## Description

1. Consider the following stochastic process:

Xt = Zt + 0.7Zt−1 + 0.2Zt−2 − 0.1Zt−3, t ∈ Z,

where {Zt}t∈Z ∼ W N(0, 1).

(a) Derive the power spectral density function of {Xt}t∈Z.

(b) Write down the normalized spectral density function of {Xt}t∈Z.

(c) Plot the normalized spectral density and comment on its behaviour.

2. (a) Show that if {Xt}t∈Z and {Yt}t∈Z are independent, stationary processes with power spectral density functions fX(ω) and fY (ω), respectively, then the process {Wt}t∈Z with

Wt = Xt + Yt

is also stationary with power spectral density function

fW (ω) = fX(ω) + fY (ω), ω ∈ (0, π).

(b) Suppose {Xt}t∈Z is an AR(1) process with Xt = −0.5Xt−1 +Zt

, and {Yt}t∈Z and {Zt}t∈Z

are two independent white noise processes with mean zero and common variance σ

2

.

Derive the power spectral density function of {Wt}t∈Z, where Wt = Xt + Yt

.

3. (This question must be completed in R Markdown; display all the R code used to perform

your data analysis.)

The Southern Oscillation Index (SOI) measures the normalised pressure difference between

Tahiti and Darwin. The data file soi.txt gives the annual SOI between 1866 and 2010.

(a) Read the data into R and coerce the data into a time series object. Plot the resulting

time series and its sample acf. Comment on what you observe. (Make sure to properly

label the axes and provide titles for the plots.)

(b) Plot the periodogram with a logarithmic vertical scale for the series, and smooth the

periodogram by setting the argument spans = sqrt(2 * length(x)), where x is your

time series object. Comment on what you observe and estimate the wavelength and

angular frequency for the dominating frequency.

(c) Build a function in R that generates the Fourier frequency ωp for a given time series and

given constant p ∈ {0, 1, …, N/2}. Document the inputs and outputs of this function, so

that another person would be able to understand how to use your function. What is the

output of your function for p = 10?

(d) To determine which Fourier frequencies are “significant”, suppose we were to fit the linear

model

Yt = a0 + ap cos(ωpt) + bp sin(ωpt) + t

, t = 1, . . . , N,

where we assume t ∼ N(0, σ2

) for all t and are independent. Let {Yt}t=1,…,N be the

time series and ωp =

2πp

N

.

Note that the model will be fitted N/2 times for each p =

1, 2, …, N/2. On fitting the above model for a given p by least squares, a test of the

“significance” of the contribution of frequency ωp is a test with null hypothesis

H0 : ap = bp = 0,

that uses the F-test statistic

Fp =

1

k−1

PN

t=1(ˆyt,p − y)

2

1

N−k

PN

t=1(yt − yˆt,p)

2

,

where k is the number of estimated coefficients in the linear model, N is the number of

observations, ˆyt,p = ˆa0 + ˆap cos(ωpt) + ˆbp sin(ωpt) and y =

1

N

PN

t=1 yt

. Asymptotically,

Fp ∼ F2,N−3.

Find all values of p that give significant Fourier frequencies at the 5% confidence level.

Hint: Use function lm() to fit the linear model. The output of this function can also be

used to extract the value of the F-statistic, or compute it directly.

(e) Give the estimated coefficients for the linear model that results from using all significant

frequencies found in part (d).

(f) Plot the data and the estimated model’s fitted values on the same plot in R. Remember

to properly label the axes, specify a legend and title for the plot.