Professor Òscar Jordà
Economics, U.C. Davis
ARE/ECN 240C Time Series Analysis
Winter 2004
PROBLEM SET 3 – DUE: FEBRUARY 12
Instructions
This problem set is divided into two parts: (1) Analytical Questions, and (2) Applied
Questions. Part 1, Analytical Questions, should be attempted by each student
individually. Part 2, Applied Questions, can be done in collaboration with another partner
or alone. I have often find collaboration in the computer room to be very useful.
However, if you rely on your partner to do your work you will ensure that you do not
learn adequate computer skills nor properly understand the material presented in class.
Please try to answer the questions rigorously by stating any implied assumptions and
ensuring all the steps to your conclusion have been properly verified.
Part I – Analytical Questions
Problem 1: Let
yt  t    t
 t ~ N (0,  2 )
and  known. Given the normality of t, the exact distribution of ̂OLS can be obtained.
Answer the following questions:
(a) What is the distribution of ̂OLS for a generic ?
(b) What is the distribution of ̂OLS for  = 0?
(c) What is the distribution of ̂OLS for  = ½? How can you rescale the problem to
obtain this distribution?
(d) What is the distribution of ̂OLS for  = -1? Hint: lim  t 2   2 / 6
T 
Problem 2:
Let the true D.G.P. be
yt  yt 1  ut
where ut is a MDS with variance  u2 and also sup t E | ut |2   for some  > 0. Also,
assume that y0 = 0. Consider the regression
yt  X t    t where X t  (1 t )'.
Denote ˆ the OLS estimator, show that
1
Professor Òscar Jordà
Economics, U.C. Davis
ARE/ECN 240C Time Series Analysis
Winter 2004
1
1
T 1 / 2 ( ˆ1  1 ) L  1 1 / 2  0W (  )d 
 
u
 1/ 2 ˆ

  1

1
/
2
1
/
3
T
(



)



W
(

)
d

2
2 

 0

Hint: the problem is easier to attack by considering the following

T 1DT ( ˆ   )  DT1  X t X t 'DT1
 T
1
where vt  s1 ut and
1
DT1  X t vt

t
T 1/ 2
DT 
 0
0 

T 3/ 2 
Problem 3:
Show that if
yt    yt 1  ut with
ut  1ut 1   2 ut 2   3ut 3   t
then yt can be expressed as
yt    yt 1  1yt 1  2 yt 2  3yt 3   t
Find the values for 1, 2, and 3,  and .
Problem 4:
The Sargan-Bhargawa (1983) statistic for a sample {y0, …,yT} is defined as
T
1
y2
2 t 0 t
SB  T
1 T
 yt2
T t 1
(which incidentally, is the reciprocal of the Durbin-Watson statistic). Show that if {yt} is
a driftless random walk, then
1
L
2
SB 

 W (r) dr
0
Hint:
2
Professor Òscar Jordà
Economics, U.C. Davis
ARE/ECN 240C Time Series Analysis
Winter 2004
yT2 p
 0
T2
t 0 yt2  t1 yt2  yT2 and
T
T
Finally, what effect, if any, will serial correlation in the residuals have on the distribution
of the SB statistic?
Problem 5:
Let {yt} be generated for t = 1, …, T by the process
iid
yt  t  S t where S t   j 1 v j and vt ~ N (0,  2 ); S 0  0
t
Consider estimating by least-squares the parameters of the model
yt    yt 1  vt
Define the scaling matrix
 T 1/ 2
CT  
 0
0 

T 3 / 2 
then show that
T
1
T 2 t 1 y t 1 
 ˆ    
CT 
  2 T
T
3
2
 ˆ  1   T t 1 yt 1 T t 1 y t 1 
 BT
1
1
 T 1 / 2 T vt 
t 1


 T 3 / 2 T y v 
t 1 t 1 t 

 T 1 / 2 T vt 
t 1


 T 3 / 2 T y v 
t 1 t 1 t 

(b) Given that
p
T 2 t 1 St 
0;
T
L
T 5 / 2 t 1 tSt 

W1
T
p
T 3 / 2 t 1 St 1vt 
0
T
L
T 3 t 1 St 

W2
T
2
where W1 and W2 are non-degenerate distributions, show that:
3
Professor Òscar Jordà
Economics, U.C. Davis
ARE/ECN 240C Time Series Analysis
Winter 2004

 1
p lim BT  
T 
 1 
2
1 

1 2 2
L
2   B and T 3 / 2 T y v 

N
(
0
,
  )

t

1
t
t 1
1 2
3
 
3 
(c) Since the asymptotic covariance between T 1/ 2 t 1 vt and T 3 / 2 t 1 yt 1vt is
T
T
1 2
 
