ATT00001 ATT00002 ATT00003 ATT00004 ATT00005 CARD.DTA Card_1995_geo_var_schooling.pdf Exam2_2014.pdf ADVANCED ECONOMETRICS Midterm 2 (Take Home) Due: Dec.25, 2014 Answer all questions. You should not discuss solutions with your peers but me. Good luck! Prof. Dr. H. Taştan , First Name:................................................... Last Name:................................................ No:................................................... 1 (20) In class we have shown that when the number of instrumental variables is larger than the number of endogenous variables the generalized IV estimator (or 2SLS) can be written as β̂IV = ( X>PzX )−1 X>Pzy where Pz = Z(Z >Z)−1Z>. In this formulation X is n×k and Z is n× l, l > k. (a) Show that β̂IV can be obtained as a solution to the following minimization problem min β Q(β) = (y−Xβ)>Pz (y−Xβ) (b) Show that when k = l the generalized IV estimator reduces to the simple IV estimator: β̂IV = ( Z>X )−1 Z>y 2 (20) Consider the following simple consumption model as a function of permanent income ci = β1 + β2y ∗ i + ui, ui ∼ iid (0,σ 2 u) where ci is the logarithm of consumption by household i, and y ∗ i is the permanent income of household i which is not observed. Instead we observe current income, yi yi = y ∗ i + vi, vi ∼ iid (0,σ 2 v) where vi is assumed to be uncorrelated with y ∗ i and ηi. We run the following regression ci = β1 + β2yi + ηi (a) Show that yi is negatively correlated with ηi. You can assume β2 > 0. (b) Evaluate the plim of the OLS estimator β̂2: β̂2 = ∑n i=1(yi − ȳ)ci∑n i=1(yi − ȳ)2 In particular, show that this plim is less than the true β2. 1 3 (30) Use card.dta to answer the following questions. Also read Card (1993), “Using Geographic Variation in College Proximity to Estimate the Return to Schooling”, NBER Working Paper. (a) Run the OLS regression of log(wage) on educ, exper, exper2, black, smsa, south, smsa66, reg662 to reg669. Comment on the coefficient estimate of educ. (b) Estimate the same model by 2SLS using nearc4 as an instrument for educ. Compare the OLS and IV coefficient estimates on educ. (Note that we partly did this in class). Carry out the Hausman test. (c) Use both nearc2 and nearc4 as instruments for educ. Run the reduced form model for educ. Compare 2SLS estimates to the results obtained in the previous section. Carry out the OID test. (d) Discuss the plausibility of Card (1993)’s econometric methodology and empirical findings. Do you agree with his conclusions? 4 (30) A continuous time model for short term interest rates may be written as a stochastic differential equation dr = (α + βr)dt + σrγ� √ dt where r is the short term interest rate, � is standard normal random variable, dt is a short time interval and α,β,γ,σ are parameters. Discrete time approximation is given as rt+1 − rt = α + βrt + �t+1 with E(�t+1) = 0, E(� 2 t+1) = σ 2 ...

1. ATT00001
ATT00002
ATT00003
ATT00004
ATT00005
CARD.DTA
Card_1995_geo_var_schooling.pdf

3. Exam2_2014.pdf
ADVANCED ECONOMETRICS
Midterm 2 (Take Home)
Due: Dec.25, 2014
Answer all questions. You should not discuss solutions with
your peers but me. Good luck!
Prof. Dr. H. Taştan ,
First Name:...................................................
Last Name:................................................
No:...................................................

