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The following is from Gelman & Hill 2007:

Suppose that, for a certain population, we can predict log earnings from log 


- A person who is 66 inches tall is predicted to have earnings of $30,000.    

Every increase of 1% in height corresponds to a predicted increase of 0.8% 
in earnings.

- The earnings of approximately 95% of people fall within a factor of 1.1
of predicted values.

1.Give the equation of the regression line and the residual standard deviation 
of the regression.

2.Suppose the standard deviation of log heights is 5% in this population. 
What, then, is the R2 of the regression model described here?

For the first question I would need a regression equation where log(Y) =intercept + 0.8*log(X) + error, where:

Y= earnings for people who have the height of 0 inches.

Beta= 0.8 increase in earnings with every inch of increase in height

I'm wondering if there's a simple way to calculate Y and the residual standard deviation...

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  • $\begingroup$ It's an odd example in several ways. Normally we'd calculate b as the change in Y associated with a change of 1 (rather than of a certain %) in X. I don't see how to handle that % change as indicated. Also please note that the unstandardized coefficient, b, is different from the standardized one, Beta. $\endgroup$
    – rolando2
    Commented Nov 11, 2011 at 11:10
  • $\begingroup$ yes I also found the % strange but I just assumed that it means a 1 unit increase $\endgroup$
    – upabove
    Commented Nov 11, 2011 at 11:19
  • $\begingroup$ Hint: The question says that log earnings can be predicted from log height. So, your answer is along the right lines but you are missing some logs here and there. $\endgroup$
    – varty
    Commented Nov 11, 2011 at 15:27
  • $\begingroup$ @varty yes I left that out so its log(earnings)=0.8*log(height)+error but I still don't know how to calculate the intercept and how to proceed to the second question $\endgroup$
    – upabove
    Commented Nov 11, 2011 at 15:30
  • $\begingroup$ @Dbr In that case you should edit the question to make it accurate/correct. You are not using all the information that has been given to you. What about the information on the earnings of someone who is 66 inches? $\endgroup$
    – varty
    Commented Nov 11, 2011 at 15:43

1 Answer 1

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I might as well answer your question albeit indirectly to help out a bit:

  1. The regression model is:

    Log(Earnings) = $\beta_0$ + $\beta_1$ Log(Height) + $\epsilon$

  2. Slope

    Since both the dependent and the independent variable are in 'logs' $\beta_1$ represents the elasticity of earnings vis-a-vis height. In other words, $\beta_1$ represents the percentage increase in earnings with every 1% increase in height.

    Thus, $\beta_1$ = 0.8.

  3. Intercept

    You know that a 66 inches tall person is predicted to earn $30,000. Notice the emphasis on the word 'prediction'. The above value is not the actual earnings of a 66 inches person but the prediction of the regression model.

    You should be able to use this information to find out the intercept.

Hopefully the above will get you started on the rest of the questions.

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  • $\begingroup$ to be honest this is all the information I have already known, I understand what the intercept and slope are etc, I'm just not sure how to calculate the intercept for someone with a log height of 0. $\endgroup$
    – upabove
    Commented Nov 11, 2011 at 16:40
  • $\begingroup$ i'm thinking 30.000/66 could be the intercept but i'm not sure.. $\endgroup$
    – upabove
    Commented Nov 11, 2011 at 16:41
  • $\begingroup$ Suppose that you do know the intercept and slope. How would you predict the earnings of a 66 inch tall person? 30000/66 is not the correct answer for the intercept. $\endgroup$
    – varty
    Commented Nov 11, 2011 at 16:42
  • $\begingroup$ I would add the intercept which is the earnings of someone with a height of 0 and add 66times the Beta1, which is 0.8*66 in this case $\endgroup$
    – upabove
    Commented Nov 11, 2011 at 16:44
  • $\begingroup$ Well, you are dealing with logs remember. So, you would calculate: beta_1 + 0.88 log(66) and that equals what now? $\endgroup$
    – varty
    Commented Nov 11, 2011 at 16:49

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