We have estimated the following equation:

(1)

TOULOUSE SCHOOL OF ECONOMICS MASTER 1 - FALL 2016

INTERMEDIATE ECONOMETRICS

TD1 Least-Squares Regression

Exercise 1.1

We have estimated the following equation:

educ d = 10.36 − .094 sibs + .131 meduc + .210 f educ

n = 722, R

²

= 0.214,

where educ is years of schooling, sibs is number of siblings, meduc is mother's years of schooling, and f educ is father's years of schooling.

1. Does sibs have the expected eect? Explain. Holding meduc and f educ xed, by how much does sibs have to increase to reduce expected years of education by one year? (A noninteger answer is acceptable here.) 2. Discuss the interpretation of the coecient on meduc .

3. Suppose that Mister A has no siblings, and his mother and father each have 12 years of education. Miss B has no siblings, and her mother and father each have 16 years of education. What is the expected dierence in years of education between Miss B and Mister A?

Exercise 1.2

A dummy variable takes on only the values 0 and 1. It is used for categorical data, such as an individual's gender. Let d

1

and d

2

be vectors of 1 and 0 , with the i -th element of d

1

equaling 1 and that of d

2

equaling 0 if the person is a man, and the reverse if the person is a woman. Consider tting the following three equations by OLS

y = α

0

+ d

1

α

1

+ d

2

α

2

+ u (1)

y = d

1

β

1

+ d

2

β

2

+ u (2)

y = γ

0

+ d

1

γ

1

+ u (3)

1

(2)

1. Can all three equations be estimated by OLS? Explain if not.

2. Compare regressions (2) and (3). Is one more general than the other?

Explain the relationship between the parameters in these equations.

Exercise 1.3

Consider the two regressions

y = β

₁

x

₁

+ β

₂

x

₂

+ u y = α

₁

z

₁

+ α

₂

z

₂

+ v

where z

₁

= x

₁

− 2x

₂

and z

₂

= 2x

₁

− 3x

₂

. Let X = [x

₁

, x

₂

] and Z dened accordingly.

1. Show that the columns of Z can be expressed as linear combinations of the columns of X , that is Z = XA . Find the elements of A .

2. Show that the matrix A

⁻¹

is

−3 −2 2 1

.

3. Show that the two regressions give the same tted value and residuals.

4. How are the OLS estimators β ˆ

i

related to the OLS estimates α ˆ

i

, for i = 1, 2?

Exercise 1.4

Of y

_i^∗

, y

_i

, x

_i

, only the pairs (y

_i

, x

_i

) are observed. In this case, we say that y

_i^∗

is a latent variable. Suppose

y

^∗_i

= x

⁰_i

β + u

_i

u

_i

|x

_i

∼ N(0, σ

²_u

) y

i

= y

_i^∗

+ v

i

where v

i

is a measurement error such that v

i

|y

_i^∗

, x

i

∼ N (0, σ

_v²

) and which is independent of u

i

.

1. Compute E[y|x] . Is β the coecient of the linear projection of y

_i

on x

_i

? 2. Let β b denote the OLS coecient from the regression of y

_i

on x

_i