Question

Suppose that we have data on ECON 333 test scores (Yi), duration for which student i studies for exam (Xi), and the major of the student, call it Di, where Di =( 1, if economics major 0, if non economics major
Consider the following model:
Yi = β0 + β1Xi + β2Di + β3DiXi + ui (1)
where Assumption 1 holds:
E (ui|Xi,Di) = 0. (2) Yi is the score between 0 and 100. Xi is the duration studied in hours, between 1 and 100. a) What is the intercept for econ-majors? b) What is the slope for econ-majors? c) What is the interpretation (in words) of β1? d) What is the interpretation (in words) for β3? Now suppose that the OLS estimate for β1 is 0.60, and its standard error is 0.2. e) Test whether β1 = 0, at the 5% and 1% significance level. Finally, we try a different model:
ln(Yi) = β0 + β1 ln(Xi) + β2Di + β3Di ln(Xi) + ui (3)
Yi = β0 + β1 ln(Xi) + β2Di + β3Di ln(Xi) + ui (4)
ln(Yi) = β0 + β1Xi + β2Di + β3DiXi + ui (5)
f) What is the interpretation of β1 in models (3), (4) and (5)?

Answer 1

Given , Yi = test scores for student i, Xi = duration of study in hours for student i , Di = dummy variable indicating 1 for econ-majors and 0 for non-econ-majors.

also the model used is ______(1)

Y; = Bo + Bi X; + B2D; + B3D;X; + u

(a)we want the intercept of econ majors, so we put Di = 1 in the above model and get
Y; = (Bo + B2) +(B1 + B3) X; + uz
. Thus the required slope is
(Bo + B2)

(b) we want the slope of econ majors, so we put Di = 1 in the above model and get
Y; = (Bo + B2) +(B1 + B3) X; + uz
. Thus the required slope is
(3 + B)

(c) in the original model (1) marked above the interpretation of $\beta_1$ is as follows : assuming all other factors are at zero one unit change in Xi will cause atleast $\beta_1$ change in Yi depending on whether the student is an econ-major or not.

(d) based on the original model $\beta_3$ interprets as follows: assuming all other factors to be constant a change in the value of Xi will cause an additional change of  $\beta_3$ in Yi depending on whether Di = 0 or 1