This continues question 18; different notation is used: part

(b) might be a little tricky. Garrett’s model includes a dummy variable for each of the 14 countries. The growth rate for country i in year t is modeled as

αi + Zitγ + $it, where Zit is a 1×10 vector of explanatory variables, including LPP, TUP, and the interaction. (In question 18, the country dummies didn’t matter, and were folded into X.) Beck (2001) uses the same model—except that an intercept is included, and the dummy for country #1 is excluded. So, in this second model, the growth rate in country i > 1 and year t is

α∗ + α∗

i + Zitγ ∗ + $it;

whereas the growth rate in country #1 and year t is

α∗ + Z1tγ ∗ + $1t .

Assume both investigators are fitting by OLS and using the same data.

(a) Why can’t you have a dummy variable for each of the 14 countries, and an intercept too?

(b) Show that γˆ = ˆγ ∗

, αˆ 1 = ˆα∗

, and αˆi = ˆα∗ + ˆα∗

i for i > 1.

Hints for (b). Let M be the design matrix for the first model; M∗, for the second. Find a lower triangular matrix L—which will have 1’s on the diagonal and mainly be 0 elsewhere—such that ML = M∗. How does this relationship carry over to the parameters and the estimates?