| Criterion | ( R^2 ) | Adjusted ( R^2 ) (( \barR^2 )) | |-----------|-----------|------------------------------------| | Formula | ( 1 - \fracRSSTSS ) | ( 1 - \fracRSS/(n-k)TSS/(n-1) ) | | Penalty for extra X | No | Yes | | Range | 0 to 1 (can increase by adding any variable) | Can be negative (if very poor model) | | Use | Measures proportion of variance explained | Model selection across different numbers of regressors |
In a log-log model, ( \beta_2 ) is the elasticity of ( Y ) with respect to ( X ): [ \beta_2 = \frac%\Delta Y%\Delta X ]
If ( \hat\beta_2 = 0.8 ), a 1% increase in ( X ) leads to a 0.8% increase in ( Y ) (holding other factors constant). This model is very common in demand analysis (price elasticity) and production functions. Q5: Distinguish between ( R^2 ) and adjusted ( R^2 ). A:
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| Criterion | ( R^2 ) | Adjusted ( R^2 ) (( \barR^2 )) | |-----------|-----------|------------------------------------| | Formula | ( 1 - \fracRSSTSS ) | ( 1 - \fracRSS/(n-k)TSS/(n-1) ) | | Penalty for extra X | No | Yes | | Range | 0 to 1 (can increase by adding any variable) | Can be negative (if very poor model) | | Use | Measures proportion of variance explained | Model selection across different numbers of regressors |
In a log-log model, ( \beta_2 ) is the elasticity of ( Y ) with respect to ( X ): [ \beta_2 = \frac%\Delta Y%\Delta X ]
If ( \hat\beta_2 = 0.8 ), a 1% increase in ( X ) leads to a 0.8% increase in ( Y ) (holding other factors constant). This model is very common in demand analysis (price elasticity) and production functions. Q5: Distinguish between ( R^2 ) and adjusted ( R^2 ). A:
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