Homework 1
Due Thursday, February 17 at 11:25 AM ET
1 Homework 1 Part 1
Problem 1.1 (4+3+3 points). Consider the time series {Xt} given by Xt = β0 + β1t + εt,
where εt is normally distributed white noise with variance σε2.
1. Compute the autocovariance function and the mean function and determine whether
{Xt} is stationary.
2. Let us say, β0 = 1,β1 = 0.1 and σε2 = 1. In the following figure, you see the sample
autocovariance function based on n = 1000 observations.
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ACF of {Xt}
0 20 40 60 80 100 Lag
ACF (cov) 600 650 700 750 800
If you would like to reproduce this figure, the following code was used.
beta 0=1 beta 1=0.1
X t=beta 0+beta 1∗(1:T)+rnorm(T)
acf(X t,lag.max=100,type=”covariance”)
Explain what assumptions are behind this autocovariance estimator and if they are
fulfilled in this case.
3. We filter the processes {Xt} as follows
Yt = Xt − 0.5Xt−1 − 0.5Xt+1. Determine whether {Yt} is stationary.
If you like to play with code, this is a way of estimating the autocovariance function with the parameters above in R again.
Y t=filter(X t,filter=c(−0.5,1,−0.5),sides=2L) acf(Y t,na.action = na.pass)
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Problem 1.2 (3+4 points).
1. Suppose we have a time series {Xt} and observations X1, . . . , X1000. In the follow-
ing figure, you see the estimated autocovariance function based on the observations X1,…,X1000 as well as based on the first half X1,…,X500 and the second half X501,…,X1000. What can you say about stationarity based on these figure?
ACF(Cov) of {Xt} based on X1,…,X1000
0 10 20 30 40 50 Lag
ACF(Cov) of {Xt} based on X1,…,X500
0 10 20 30 40 50 Lag
ACF(Cov) of {Xt} based on X501,…,X1000
0 10 20 30 40 50 Lag
ACF (cov) ACF (cov) ACF (cov) 150 160 170 180 190 200 210 150 160 170 180 190 200 210 720 740 760 780 800 820 840
2. In the file hm1.csv you find n = 1000 realizations of a time series, say {Yt}. 3
In the following figure, you see a plot of the data.
Yt 0 5 10 15 20
600 800 1000
Argue, if that data is stationary. If not, how could it be transformed (e.g., detrending, power transformation) to stationarity?
You do not need to perform the transformation but in case you would like to play with the data, you find in Canvas a csv file. If you placed the csv-file in your working directory (you can get your working directory in R by getwd()), you can load it as follows.
Y t=read.csv(file=”hw1.csv”)$x
Problem 1.3 (3 points). In a few sentences, describe the kind of data you are interested in
analyzing using the tools from this course.
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Problem 1.4 (5 points). Determine whether the following time series are weakly or strictly stationary. Here, (et )t∈Z and (εt )t∈Z are white noises; et ∼ N (0, 1) and εt ∼ Exp(1) − 1, i.e. εt + 1 ∼ Exp(1). Furthermore, (et)t∈Z and (εt)t∈Z are mutually independent.
1. Xt =et1{1,3,5,…}(t)+εt1{2,4,6,…}(t),
2. Xt = e1 cos(ωt) + e2 sin(ωt), ω ∈ (−π, π], 3. Xt = et sin(at), a ∈ (−π, π],
4. X = Z , Z random variable with density f (x) = 1 1 (Cauchy distributed).
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