EXAM 2
KU Math 611, Spring 2021 Zsolt Talata April 14, 2021
You have to skip one problem. The first line of the submission must state which problem you
skip. Each of the remaining four problems is worth 10 points. The skipped problem may be submitted
for an extra 6 points. No communication with other students or persons. All solutions must be submitted
in at most two PDF files (theoretical and computational) containing letter-size portrait-orientation pages,
and uploaded to BlackBoard immediately at submission.
Show your work in full detail. No credit for answers unsupported by work. Show the derivations step
by step and justify the steps when you use other results, for example, “by the chain rule”, “because Z[j]’s
are independent”, etc. Solutions need to be written on a paper and scanned into a PDF file.
For the computational problems, obtain the solution by importing only the NumPy, Matplotlib and
Statsmodels libraries. The solution needs to include all relevant Python input and output.
P. 1 Let f(0) = 1, f(1) = f(−1) = 34 and f(h) = 0 for |h| > 1. Prove that there does not exist any time
series whose acf is f .
P. 2 Let {Xt} be a q-variate time series defined by Xt = ΦXt−1 +Zt + ΘZt−3, where Zt ∼WN(0,Σ), and
Φ,Θ are q × q parameter matrices. (Notice that {Xt} is not ARMA(1, 1) because of Zt−3 instead of
Zt−1 in the definition.)
(a) Show that X is a linear process by identifying the filter coefficient matrices.
(b) Using the theorem on the acvf of linear processes, calculate the acvf of X at lag 0 in terms of
Σ,Φ,Θ.
P. 3 For the attached data particles.txt,
(a) Remove the constant trend and check if the residual process is a white noise. Why?
(b) Remove the linear trend and check if the residual process is a white noise. Why?
P. 4 Let {Xt} be a univariate time series defined by Xt = Zt+θZt−2, where Zt ∼WN(0, σ2), and σ2, θ are
scalar parameters. (Notice that {Xt} is not MA(1) because of Zt−2 instead of Zt−1 in the definition.)
(a) Using the Bartlett formula, for the process X find W [i, i] in terms of ρX for i = 3, 4.
(b) For the attached data gap.txt remove the linear trend and assume that the residual process is
X. Then calculate the 95% confidence intervals for the acf at lags 3 and 4. Finally, based on
the two confidence intervals check if the assumption X is reasonable. Why?
P. 5 Predict the next value of the attached stationary time series sample hydra.txt based on the last two
observations.
(a) Calculate the sample acvf from the entire sample.
(b) Using the sample acvf at the necessary lags, create the matrix Γ and the vector g.
(c) Calculate the vector a of optimal coefficients, and use them to calculate the 31st value of the
process.
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