QUESTION 1

(four parts worth five marks each)

Suppose that the causal equation for a variable is

= 0  + 11, + 22, + 33, + ,   where ( |1, , 2, , 3, ) = 0, and 0  ≠ 0, 1  ≠ 0, 2  ≠ 0, 3  ≠ 0.

As 3, is unobserved, the corresponding regression equation is

( |1, , 2, ) = 0  + 11, + 22, .

(1. 1)

(1.2)

Consider  also  the  following  conditional  expectations,  in which 1, and 2, are potential instrumental variables (IVs):

(3, |1, , 2, ) = 0  + 11, + 22, ,                        with 1 ≠ 0, 2  = 0;

(1, |1, , 2, , 2, ) = 0  + 1 1, + 2 2, + 32, , with 1  ≠ 0, 2  ≠ 0, 3  ≠ 0; (3, |1, , 2, , 2, ) = 0  + 1 1, + 2 2, + 32, , with 1  = 0, 2 ≠ 0, 3  = 0; ( |1, , 2, , 2, ) = 0.

(a) (TRUE/FALSE) Given the information above, state whether each statement below is

true or false. Note: there is no need to explain your answers, but please write “true” or “false” in full in the answer booklet for each one.

1. 1  = 1

2. 2  = 2

3. ( |1, , 2, ) depends on 3,

4. 1, is a valid IV

5. 2, is a valid IV

(b) (TRUE/FALSE) Supposing that (1. 1) was estimated consistently by two-stage least

squares (2SLS) and that any IV validity testing was based on regressions compatible with such consistent estimation, state whether each statement below is true or false. Note: as in (a) above.

1. Only one first-stage regression would have been estimated

2. Only one regressor would have been included in the first-stage regression(s)

3. Both potential IVs would have been used

4. At least one overidentifying restriction would have been tested

5. Testing for IV relevance would have involved a joint significance test

Answer parts (c) and (d) below based on the following: suppose that for some sample of individuals  (indexed by = 1, … , ) , the variables  in  (1. 1)  are: = log(hourly wage ) ; 1, = years of education ; 2, = years of work experience ; 3, = ability/intelligence .

(c) Suggest, and briefly describe, a variable that could play the role of each of 1, and 2, and briefly explain your answers.

(d) If (1. 1) is estimated by OLS, for which of its causal coefficients would one obtain a biased estimate, and what sign would you expect this bias to have and why?

QUESTION 2

(two parts worth five marks each)

(a) Given cross-section data on , and , suppose that the objective is to obtain a

consistent estimate of 1  in the following model:

= 0  + 1 + ,                                                  (2. 1)

= 0  + 1 + 2 + ,                                             (2.2)

where and are i.i.d. error terms.

In the absence of any parameter restrictions, explain how you would meet the stated objective.

(b) Given time-series  data  on ,  suppose that the  objective is to  obtain  an unbiased

estimate of 1  in the following model:

= 0  + 1 −1 + ,                                                 (2.3)

where is an i.i.d. error term.

Would you expect OLS estimation of (2.3) to meet the objective, and why (or why not)?

QUESTION 3

(two parts worth five marks each)

Consider the following formulation of a panel data model:

, = + [ , + (1 − ) ] + , .

The  same  set  of individual  entities  (such  as people,  firms  or countries), indexed by = 1, . . . , ,  is  represented  in  the  sample  in  each  time period,  indexed  by =  1, . . . , .  In addition, , is an i.i.d. error term and denotes a time-invariant individual fixed effect that is correlated with both , and .

(a) Suppose that = 1.

How would you obtain a reliable estimate of , and why?

(b) Suppose that = 0.

How would you obtain a reliable estimate of , and why?