Assignment #4 – ECON 323 – Fall 2020
Professor Mikal Skuterud
Date assigned: October 6, 2020 at 7:00
Date due: October 19, 2020 at 23:30
1. Consider the time-series process {𝑦𝑦𝑡𝑡} defined by the data generating process 𝑦𝑦𝑡𝑡 = 𝑧𝑧 +
𝑒𝑒𝑡𝑡 for all 𝑡𝑡 = 1, … , where {𝑒𝑒𝑡𝑡} is an i.i.d. normally distributed sequence with mean zero
and variance 𝜎𝜎𝑒𝑒
2 = 4. The random variable 𝑧𝑧 does not change over time (so it is
uncorrelated with 𝑒𝑒𝑡𝑡) and comes from a discrete uniform distribution in the closed
interval [0;100].
a. Produce 20 realizations of this time-series process with 𝑛𝑛 = 100. Graph all 20
realizations in a graph similar to those produced in the section 2(2) slides.
Interpret your graph. Does it appear that the process is stationary? Does it
appear to be weakly dependent? Justify your answer.
b. Analytically (i.e., using a pen and paper instead of R) find the expected value and
variance of 𝑦𝑦𝑡𝑡. Do your answers depend on 𝑡𝑡? Interpret.
c. Find 𝐶𝐶𝐶𝐶𝐶𝐶(𝑦𝑦𝑡𝑡, 𝑦𝑦𝑡𝑡+ℎ) for any 𝑡𝑡 and ℎ. Is 𝑦𝑦𝑡𝑡 covariance stationary? Justify your
answer?
d. Find 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶(𝑦𝑦𝑡𝑡, 𝑦𝑦𝑡𝑡+ℎ). Does 𝑦𝑦𝑡𝑡 appear to be weakly dependent? Justify your
answer.
2. Consider the time-series process 𝑦𝑦𝑡𝑡 defined by the data generating process 𝑦𝑦𝑡𝑡 = cos(𝑡𝑡𝑡𝑡)
for all 𝑡𝑡 = 1, … , where 𝑤𝑤 is a random variable with a continuous uniform distribution in the
open interval (0; 2𝜋𝜋) and cos(⋅) is the cosine function.
a. Produce 10 realizations of this time-series process with 𝑛𝑛 = 50. Graph all 10
realizations in a graph. Interpret your graph. Does it appear that the process is
covariance stationary? Does it appear to be weakly dependent? Justify your
answer.
b. Using R or your pen and paper or any other resource, find the expected value,
variance, and covariances of 𝑦𝑦𝑡𝑡 to determine whether the process is covariance
stationary. Justify your answer.
c. Is the process strongly (sometimes called “strictly”) stationary? Why or why not?
Justify your answer.