislead Remember: we are using sample data –About 5% of the time, our sample will include random observations of X’s that result in betahat’s that meet classical hypothesis tests –Or the beta’s may be important, but the sample data will randomly include observations of X that do not meet the statistical tests That’s why we rely on theory, prior hypotheses, and replication