STA457H1 S-LEC5101 Time Series Analysis Quiz 1
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Department of Statistical Sciences
Time Series Analysis
Time Series Analysis
STA457H1 S-LEC5101
STA457H1/STA2202H S-LEC0101 & STA457H1S LEC2001
Quiz 1
Q1. Consider the observed time series xt = xt − 1 + wt , where wt is i.i.d N (0, 1), t =
1, 2, . . ., and x0 = 0.
(a) Find the maean and the autocorrelation function ρx (t, s), where 1 < t < s for {xt }. (12 pts)
Solutions:
x1 = w1
x2 = w1 + w2
x3 = w1 + w2 + w3
.
.
.
xt = w1 + w2 + . . . + wt (2 pts)
µt = E[w1 + w2 + . . . + wt]
= E[w1] + E[w2] + . . . + E[wt] = 0
So that µt = 0 for all t. (2 pts)
Var(xt ) = Var[w1 + w2 + . . . + wt]
= Var[w1] + Var[w2] + . . . + Var[wt] = tσw(2) (2 pts)
Suppose that 1 < t < s.
Cov(xt , xs ) = Cov[w1 + w2 + . . . + wt , w1 + w2 + . . . + ws]
= Cov(wi , wj )(2 pts)
However, these covariance are zero unless i = j, in which case they equal
Var(wi ) = σw(2)
Therefore, Cov(xt , xs ) = tσw(2) for 1 < t < s. (2 pts)
The autocorrelation funtion for the random walk is now easily obtained as
ρx (t, s) = = = ← (2 pts)
(b) Find ρx (1, 2), ρx (8, 9), ρx (1, 25), ρx (1, 40), and interpret your results. (6 pts)
Solutions:
ρx (1, 2) = ← = 0.707 (1 pt)
ρx (8, 9) = ← = 0.943 (1 pt)
ρx (1, 25) = ← = 0.200(1 pt)
ρx (1, 40) = ← = 0.158 (1 pt)
The values of x at neighboring time points are more and more strongly and positively correlated as time goes by. On the other hand, the values of x at distant time points are less and less correlated. (2 pt)
(c) Is xt stationary? Why or why not? (2 pts)
Solutions:
It is not a stationary process, since the Cov(xt , xs ) does depend on time t. (2 pts)
You must not copy solutions from the internet or written descriptions from anyone or anywhere else without reporting the source within your work. This includes copying from solutions provided to previous semesters of this course.
There will be a penalty for those who submit up to 30 minutes late and quizzes/tests/final submitted more than 30 minutes late will not be accepted.
Penalty |
# of minutes late |
5% 10% 20% 30% 40% 50% 100% |
1-5 6-10 11-15 16-20 21-25 > 26-30 > 30 |
2022-02-10