Question

1. Suppose you’re interested in estimating the effect of family income on health status. Youroriginal model is given as follows: health index = ?0 + ?1family income + ?

(a) How would you change the model if you think that the effect of family income on health can be different for immigrant and non-immigrant?

(b) In the modified model in (a), how would you test whether there is a differential effect of family income on health status depending on immigrant status? State the null hypothesis and alternative hypothesis.

(c) How would you change the model if you think that the effect of family income on health can be different for people with different ages?

(d) In the modified model in (d), how would you test whether there is a differential effect of family income on health status depending on ages? State the null hypothesis and alternative hypothesis.

Answer 1

Q1 a)

In order to check the difference in effect of family income on health for immigrants and non immigrants, we can add a dummy variable for immigrants, where the value of the variable will be 1 for an immigrant and 0 for a non immigrant and an interaction term.

The equation would be as follows:

health index = ?0 + ?1family income + ?2 immigrants + ?3(family income*immigrants) + ?

E(health index) = ?0 + ?1family income if immigrants = 0

E(health index) = (?0 + ?2) + (?1 +?3) family income if immigrants = 1

The additional effect of family income on health for an immigrant is given by ?3.

b) . In order to see the differential effect one needs to see the effect of family income on health index irrespective of immigration status and the new effect after introducing the dummy variable for immigration.

Thus the null hypothesis is:

H0: ?2 + ?3 = 0

H1: ?2 + ?3 0

c)

In order to check the difference in effect of family income on health based on age, we can add a control variable (Age) for the age of the respondent, where the value of the variable will depend on the respondent's age. It is best to categorise the age variable for different age groups ( eg. 0-10, 10-20, 20-30,......). This categorisation would allow each group to have a single value and hence allow to see the change in effect of family income on health for different age groups.

The equation would be as follows:

health index = ?0 + ?1family income + ?2Age + ?3(family income*Age) + ?

?3 would give the additional effect of family income on health by each age group.

d) In order to test for the differential effect the null hypothesis would be:

H0:  ?2 + ?3 = 0

H1:  ?2 + ?3 0