STSCI5550 Exam1

Time Series 时间序列代写 / CS代考

Name Net ID
Applied Time Series Analysis, Spring 2023 Exam 1
February 23, 2023
• During the exam, you are allowed to refer to the course materials and the text books listed in the syllabus. You are not allowed to use other resources.
• The exam ends at 12:40 PM and has to be submitted in Canvas by 12:40 PM. Do the problems that you find easiest first.
• To receive credit for a problem, show your work.
• The point value of each sub-question is given in parentheses, e.g. (8 points).
• This exam is worth 100 points. The maximum you can score is 100.
• Good luck!
Academic Integrity
I, , certify that this work is entirely my own. I will not look at any of my peers’ answers or communicate in any way with my peers. I will not use any resource other than the course materials and the text books listed in the syllabus. I will behave honorably in all ways and in accordance with Cornell’s Code of Academic Integrity.
Signature of Student

Match the time series models (a)-(d) to their names. (8 points)
xt =0.5+xt−1 +εt εt ∼wn0,σε2,
xt = 13 (εt−1 + εt + εt+1) εt ∼wn0,σε2
xt = −0.9xt−1 + εt εt ∼wn0,σε2,
εt ∼wn0,σε2
Autoregressive model of order one, i.e. AR(1)
Random walk with drift
White noise
Filtered white noise

Match the plotted time series to the time series models. (8 points)
0 50 100 150 200 250 300 0 50 100 150 200 250 300 Index Time
0 50 100 150 200 250 300 0 50 100 150 200 250 300 Time Index
Xt -4 -2 0 2 4 6
Xt -3 -2 -1 0 1 2
Xt -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
AR(1) with −1 < φ < 0 White noise Filtered white noise Random walk with drift Match the sample ACFs to the time series models. (8 points) 0 5 10 15 20 0 5 10 15 20 Lag Lag 0 5 10 15 20 0 5 10 15 20 Lag Lag ACF 0.0 0.2 0.4 ACF 0.0 0.5 1.0 ACF ACF 0.0 0.2 0.4 0.6 0.8 1.0 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 AR(1) with −1 < φ < 0 White noise Filtered white noise Random walk with drift Problem 2 In the following figure, you see the Consumer Price Index (CPI) for all urban con- sumers in the US. Let es denote the CPI in the following as the time series {Yt}. • Argue if {Yt} is stationary or not. (4 points) • If {Yt} is not stationary, explain how you would transform this time series {Yt} into a time series {Xt} for which it is more reasonable to argue that it is stationary. (8 points) Consumer Price Index for All Urban Consumers 1950-01-01/2022-12-01 Problem 3 Let {Xt} be a stationary time series with mean μX and autocovariance function γX. Let {Yt} be another stationary time series with the same mean, i.e., E(Yt) = μX for all t and the same autocovariance function, i.e., cov(Ys, Yt) = γX (|s − t|) for all s, t. Additionally, we have cov(Xt, Ys) = 0 for all t, s. Now, define two other time series {Zt}, {Wt} where Zt= √2 , Xt − Yt • Compute the mean and autocovariance function of {Zt} and {Wt} and argue if the processes are stationary time series or not. (12 points) • Compute the cross-covariance function of {Zt} and {Wt}, that is cov(Zs, Wt). Are {Zt} and {Wt} jointly stationary? (10 points) Problem 4 We have two time series {Xt} and {Yt} given by Xt = aXt−1 + εt where {εt} and {et} are two white noises which are mutually uncorrelated to each other, i.e., cov(εs, et) = 0 for all s, t. Based on the following figures, answer (with explanation) the following questions: • Whatisthesignofa,i.e.,a>0ora<0? (4points) • What is l and what is the sign of β? (8 points) Yt = βXt−l + et, 浙大学霸代写 加微信 cstutorcs 0 50 100 150 200 Index 0 50 100 150 200 Index Yt Xt 0 2 4 -6 -4 -2 0 2 4 6 Figure 1: Plot of {Xt} and {Yt} Programming Help, Add QQ: 749389476 0 5 10 15 20 Lag 0 5 10 15 20 Lag ACF ACF 0.5 1.0 -0.5 0.0 0.5 1.0 Figure 2: Sample Autocorrelations of {Xt} and {Yt} -4 -2 0 2 4 Y(t) -4 -2 0 2 4 Y(t-1) -4 -2 0 2 4 Y(t-2) -4 -2 0 2 4 Y(t-3) -4 -2 0 2 4 Y(t-4) -4 -2 0 2 4 Y(t-5) -4 -2 0 2 4 Y(t-6) -4 -2 0 2 4 Y(t-7) -4 -2 0 2 4 Y(t-8) X(t) -6 -4 -2 0 2 4 6 X(t) -6 -4 -2 0 2 4 6 X(t) -6 -4 -2 0 2 4 6 X(t) -6 -4 -2 0 2 4 6 X(t) -6 -4 -2 0 2 4 6 X(t) -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 X(t) -6 -4 -2 0 2 4 6 X(t) -6 -4 -2 0 2 4 6 Figure 3: Lag Plot of {Yt} and {Xt} (scatter plot and cross-correlation between Xt and Yt−h, h = 0,...,8) Code Help
-6 -4 -2 0 2 4 6 X(t)
-6 -4 -2 0 2 4 6 X(t-1)
-6 -4 -2 0 2 4 6 X(t-2)
-6 -4 -2 0 2 4 6 X(t-3)
-6 -4 -2 0 2 4 6 X(t-4)
-6 -4 -2 0 2 4 6 X(t-5)
-6 -4 -2 0 2 4 6 X(t-6)
-6 -4 -2 0 2 4 6 X(t-7)
-6 -4 -2 0 2 4 6 X(t-8)
Y(t) -4 -2 0 2 4
Y(t) -4 -2 0 2 4
Y(t) -4 -2 0 2 4
Y(t) -4 -2 0 2 4
Y(t) -4 -2 0 2 4
Y(t) -4 -2 0 2 4
-4 -2 0 2 4
Y(t) -4 -2 0 2 4
Y(t) -4 -2 0 2 4
Figure 4: Lag Plot of {Xt} and {Yt} (scatter plot and cross-correlation between Yt and Xt−h, h = 0,…,8)

Problem 5 Let {Xt} be a stationary time series with autocovariance function γX and mean μX. Let {Zt} be the time series obtain by taking first difference of {Xt}, that is
Zt = Xt − Xt−1.
Let ρZ(h) = cov(Zt+h,Zt) be the auto-correlation function of {Zt}.
1. Express ρZ(h) in terms of γX and μX. (14 points)
2. If Xt = εt, where εt ∼ iidN (0, 1), what value do we have for ρZ (1).(4 points)

3. IfX =aX +ε and|a|<1,wehavethatγ (h)= ah . t t−1 t X 1−a2 LetXt =bXt−1+εt andb∈(0,1). DeriveinthiscaseρZ(h)and ρZ(h). (8points). ρX (h) Let us say, we reduce the dependency by taking first difference if |ρZ(h)| < 1 for all h. ρX (h) Determine for which values of b we reduce the dependency in this setting.(4 points) SCRATCH PAGE Graders will not review this page! SCRATCH PAGE Graders will not review this page! SCRATCH PAGE Graders will not review this page!