Homework 4, Due Monday, March 18th

Computational Problems:
(1) Consider the series rec of the recruitment (number of new fish) measured monthly at a pacific ocean fishery over
the period 1950–1987, and salttemp of soil temperatures measured at a series of locations in an agricultural
field.
(a) Plot the series, as well as their periodograms. Identify frequencies at which the series appear to be oscillat-
ing strongly.
(b) Compute in each case a non-parameteric spectral density estimator for each series. Re-identify frequen-
cies at which the series appear to be oscillating strongly. Investigate to what extend these esimators are
influenced by the choice of the bandwidth and/or kernel used.
(c) Compute in each case a parametric spectral density estimator using a pure autoregressive model with order
selected using AIC. Re-identify frequencies at which the series appear to be oscillating strongly.
(2) Generate a times series of the form
xt = U1 sin(2π0.003t) + U2 sin(2π0.005t) + 2U3 cos(2π0.01t) + wt ,
iid
for t = 1, ..., 1000, where Ui ∼ N (0, 1), and wt is a Gaussian white noise with variance 0.52 . Then
(a) Plot the time series with the sum of the first three terms (i.e. xt − wt ) superimposed.
(b) Plot the periodogram of xt .
(c) Use the DFT/inverse DFT and thresholding based on the periodogram to approximately filter out the white
noise component. Plot the resulting filtered series, and superimpose the sum of the first three terms once
again. Comment on how effective your filtering method was for filtering out wt , and try to explain any
deficiencies.

Theoretical Problems:

(1) In applications, we will often observe series containing a signal that has been delayed by some unknown time
D, i.e.,
xt = st + Ast−D + nt
where st and nt are stationary and independent with zero means and spectral densities fs (ω) and fn (ω), respec-
tively. The delayed signal is multiplied by some unknown constant A.
(a) Prove
fx (ω) = 1 + A2 + 2A cos(2πωD) fs (ω) + fn (ω)
 

(b) How could the periodicity expected in the spectrum derived in (a) be used to estimate the delay D ? (Hint:
Consider the case where fn (ω) = 0; i.e., there is no noise.)
(2) Suppose xt and yt are stationary zero-mean time series with xt independent of ys for all s and t. Consider the
product series
zt = xt yt
Prove the spectral density for zt can be written as
Z 1/2
fz (ω) = fx (ω − ν)fy (ν)dν
−1/2

Challenge Problems:

2
1) Suppose xt = wt − 2wt−1 , where wt is a mean zero white noise sequence with variance σw . Given  > 0, find an
integer k and constants a0 , ..., ak , with a0 = 1, such that if fy is the spectral density of the process

k
X
yt = aj xt−j ,
j=0
then

sup |fy (ω) − var(yt )| < .

−1/2≤ω≤1/2
1
2

2) Suppose wt is a weak white noise. Show that if φ(z) and θ(z) are polynomials with no common zeros and φ(z) = 0
for some z ∈ C such that |z| = 1, then the ARMA equations
φ(B)xt = θ(B)wt ,
have no weakly stationary solution.