Question

Suppose that a random sample of 200 40-year old men is selected from a population and the annual income of each man is recorded. Suppose I am also able to obtain the annual income of each man's father when the father was 40. A regression of the former on the latter yields W? = 25 +0.5W where Wę and W are the annual income in thousands of dollars of man i and his father, respectively, both measured in 1980 dollars. (i) Explain what the estimates (25 for the intercept and 0.5 for the slope) mean intuitively. (ii) What is the predicted annual income for someone whose father earned $50,000? (iii) Using the estimates from this regression, what is the effect of a $20,000 increase in the income of the father on the son's earnings? (iv) I can convert both the dependent and independent variables in the data to 2013 dollars by multiplying by 2.83. How would the estimates of the slope and intercept change? Use the formulas for ſo and ß1 to figure this out.

Answer 1

(i)
Interpretation of intercept -
Annual income of man is 25 thousand dollars when the mean annual income of his father was 0 dollars at age 40.

Interpretation of slope -
Annual income of man will increase by 0.5 thousand dollars when the mean annual income of his father at age 40 will increase by one thousand dollar.

(ii)
For annual income of father = $50,000
$W_i^f$ = 50

$\hat{W_i^c} = 25 + 0.5 W_i^f = 25 + 0.5 * 50 = 50$

Thus, predicted annual income of man is 50 thousand dollars.

(iii)

For a $20,000 increase in the father, $W_i^f$ will increase by 20.

Effect on annual income of man = 20 * 0.5 = 10

Thus, annual income of man will increase by $10,000 with $20,000 increase in the annual income of father.

(iv)

Let the estimated regression equation be,

$\hat{W_i^c} = \beta_0+ \beta_1 W_i^f$

By converting the data to 2013 dollars by multiplying by 2.83, we get

$\hat{W_i^c} * 2.83 = \beta_0+ \beta_1 W_i^f * 2.83$

$\Rightarrow \hat{W_i^c} = \beta_0 /2.83+ \beta_1 W_i^f$

Thus, for the new regression equation, slope will remain same as $\beta_1 = 0.5$

The intercept will change as, $\beta_0 = 25 / 2.83 = 8.83$