2022 Practice Final Exam
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2022 Practice Final Exam
1. (10 points) What are the model assumptions of the ARIMA(p,d,q) model?
2. (30 points) Explain the following terms briely
1. The seasonal ARMA model.
2. The ACF and the sample ACF.
3. Partial autocorrelation function.
4. The minimum mean error forecast.
1. Using the following command:
> library(TSA)
> data(robot),
we can retrieve a data ile named robot in R. The data ile contains a time series obtained from an industrial robot. The robot was put through a sequence of maneuvers, and the distance from a desired ending point was recorded in inches. This was repeated 324 times to form the time series.
(a) Estimate the parameters of an AR(1) model for these data.
(b) Estimate the parameters of an IMA(1,1) model for these data.
(c) Compare the results from parts (a) and (b) in terms of AIC.
2. Consider the daily log return of INTC. Summary statistics of the returns and some sum- mary outputs are attached.
> da=read.table("m-intc7303.txt",header=T)
> intc=log(da[,2]+1)
> x=ts(intc,frequency=12,start=c(1973,1))
> Box.test(intc,lag=10,type=’Ljung’)
Box-Ljung test
data: intc
X-squared = 15.8261, df = 10, p-value = 0.1047
> Box.test(intc^2,lag=10,type=’Ljung’)
Box-Ljung test
X-squared = 59.7216, df = 10, p-value = 4.091e-09
> basicStats(intc)
intc
nobs 372.000000
NAs 0.000000
Minimum -0.595422
Maximum 0.485508
1 . Quartile -0.044997
3 . Quartile 0.103822
Mean 0.017990
Median 0.021897
Sum 6.692218
SE Mean 0.006934
LCL Mean 0.004355
UCL Mean 0.031625
Variance 0.017886
Stdev 0.133738
Skewness -0.598995
Kurtosis 2.889686
. Let μ be the mean of the log return. Test to see if μ is signiicantly diferent from 0 at 5% level.
. Let ρi be the lag i autocorrelation of the return. We are testing if all the ρi = 0. What’s the test statistics, draw your conclusion.
3. Refer to the weekly interest rate example discussed in the lecture and the outputs (regres- sion with time series errors).
. Write down the itted linear regression. What is the R2 of the linear regression? Is the itted model adequate? Why?
. Let c1 and c3 be the diferenced series. Consider the linear regression between these two variables. What is the itted model? What is the residual standard deviation of the model?
. The residuals of the prior linear regression show certain serial correlations. A linear regression model with time series errors is employed. Write down the itted model. Based on the available output, is this model adequate? Why?
4. A stationary time series Xt follows (1 - 0.3B2)Xt = 0.2 + (1 + 0.2B4)Zt. What is the mean of Xt?
5. Rewrite the following time series models using the backshift operator B and diference operator Δ and in terms of the series xt and Zt.
1. xt = φ 1xt-1 + Zt + θ1Zt-1 .
2. y t = φ 1yt-1 + Zt + θ1Zt-1 and y t = xt - xt-1 .
Rewrite the following without the operators
(1 - φ 1B)Δ4xt = (1 + θ1B)Zt.
6. Make 1 step ahead forecast for the following models. Assume φ(ˆ)1 = 0.7, θ(ˆ)1 = 0.6,xn-2 = 30,
xn-1 = 20, xn = 25,ˆ(z)n-2 = 2, ˆ(z)n-1 = 1, ˆ(z)n = 3
1. (Xt - μ) = Zt + θ1Zt-1
2. (1 - φ 1B)(Xt - μ) = (1 - θ 1B)Zt.
7. Suppose Xt is an stationary AR(1) process: Xt = 0:8Xt-1 + Zt. Compute the ACF and ACVF for Xt.
8. Also review the midterm exam. and the midterm review.
2023-12-25
Time series modeling