4. 1 (20) In class we have shown that when the number of
instrumental variables is larger than the number
of endogenous variables the generalized IV estimator (or 2SLS)
can be written as
β̂IV =
(
X>PzX
)−1
X>Pzy
where Pz = Z(Z
>Z)−1Z>. In this formulation X is n×k and Z is n× l, l > k.
(a) Show that β̂IV can be obtained as a solution to the following
minimization problem
min
β
Q(β) = (y−Xβ)>Pz (y−Xβ)
(b) Show that when k = l the generalized IV estimator reduces
to the simple IV estimator:
β̂IV =
(
Z>X
)−1
Z>y
2 (20) Consider the following simple consumption model as a
function of permanent income

5. ci = β1 + β2y
∗
i + ui, ui ∼ iid (0,σ
2
u)
where ci is the logarithm of consumption by household i, and y
∗
i is the permanent income of household
i which is not observed. Instead we observe current income, yi
yi = y
∗
i + vi, vi ∼ iid (0,σ
2
v)
where vi is assumed to be uncorrelated with y
∗
i and ηi. We run the following regression
ci = β1 + β2yi + ηi
(a) Show that yi is negatively correlated with ηi. You can
assume β2 > 0.
(b) Evaluate the plim of the OLS estimator β̂2:
β̂2 =
∑n
i=1(yi − ȳ)ci∑n
i=1(yi − ȳ)2

6. In particular, show that this plim is less than the true β2.
1
3 (30) Use card.dta to answer the following questions. Also read
Card (1993), “Using Geographic
Variation in College Proximity to Estimate the Return to
Schooling”, NBER Working Paper.
(a) Run the OLS regression of log(wage) on educ, exper,
exper2, black, smsa, south, smsa66, reg662
to reg669. Comment on the coefficient estimate of educ.
(b) Estimate the same model by 2SLS using nearc4 as an
instrument for educ. Compare the OLS
and IV coefficient estimates on educ. (Note that we partly did
this in class). Carry out the
Hausman test.
(c) Use both nearc2 and nearc4 as instruments for educ. Run the
reduced form model for educ.
Compare 2SLS estimates to the results obtained in the previous
section. Carry out the OID
test.
(d) Discuss the plausibility of Card (1993)’s econometric
methodology and empirical findings. Do
you agree with his conclusions?
4 (30) A continuous time model for short term interest rates
may be written as a stochastic differential
equation

7. dr = (α + βr)dt + σrγ�
√
dt
where r is the short term interest rate, � is standard normal
random variable, dt is a short time
interval and α,β,γ,σ are parameters. Discrete time approximation
is given as
rt+1 − rt = α + βrt + �t+1
with
E(�t+1) = 0, E(�
2
t+1) = σ
2r
2γ
t
In this question we will consider the restricted version of the
model in which γ = 1/2 (for details, see
Chan et al 1992). Assume that γ is known so that the parameter
vector is given by θ = (α, β, σ)>.
The vector of population moment conditions is given as
E(f(θ,rt)) = 0 where
f(θ,rt) =
�t+1
�t+1rt

8. �2t+1 −σ2rt
(�2t+1 −σ2rt)rt
with �t+1 = rt+1 − rt −α −βrt. STATA data file intrate.dta
contains 294 weekly observations on
1-month average interest rates on bank deposits. You need to
use tsset date to use the data set.
Also note that, in the data set, r denotes rt+1 and rlag denotes
rt.
(a) Using this data, and the population moment conditions
described above, estimate the parameter
vector using GMM. You may set initial values as θ0 = (0, 0, 1)
>. Choose initial weighting matrix
as identity, and also use HAC Newey-West procedure for the
long run covariance matrix with
bandwidth set to 5.
(b) Compute the Hansen J-test. Interpret the result.
2
intrate.dta
Short term int rate Chan et al GMM est.pdf

10. ATT00001
ATT00002
ATT00003
ATT00004
ATT00005
CARD.DTA
Card_1995_geo_var_schooling.pdf

12. Exam2_2014.pdf
ADVANCED ECONOMETRICS
Midterm 2 (Take Home)
Due: Dec.25, 2014
Answer all questions. You should not discuss solutions with
your peers but me. Good luck!
Prof. Dr. H. Taştan ,
First Name:...................................................
Last Name:................................................
No:...................................................
1 (20) In class we have shown that when the number of
instrumental variables is larger than the number

