Homework # 3 (100 points) STAT-4534 – Spring 2019 Due on Friday, March, 29th Applied Time Series
Answer all the questions completely. You are allowed to work with other students but you MUST provide your OWN answers.
Name: __________________ ID: __________________ ____________________________________________________________________________________________________________________________
Problem1:
1. A researcher has a time series that has accounted for trend and seasonality, and now calculates the theoretical autocovariance values for lags t = 0, 1, … , 9
Could this be a valid covariance structure in a stationary time series? Why or why not? 2. Say 𝛾 2 = 1.923 in the auto covariance structure above. Calculate the
autocorrelation function ρ(𝑡) for t = 0, 1, … , 9 Problem 2:
1. Express the following model using Backshift operator notation.
y! −0.5y!!! +0.9y!!! = ε! +0.2ε!!! −0.1ε!!! −0.5ε!!!
2. Simulate a time series realization using the following code: set.seed(88)
y=rnorm(100)
3. Does the autocorrelation function and partial autocorrelation function suggest that your series is a white noise process? Provide plots of both functions to support your conclusion. Also run the Box- Ljung test specifying your null and alternative hypotheses, the p-value and your decision.
t
γ(t)
t
γ(t)
0
2.638
5
-0.3298
1
2.194
6
0.0473
2
2.894
7
-0.0193
3
1.852
8
-0.1426
4
0.4740
9
0.1193
1
Homework # 3 (100 points) STAT-4534 – Spring 2019 Due on Friday, March, 29th Applied Time Series
Problem 3
Given the following ACF and PACF plots, define what sort of AR(p) or MA(q) model we are working with.
1.
Series y
Series y
0 5 10 15 20
Lag
5 10 15 20
Lag
2.
Series y
Series y
0 5 10 15 20
Lag
5 10 15 20
Lag
2
-0.2 0.0
0.2 0.4
-0.2 0.0
0.2 0.4
ACF
ACF
-0.2 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
Partial ACF
Partial ACF
Homework # 3 (100 points) Due on Friday, March, 29th
STAT-4534 – Spring 2019 Applied Time Series
3.
Series y
Series y
0 5 10 15 20
Lag
5 10 15 20
Lag
4.
Series y
Series y
0 5 10 15 20
Lag
5 10 15 20
Lag
3
-0.2 0.0
0.2 0.4
0.6 0.8
1.0
-0.5
0.0
0.5
1.0
-0.2
0.0
0.2
0.4
-0.4
-0.2
0.0 0.2
ACF
ACF
Partial ACF
Partial ACF
Homework # 3 (100 points) STAT-4534 – Spring 2019 Due on Friday, March, 29th Applied Time Series
Problem 4
In this problem, we’re going to work with an MA(1) process that we will simulate. Simulate the MA(1) process using the following code:
set.seed(161616) y=arima.sim(model=list(ma=c(0.89)),n=100)
So our process is y! = 𝜖! + 0.89 𝜖!!!
1. Is this process stationary? Why?
2. Is this process invertible? Why?
3. Provide the ACF and PACF plots of the process. Do they support the conclusion
that this is an MA(1) process?
4. Fit the model with no intercept included. What is the estimate of θ!?
5. One might argue from the PACF seen in Part 3 that this could be fit with an AR(3)
process. Which model would you choose between MA(1) and AR(3) based on BIC?
6. Given that 𝜖 !”” = 1.7479, calculate the predicted points a 5-step ahead forecast (if
possible) without using R.
Problem 5
In this problem, we’re going to work with an AR(2) process that we will simulate. Simulate the AR(2) process using the following code:
set.seed(8) y=arima.sim(model=list(ar=c(0.9,-0.7)),n=100)
Soourprocessisy! =0.9y!!! −0.7y!!! +𝜖!
1. Is this process stationary? Why?
2. Is this process invertible? Why?
3. What is the formula for the theoretical ACF of y! using the Yule-Walker equation?
4. Provide the ACF and PACF plots of the process. Do they support the conclusion
that this is an AR(2) process?
5. Fit the model with an intercept included. Write the estimated model.
6. Provide a plot of the time series y!, the fitted values 𝑦! and a 5-step ahead forecast.
4