PAPER CODE: ECO3 10 / 20 – 21 /S 2 /Final Exam
security – 这道题目是利用security进行的编程代写任务, 是有一定代表意义的security等代写方向
University
PAPER CODE: ECO3 10 / 20 – 21 /S 2 /Final Exam
Problem A For questions 1 to 2 [ 20 MARKS]
Based on the information below:
# R output:
Call:
arima(x = y, order = c(0, 1, 1), seasonal = list(order = c(0, 1, 1), period = 4))
Coefficients:
ma1 sma
- 3872 – 0.3 765
s.e. 0.0 645 0. 1804
sigma^2 estimated as 0. 0564 : log likelihood = 81.93, aic = - 189. 23
- Please tell what the seasonal coefficient is and what the standard
deviation of!
"
is? [1 0 MARKS]
- Please write down the fitted model. [1 0 MARKS]
Problem B For questions 3 to 4 [ 20 MARKS]
Consider a stationary four-dimensional time series &
"
. Let
)
####### =
####### +,-(&
"
####### ,&
" 0 )
) be the lag- 1 auto-covariance matrix of &
"
. More specifically,
we have
####### &
"
####### =
#######
#######
#######
#######
####### &
),"
####### &
5 ,"
####### &
6 ,"
####### &
7 ,"
#######
#######
#######
#######
#######
)
####### =
#######
#######
#######
#######
#######
))
####### ( 1 )
)
####### ( 1 )
)
####### ( 1 )
)
####### ( 1 )
#######
5)
####### ( 1 )
55
####### ( 1 )
56
####### ( 1 )
57
####### ( 1 )
#######
6)
####### (
####### 1
####### )
#######
65
####### (
####### 1
####### )
#######
66
####### (
####### 1
####### )
#######
67
####### (
####### 1
####### )
#######
7)
####### (
####### 1
####### )
#######
75
####### (
####### 1
####### )
#######
76
####### (
####### 1
####### )
#######
77
####### (
####### 1
####### )
#######
#######
#######
#######
- Write down the meanings of
))
####### (
####### 1
####### )
####### ,
77
####### (
####### 1
####### )
####### . [10 MARKS]
- Write down the meanings of
)
( 1 ) and
75
####### ( 1 ). [10 MARKS]
Problem C For questions 5 to 6 [ 20 MARKS]
Based on the R output of the tests and the figure listed below:
# R output:
University
PAPER CODE: ECO3 10 / 20 – 21 /S 2 /Final Exam
> t.test(x)
One Sample t-test
data: x
t = 3.5818, df = 431, p-value = 0.
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
0.009990214 0.
sample estimates:
mean of x
0.
>Plot(x)
> adfTest(x,lags=1 5 ,type=c("##"))
Title:
Augmented Dickey-Fuller Test
Test Results:
PARAMETER:
Lag Order: 1 5
STATISTIC:
Dickey-Fuller: -1. 9456
P VALUE:
- 7421
Please answer the questions:
- Should we keep the drift for the random walk? Should we keep the time
trend for the random walk? [10 MARKS]
- Is this time series stationary? Why? And what should you do to get a
stationary time series? [10 MARKS]
University
PAPER CODE: ECO3 10 / 20 – 21 /S 2 /Final Exam
Problem D For questions 7 to 8 [ 20 MARKS]
Suppose that the daily log return of a security follows the model:
####### <
"
####### = 0. 02 + 0. 3 <
" 05
####### +!
"
where
####### {
####### !
"
####### }
is a Gaussian white noise series with mean 0 and variance 0.0 5.
Assume that <
)CC
= 0. 022 , and <
DD
= 0. 015. Please answer the questions:
- Compute the 1-step ahead forecast of the return series at the forecast
origin G= 100. What are the associated standard deviation of the
forecast error?
- Compute the 2-step ahead forecast of the return series at the forecast
origin G= 100. What are the associated standard deviation of the
forecast error?
Problem E For questions 9 to 10 [ 20 MARKS]
Based on the R output below:
# R output:
Title:
GARCH Modelling
Call:
garchFit(formula = ~garch(3, 0), data = abcd, trace = F)
Mean and Variance Equation:
data ~ garch(3, 0)
[data = abcd]
Conditional Distribution:
norm
Coefficient(s):
mu omega alpha1 alpha2 alpha
0.011852 0.010588 0.237149 0.072747 0.
Std. Errors:
based on Hessian
Error Analysis:
Estimate Std. Error t value Pr(>|t|)
University
PAPER CODE: ECO3 10 / 20 – 21 /S 2 /Final Exam
mu 0.011852 0.005640 2.102 0.0356 *
omega 0.010588 0.001284 8.249 2.22e-16 ***
alpha1 0.237149 0.114734 2.067 0.0387 *
alpha2 0.072747 0.046990 1.548 0.
alpha3 0.053080 0.046526 1.141 0.
---
Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
Log Likelihood:
291.8891 normalized: 0.
Standardised Residuals Tests:
Statistic p-Value
Jarque-Bera Test R Chi^2 192.312 0
Shapiro-Wilk Test R W 0.9638102 7.82629e- 09
Ljung-Box Test R Q(10) 9.731163 0.
Ljung-Box Test R Q(15) 18.39307 0.
Ljung-Box Test R Q(20) 19.28568 0.
Ljung-Box Test R^2 Q(10) 27 .2857 0. 0182258
Ljung-Box Test R^2 Q(15) 26.92023 0.
Ljung-Box Test R^2 Q(20) 27.70902 0.
LM Arch Test R TR^2 24.88263 0.
Information Criterion Statistics:
AIC BIC SIC HQIC
- 1.328190 – 1.281102 – 1.328454 – 1.
Please answer the questions:
- Does the standardized residuals follows a normal distribution? Why? Is
this GARCH model adequate? Why? [ 10 MARKS]
- How to improve these model? [ 10 MARKS]
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