2
show that:
 T 1/ 2 T vt 
0 2 
L
t 1



N
 ;  B  and therefore
T
 T 3 / 2  y v 
0

t 1 t 1 t 

 ˆ    L
0

CT 
 N  ;  2 B 1 
 
 ˆ  1 
0

Carefully state any theorems and assumptions you make.
Hints:
St  St 1  vt . Also
2
T
1
1
S  L
S 
L
T   t 1  
  W ( r )dr so T 1   t 1  

W 2 ( r )dr

0
0
t 1  T 
t 1  T 
1
T
lim T
( n 1)
T 
T
t
n
 ( n  1) 1
t 1
T
1
t 1
0
L
T 1 / 2  vt 

 dW ( r )  W (1) ~ N (0,1)
T
1
1
L
T 3 / 2  tvt 

rdW ( r ) ~ N (0, )

0
3
t 1
T
1
t 1
0
T
L
p
3 / 2
T 1  S t 1vt 

 W ( r )dW ( r ) so that T  St 1vt  0
t 1
T
T
T
1
t 1
t 1
t 1
0
L
T 3 / 2  yt 1vt  T 3 / 2   (t  1)vt  T 3 / 2  S t 1vt 

  rdW ( r )
Alternativ ely, by convention al methods,
 1 / 2 T

 1 / 2 T

 T  vt 
 T  vt 
0

L
L
t 1
t 1






N  ;  2 B 
T
T
 3 / 2

 3 / 2

0

tvt 
tvt 


T
 T
t 1
t 1




4
Professor Òscar Jordà
Economics, U.C. Davis
ARE/ECN 240C Time Series Analysis
Winter 2004
II Applied Questions
Problem 1: Empirical Properties of the t-test near unit roots
Consider the following D.G.P.
yt  yt 1  et
et ~ N (0,1)
y0  0
where  can take on values: (i)  = 1.0; (ii)  = 0.9; (iii)  = 0.5. We will investigate the
distributional properties of ̂OLS with a Monte Carlo experiment. Let M = 1,000 be the
number of replications in the Monte Carlo. Therefore, you need to generate 1,000
replications of series generated by the D.G.P. presented above, for each value of  with a
final sample size T = 50 where you disregard the first 100 observations of each series.
Thus, you will have 1,000 samples of size T = 50 for each possible value of . You will
need to estimate 1,000 OLS regressions for each value of  and calculate the following
magnitudes:
(1) The asymptotic bias:
1
M

M
r 1
( ˆOLS   )

M
(2) The Monte Carlo standard error of ̂OLS , that is, ˆ ˆ 
(3) The Monte Carlo average fit:
1
M

M
r 1
r 1
( ˆOLS  ˆOLS )
M 1
R2
(4) The size and the power of the usual t-test for the following nulls:
(a) H0:  = 1. That is, calculate the size by checking the rejection frequency of
this test when the true model is with  = 1, and check the power by calculating
the rejection frequency when  = 0.9, and 0.5.
(b) H0:  = 0.9
(c) H0:  = 0.5
Comment on the power and size distortions of the test for each null hypothesis.
(5) Plot a histogram for ̂OLS for each value of  = 1, 0.9, 0.5
Comments:

Turn in the GAUSS code you write to do this exercise. If working with other
partners, just turn in one copy with the names of the persons in the group.
5
Professor Òscar Jordà
Economics, U.C. Davis
ARE/ECN 240C Time Series Analysis
Winter 2004




In a table, report the results to questions (1)-(3) with appropriate comments (no
more than one paragraph).
Turn in a table for the results in (4) with comments.
Turn in one page with the plot of the 3 histograms described in (5).
No handwriting please.
Problem 2: Nonsense Regressions
Consider a Monte Carlo exercise with the following D.G.P.s
yt  yt 1   t
 t ~ N (0,1)
y0  0
xt  xt 1  ut
ut ~ N (0,1)
x0  0
with E(t, us) = 0 for any t and s. Let T = 100 (after disregarding the first 100
observations) and consider three possible values of : (i) 1, (ii) 0.5, (iii) 0. You will need
to do 1,000 replications for each value of  and then estimate the following OLS
regression:
yt   0  1 xt  vt
from which you should calculate the following:
(1) Asymptotic biases for 0 and 1:
1
M

M
r 1
( ˆi   i ) , i = 0, 1. Notice that i = 0 given
the DGP.
(2) Empirical sizes for the t-statistic for each of the following null hypotheses:


H0: 0 = 0
H0: 1 = 0
(3) Display the histogram for ˆ0 and ˆ1 for each value of .
Comments:
Follow similar protocols as in the previous problem.
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