13. of endogenous variables the generalized IV estimator (or 2SLS)
can be written as
β̂IV =
(
X>PzX
)−1
X>Pzy
where Pz = Z(Z
>Z)−1Z>. In this formulation X is n×k and Z is n× l, l > k.
(a) Show that β̂IV can be obtained as a solution to the following
minimization problem
min
β
Q(β) = (y−Xβ)>Pz (y−Xβ)
(b) Show that when k = l the generalized IV estimator reduces
to the simple IV estimator:
β̂IV =
(
Z>X
)−1
Z>y
2 (20) Consider the following simple consumption model as a
function of permanent income
ci = β1 + β2y
∗
i + ui, ui ∼ iid (0,σ

14. 2
u)
where ci is the logarithm of consumption by household i, and y
∗
i is the permanent income of household
i which is not observed. Instead we observe current income, yi
yi = y
∗
i + vi, vi ∼ iid (0,σ
2
v)
where vi is assumed to be uncorrelated with y
∗
i and ηi. We run the following regression
ci = β1 + β2yi + ηi
(a) Show that yi is negatively correlated with ηi. You can
assume β2 > 0.
(b) Evaluate the plim of the OLS estimator β̂2:
β̂2 =
∑n
i=1(yi − ȳ)ci∑n
i=1(yi − ȳ)2
In particular, show that this plim is less than the true β2.

15. 1
3 (30) Use card.dta to answer the following questions. Also read
Card (1993), “Using Geographic
Variation in College Proximity to Estimate the Return to
Schooling”, NBER Working Paper.
(a) Run the OLS regression of log(wage) on educ, exper,
exper2, black, smsa, south, smsa66, reg662
to reg669. Comment on the coefficient estimate of educ.
(b) Estimate the same model by 2SLS using nearc4 as an
instrument for educ. Compare the OLS
and IV coefficient estimates on educ. (Note that we partly did
this in class). Carry out the
Hausman test.
(c) Use both nearc2 and nearc4 as instruments for educ. Run the
reduced form model for educ.
Compare 2SLS estimates to the results obtained in the previous
section. Carry out the OID
test.
(d) Discuss the plausibility of Card (1993)’s econometric
methodology and empirical findings. Do
you agree with his conclusions?
4 (30) A continuous time model for short term interest rates
may be written as a stochastic differential
equation
dr = (α + βr)dt + σrγ�
√
dt

16. where r is the short term interest rate, � is standard normal
random variable, dt is a short time
interval and α,β,γ,σ are parameters. Discrete time approximation
is given as
rt+1 − rt = α + βrt + �t+1
with
E(�t+1) = 0, E(�
2
t+1) = σ
2r
2γ
t
In this question we will consider the restricted version of the
model in which γ = 1/2 (for details, see
Chan et al 1992). Assume that γ is known so that the parameter
vector is given by θ = (α, β, σ)>.
The vector of population moment conditions is given as
E(f(θ,rt)) = 0 where
f(θ,rt) =
�t+1
�t+1rt
�2t+1 −σ2rt
(�2t+1 −σ2rt)rt

17. with �t+1 = rt+1 − rt −α −βrt. STATA data file intrate.dta
contains 294 weekly observations on
1-month average interest rates on bank deposits. You need to
use tsset date to use the data set.
Also note that, in the data set, r denotes rt+1 and rlag denotes
rt.
(a) Using this data, and the population moment conditions
described above, estimate the parameter
vector using GMM. You may set initial values as θ0 = (0, 0, 1)
>. Choose initial weighting matrix
as identity, and also use HAC Newey-West procedure for the
long run covariance matrix with
bandwidth set to 5.
(b) Compute the Hansen J-test. Interpret the result.
2
intrate.dta
Short term int rate Chan et al GMM est.pdf