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Economics 618B: Time Series Analysis
Department of Economics
State University of New York at Binghamton
Problem Set #1
1. Generate
n
= 500
random numbers from both the uniform
1
b
−
a
(
U
[0
,
1]
, uniform be-
tween zero and one) and exponential
λ
exp (
−
λx
)
(set
λ
= 2
and let
x
∼
U
[0
,
1]
)
distributions.
Plot the histograms of each of the variables.
What are the true and
estimated means and variances?
2. Generate a random variable with
n
= 500
from the mixed normal distribution with
P
[
x
∼
N
(
−
3
,
1)] =
P
[
x
∼
N
(3
,
1)] = 0
.
5
. Plot the histogram. Note that this should
be a bimodal distribution. Do not add the two variables together. This will lead to a
unimodal density with mean near zero.
3. Generate a random variable with
n
= 500
from the Student-t distribution with 5
degrees of freedom,
t
5
. Plot the histogram.
4. Generate a random variable with
n
= 500
from the
χ
2
distribution with 1 degree of
freedom. Plot the histogram.
5. Generate a random variable with
n
= 500
from the Normal distribution with mean 1
and variance 0.5. Plot the histogram.
6. This problem will show you how the central limit theorem works. Using a programming
software show how the CLT works for the sample mean with the
fi
ve separate distrib-
utions. Use
n
= 10
,
50
,
and
100
with
m
= 100
,
200
, and
500
Monte Carlo replications.
Plot histograms of the sample means for each distribution and each sample size.
(a) Standard normal,
N
(0
,
1)
(b) Uniform,
U
[0
,
1]
(c) Mixed Normal,
P
[
x
∼
N
(
−
3
,
1)] =
P
[
x
∼
N
(3
,
1)] = 0
.
5
(d) Exponential,
λ
exp (
−
λx
)
(set
λ
= 2
and let
x
∼
U
[0
,
1]
)
(e) Student-
t, t
5
7. This problem is to conduct a Monte Carlo experiment to examine the
fi
nite sample per-
formance of a test in size (
P
(
reject H
0
:
H
0
is true
)
) and power (
P
(
reject H
0
:
H
0
is false
)
).
(a) Size: Generate a random sample
x
1
, x
2
, . . . , x
n
of size
n
= 50
, with
μ
= 0
and
σ
2
= 1
from two distributions, (i) normal and (ii) uniform. Pretend you do not
know the mean and variance. Then test the null hypothesis that
μ
= 0
against
the alternative that it is not equal to zero. Use Monte Carlo experiments with
500
replications to evaluate the size of the test statistic t.
Use the asymptotic
critical value at the 5% level (1.96). Repeat with
n
= 100
and
200
. What do you
fi
nd?
(b) Power: Repeat part (a) with
μ
= 0
.
1
,
0
.
3
,
−
0
.
1
, and
−
0
.
3
. Plot the power function
(include
μ
= 0
in the plot).
1
Economics 618B: Time Series Analysis
Department of Economics
State University of New York at Binghamton
Problem Set #2
1. The
fi
le entitled SIM_2.XLS contains simulated data sets. The
fi
rst series, denoted
Y1, contains 100 values of a simulated AR(1) process. Use this series to perform the
following tasks.
(a) Plot the sequence against time. Does the series appear to be stationary?
(b) Plot the ACF and PACF.
(c) Estimate the AR(1), AR(2), ARMA(1,1), ARMA(1,(1,4)), ARMA(2,1) and ARMA(2,(0,4))
models.
(d) Estimate the series as both an AR(2) and ARMA(1,1) process without an inter-
cept.
(e) Use MSE, AIC and SBC to choose the best model from parts (c) and (d).
(f) Obtain the one-step-ahead forecast and one-step-ahead forecast error from model.
Which model performs the best? Is this surprising?
(g) The second series in SIM_2.XLS, denoted Y2, contains 100 values of a simulated
ARMA(1,1) process. Repeat steps (a-f) with the series Y2.
(h) The third series in SIM_2.XLS, denoted Y3, contains 100 values of a simulated
AR(2) process. Repeat steps (a-f) with the series Y3.
2. The
fi
le QUARTERLY.XLS contains demand deposits reported by commercial banks
(
DDNSA
). The series is not seasonally adjusted. The series is quarterly averages over
the period 1960:Q1 to 2002:Q1. Be sure to brie
fl
y explain the relevance of each step.
(a) Plot the
DDNSA
sequence against time. Does the series appear to be stationary?
(b) Plot the ACF and PACF of
DDNSA
.
(c) Create the growth rate series
log (
DDNSA
t
/DDNSA
t
−
1
)
and plot this series
against time. Does the series appear to be stationary?
(d) Plot the ACF and PACF of
log (
DDNSA
t
/DDNSA
t
−
1
)
.
(e) Seasonally di
ff
erence demand deposits using
log (
DDNSA
t
/DDNSA
t
−
4
)
. Does
this series appear to be stationary?
(f) Plot the ACF and PACF of
log (
DDNSA
t
/DDNSA
t
−
4
)
2
Economics 618B: Time Series Analysis
Department of Economics
State University of New York at Binghamton
Problem Set #3
1. The
fi
le QUARTERLY.XLS contains data on the U.S. Producer Price Index (
PPI
)
and M1 (
M
1
NSA
). Here the goal is to model both simultaneously using a VAR. The
series are quarterly averages over the period 1960:Q1 to 2002:Q1.
(a) Construct the rate of growth of the money supply and the in
fl
ation rate as mea-
sured by the PPI as
m
t
=
log (
M
1
NSA
t
/M
1
NSA
t
−
1
)
π
t
=
log (
PPI
t
/PPI
t
−
1
)
(b) Plot each series. Is there evidence of seasonality?
(c) Estimate the VAR with seasonal dummies
D
1
, D
2
, and
D
3
where
D
i
= 1
in the
i
-th quarter of each year and zero otherwise
m
t
=
a
1
+
δ
11
D
1
+
δ
21
D
2
+
δ
31
D
3
+
a
11
m
t
−
1
+
a
21
π
t
−
1
+
e
1
t
π
t
=
a
2
+
δ
12
D
1
+
δ
22
D
2
+
δ
32
D
3
+
a
12
m
t
−
1
+
a
22
π
t
−
1
+
e
2
t
(d) Estimated the VAR model in (b) using 12 lags of each variable and save the
residuals.
(e) Explain why the estimation in (d) cannot begin earlier than 1963:Q2.
(f) Estimated the VAR model in (b) using 8 lags of each variable and save the resid-
uals.
(g) Construct a likelihood ratio test for the null hypothesis of 8 lags versus 12 lags.
(h) Repeat the procedure to see if it is possible to further restrict the system to 4 lags
of each variable.
(i) Determine whether
m
t
Granger causes
π
t
.
(j) Determine whether
π
t
Granger causes
m
t
.
(k) Show that each variable Granger causes itself.
(l) Some argue that it would be better to estimate the VAR using log-levels of the
variables instead of the logarithmic
fi
rst di
ff
erences. Estimate an eight-lag model
with seasonal using the log-levels of the variables.
(m) Do the causality tests using undi
ff
erenced data di
ff
er from those using di
ff
erenced
data?
3
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