Effect of years of education on earnings.
Problem Set I
Problem 1. Consider an earnings function with the dependent variable y monthly
usual earnings and as independent variables years of education x1, gender x2
coded as 1 if female and 0 if male, and work experience in years x3. We are
interested in the partial effect of years of education on earnings. We consider
the following possible relations (that are assumed to be exact)
y =β0 + β1×1 + β2×2 + β3×3 (1)
y =β0 + β1×1 + β2×2 + β3×3 + β4x
2
1
(2)
y =β0 + β1×1 + β2×2 + β3×3 + β4x1x2 (3)
We are interested in the partial, i.e. ceteris paribus, effect of x1 on earnings y.
(i) Use partial differentiation to find the partial effect in the three specifications above.
(ii) For which specifications are the partial effects constant, i.e. independent
of the level of x1, x2, x3? If not constant how does the partial effect change
with x1, x2, x3?
(iii) If we have data that allow us to estimate the regression coefficients, how
would you report the partial effects if they are not constant and you still
want to report a single number?
(iv) Can you use partial differentiation to find the partial effect of x2? Why
(not)?
(v) Often work experience is not directly observed, but measured as AGEYEARS OF EDUCATION – 6. Does this change your answers to (i) and
(ii)?
Problem 2. Instead of partitioning the n×K matrix X as a row of columns as in
lecture 2 we can partition the matrix as a column of rows. Define xi
, i = 1, . . . , n
as the i-th row of X represented as a K × 1 column vector.
(i) Give the accordingly partitioned X matrix.
(ii) Use the partitioned matrix to obtain an expression for X0X written as a
sum.
(iii) Do the same for X0y.
(iv) Combine the answers to get an expression for βˆ that only depends on
sums.
(v) Can you replace the sums by sample averages, i.e. divide them by n?
1
Problem 3 Let x be an n vector and A be an n×n matrix. A quadratic function
of x is f(x) = x
0Ax. Show that A can be assumed to be symmetric. Hint: If
A is not symmetric replace it by A
0+A
2
. Is this matrix symmetric? What is the
quadratic function with this matrix? What is the vector of partial derivatives
of f with respect to x? We organize the partial derivatives in a column vector.
Hint: write the quadratic function as a sum and determine the derivatives of
this sum.
Problem 4 Prove the partial regression formula in lecture 2. Hint: Write the
normal equations in partitioned form and solve first for βˆ
2 as a function of
βˆ
1. Substitute this solution and solve for βˆ
1. Show that the partial regression
formula still holds if we replace y
∗ by y, i.e. if we do not ’purge’ the dependent
variable.
SAMPLE SOLUTION