Prof. Bryan Caplan

bcaplan@gmu.edu

http://www.gmu.edu/departments/economics/bcaplan

Econ 637

Spring, 1998

General Comments: This packet contains sample computing programs for BOTH HW#2 and HW#3, in addition to suggested answers for math problems from HW#3. Don't forget to pick up your graded HW - it's ready outside my door. Have a good break, and don't forget the Monday the 16^th midterm.

2.5. You have . Guessing that this is the equation for a hyperbola, I try to rewrite it in the form xy=K to double-check and find the asymptotes:

, so . Just define Y*=100/(100-Y); then:

. Then the asymptotes appear at x=0 and (Y*-a )=0.

2.6. Function #1: . Taking the reciprocals of both sides:

, so . Then let Y*=1/y, and X*=-1/x, so you have Y*=a +b x*.

Properties: x¹ 0; y¹ 0.

Function #2: . Experimenting, I multiply both sides of the equation by the denominator of the right-hand side:

; then , so . Now just take logs of both sides of the equation: . Then just define the left-hand side of the equation as Y*.

Properties: This is a semilog transformation (p.46). Since ln (y/(1-y))=ln y - ln (1-y), it has asymptotes at y=0 and y=1.

4.4. . For convenience, I switch to b 's and b's:

. Then we must prove that:

. I assume that the three educational classes are supposed to be mutually exclusive and jointly exhaustive - everyone is supposed to be in one and only one class. Now just apply the equations from Ex 3.2:

Since everyone is only in one class, the cross-products of the dummies are always zero:

Since the dummies are either 0 or 1, ; similarly, since the dummies are 0 or 1, , where is the sum of the values of Y for group i.

Now designate the number of persons in each class as n1, n2, and n3, so that n1+n2+n3=n. Then:

Notice now that gives the total value of the Y's of group 2; since n2 is the number of members of group 2, , the average value of Y for group 2!

Thus:

The same goes for group 3: .

Now plug these two equations into the 1st equation:

Then:

; so

; . Plugging this back in to and , we find that:

QED.

9.9.

(a) Since we want "reduced form" estimates, we want to solve for either y1t or y2t as functions of z1t, z2t, and z3t. I'll set it up both ways at once, although you only had to do it one way or the other.

First, use matrix notation to solve for Y as a function of Z:

The formula for the coefficients of a regression of Y on Z yields is just :

Inverting Z'Z (the first matrix) in Gauss, and subbing in:

In other words, the best fit of a regression of y1t on the Z's is:

y1t=z1t+2z2t-z3t

Similarly:

z2t=z1t+2z2t+z3t.

Note: You could also have solved this problem by subbing the first equation into the second (or the second into the first) to solve for y2t as a function of the Z's, then regressing y2t on the Z's.

(c) To get the 2SLS estimates of the 2^nd equation's parameters, one first regresses y1t and z3t on ALL of the exogenous variables (z1t, z2t, z3t). The fitted value for z3t is naturally just z3t. Furthermore, as shown in (a), the fitted values for Y* are: , so the fitted values for Y1* are: .

The second stage of the two-stage process simply requires us to regress y2t on y1* and z3t.

Using OLS to get the coefficients on y1* and z3t:

Plugging in for y1*:

Cancelling:

Now just plug in from the table provided:

Plug in for inv(Z'Z) from Gauss:

So

Simplifying:

, so .