Calculate regression parameters by hand

Question

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...

Answer 1

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.