Final Exam 01/06/2010
Name: Number:
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SECTION A: Multiple Choice and Short Questions (30 points)
(45 minutes)
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IMPORTANT: o
In the exam, consider that a growth rate of any variable y t corresponding first difference of its logarithm. That is: (y t
- y t-k
)/ y
is approximated by the t-k
≈ log y t
– log y t-k
. o In the Multiple Choice questions, mark the correct statements writing ‘TRUE’ at the end of the statement. Any other way of answering will be invalid. Look at the following example:
Today, 1June 2010, it is the final exam of Econometrics II. Mark the correct statements: a) The exam will be the first Tuesday of June. TRUE b) All the students will take an oral exam of Econometrics II. c) The exam is in Leganés. d) The exam lasts two hours and a half. TRUE e) After the first two hours and a half, the exam will continue in the dining hall.
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A.1. Unit 1 (10 points)
(points of multiple choice x value of the short question)
Multiple Choice (5 points)
Given a stationary time series, y t
, one knows that its sample autocorrelation coefficients
(sample ACF coefficients) are not significantly different from zero, except for the first lag.
Mark the correct statements: a) The marginal mean of y t
is zero. b) The conditional mean of y is zero. t c) There is no temporal dependence between y t
and its past values. d) y can be generated by a MA(1) process. t e) y can be generated by an AR(1) process. t

Short Question (2 if your answer is correct; 1 if it isn’t)
Justify your answer d) in the previous multiple choice question.
Option d) is true.
If y follows a MA(1) process, only its first sample ACF coefficient is t significantly different from zero, and the others, {
ρ
} , are not significantly different j j
>
1 from zero.
A.2. Unit 2 (10 points)
(points of multiple choice x value of the short question)
Multiple Choice (5 points)
The logarithmic transformation of the quarterly index of industrial production index (IP t
), that is shown in the following graph,
LOG(IP)
4.9
4.8
4.7
4.6
4.5
4.4
86 87 88 89 90 91 92 93 94 95 96 97 is modelled as follows:
Δ
4 log IP t
=
0 .
007
+
0 .
832
Δ
4 log IP t
−
1 where
ε t
is a white noise process. Mark the correct statements:
+ ε t
(1)
98 a) A property of the variable log IP t
is to have systematic growth with an annual growth mean rate that is constant. b) The annual growth mean rate of IP t
is 2% c) The stationary transformation of IP t
is Δ
4 log IP t
. d) The variable log IP t
has seasonality. e) The variable
Δ log IP t
is stationary.
Short Question (2 if your answer is correct; 1 if it isn’t)
Justify your answer e) in the previous multiple choice question.
Option e) is false. Note that Δ
4
= Δ
( 1
+
L
+
L
2 +
L
3
) , then in model (1) we have:
( 1
+
Δ
L log
+
IP t
L
2
=
+
L
3
−
(
Δ
)
Δ log log IP t
IP t
−
1
+
=
Δ
0 .
007 log
+
IP t
−
2
0 .
832
Δ
+ Δ log
4 log
IP t
−
3
IP t
−
1
)
+
+ ε t
0 .
007
, i.e.,
+
0 .
832
Δ
4 log IP t
−
1
+ ε t
, where the first term on the right is a polinomial that has a seasonal unit root. Then, Δ log IP t
is non-stationary (seasonal non-stationarity). The stationary transformation is given by
Δ
4 log IP t
.

A.3. Unit 3 (10 points)
(points of multiple choice x value of the short question)
Multiple Choice (5 points)
Given two variables, x and t y , which are stationary, one is interested in analyzing the dynamic t relationship between them. We test for Granger causality between x and t y , including the t temporal dependence up to lag 2. The results are given by the next output:
According to these results, mark the correct statements. a) The null hypothesis associated to the p-value 0.0042 is that neither x t-1 provides statistically significant information about y t
.
nor x t-2 b) The null hypothesis associated to the p-value 0.3514 is that neither y t-1 provides statistically significant information about x t
.
nor y t-2 c) The Granger causality tests should be performed on the first differences of the variables. d) In order to analyze the dynamic relation between x t
and y t
, a bivariate VAR model for x t
and y t
where the VAR coefficients matrices are diagonal can be used. e) In order to analyze the dynamic relation between x t
and y t
, the bivariate VAR model for x t
and y t
should incorporate a cointegration relationship between x t
and y t
.
Short Question (2 if your answer is correct; 1 if it isn’t)
Justify your answer e) in the previous multiple choice question.
Option e) is false because the variables x t
e y t
are stationary, and for analyzing the dynamic relationship between the two variables, one should not take into account a cointegration relationship between them.