HW6
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School
University of Illinois, Urbana Champaign *
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Course
522
Subject
Industrial Engineering
Date
Dec 6, 2023
Type
Pages
7
Uploaded by CoachSteel12299
IE 522 HW06
1.
The file TSLA.csv on canvas contains TSLA stock prices in a certain period. Assume that the daily log
returns of TSLA are i.i.d.
•
1.1. (0.5 point) Compute the daily log returns of TSLA using adjusted close prices. Denote the true
distribution of TSLA’s log return by
F
.
F
is unknown. You’ll use the empirical cdf
ˆ
F
to estimate
F
.
Plot the empirical cdf
ˆ
F
.
1
•
1.2. (0.5 point) You want to estimate the true kurtosis
θ
of
F
, which is unknown. Give a point estimate
of
θ
by computing the sample kurtosis of the
n
= 1257
daily log returns.
•
1.3. (0.5 point) A point estimate is not enough. You want to construct a confidence interval for
θ
.
You need to learn more about the sample kurtosis
T
=
g
(
X
1
,
· · ·
, X
n
) =
1
n
QQQQQQQ
n
i
=1
(
X
i
−
¯
X
n
)
4
(
1
n
QQQQQQQ
n
i
=1
(
X
i
−
¯
X
n
)
2
)
2
,
n
= 1257
.
As a function of a random sample
{
X
1
,
· · ·
, X
n
}
from
F
,
T
is random. What you computed in 1.2 is
only a numeric realization of
T
. You want to find out the sampling distribution of
T
. This sampling
distribution is unknown, but can be estimated using resampling. Use resampling to simulate
B
= 5000
values of the sample kurtosis from the empirical cdf
ˆ
F
. Construct a histogram of the sample kurtoses
(use 30 bins).
2
•
1.4. (0.5 point) Estimate the standard error of the sample kurtosis
T
.
•
1.5. (0.5 point) Estimate the probability that the sample kurtosis is greater than or equal to 6.
•
1.6. (0.5 point) Estimate the 0.025 quantile and 0.975 quantile of
T
.
•
1.7. (0.5 point) Construct the 95% approximate confidence interval for the kurtosis
θ
of TSLA’s daily
log return.
3
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2.
The file TSLANVDA.csv on canvas contains prices of TSLA and NVDA over a certain period. Compute
daily log returns for both stocks using adjusted close prices. Denote TSLA returns by
{
X
1
,
· · ·
, X
n
}
with mean
µ
tsla
, NVDA returns by
{
Y
1
,
· · ·
, Y
n
}
with mean
µ
nvda
. Let
D
i
=
X
i
−
Y
i
,
1
≤
i
≤
n,
be the
differences of the log returns of the two stocks.
•
2.1. (0.5 point) Conduct a runs test. At significance level
α
= 5%
, is there strong enough evidence that
D
i
’s are not independent?
4
•
2.2. (1 point) Conduct normality tests using the Shapiro-Wilk test and Jarque-Bera test for TSLA’s
return. At significance level
α
= 5%
, is there strong enough evidence that the return of TSLA is not
normal? Construct a normal plot for TSLA’s return. Is TSLA’s return non-normal statistically? or
practically? or both? or neither?
5
•
2.3. (0.5 point) Suppose you want to test whether there is strong enough evidence that
µ
tsla
< µ
nvda
.
Write down the null hypothesis
H
0
and the alternative hypothesis
H
1
.
•
2.4. (0.5 point) Assume that
D
i
’s are i.i.d. and the sample size is large enough for the central limit
theorem to apply. Give the test statistic for question 2.3 and the corresponding sampling distribution
when
H
0
is true.
•
2.5. (1 point) For significance level
α
, determine the criteria for rejecting
H
0
in support of
H
1
in terms
of
¯
D
n
: i.e., for what values of
¯
D
n
, will
H
0
be rejected? For the data given, will
H
0
be rejected at
significance level
α
= 5%
?
•
2.6. (1 point) Construct the one-sided confidence interval for
µ
tsla
−
µ
nvda
corresponding to the testing
in 2.3. Is the confidence interval consistent with your conclusion in question 2.5?
6
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•
2.7. (1 point) Give the formula for the p-value of the testing in question 2.3. Compute the p-value.
Does the p-value confirm your conclusion in question 2.5?
•
2.8. (0.5 point) When does the type I error occur in the hypothesis testing in question 2.5? What’s the
probability of type I error there?
•
2.9. (0.5 point) If
H
0
is rejected, does that mean
H
1
is true? If
H
0
is not rejected, does that mean
H
0
is true?
7