Chapter 7
Consistency and and asymptotic
normality of estimators
In the previous chapter we considered estimators of several dierent parameters. The hope
is that as the sample size increases the estimator should get closer to the parameter of
interest. Wh
50 100
0
sunsportts
1700
1750
1800
1850
1900
1950
2000
Time
0.4 0.0 0.4 0.8
ACF
Series sunspot
0
10
20
30
40
Lag
Figure 1: Plot of the sunspot data
Solutions 6 (STAT673)
(5.1) In this exercise we analyze the Sunspot data found on the course website. In th
Solutions 3 (STAT673)
(1.8)
(i) Show that the function c(u) = exp(a|u|) where a > 0 is a positive semi-denite
function.
The Fourier transform is
exp(a|u|) exp(iu)du
f () =
0
=
exp(a|u|) exp(iu)du
exp(a|u|) exp(iu)du +
=
a2
0
2a
,
+ 2
which is clearly posi
Solutions 4 (STAT673)
(3.1) Recall the AR(2) models considered in Exercise (2.4). Now we want to derive their
ACF functions.
(i) (a) Obtain the ACF corresponding to
2
7
Xt = Xt1 Xt2 + t ,
3
3
where cfw_t are iid random variables with mean zero and varian
Solutions 8 (STAT673)
(8.1) (a) Simulate an AR(2) process and run the above code using the sample size
2
q
q
q
q
1
q
q
q
q
2 1 0
sqrt(n) * Re(dftcov1[1:30])
(i) n = 64 (however use k<-kernel("daniell",3)
A plot if given in Figure 1. In this simulation we
Solutions 2 (STAT673)
(1.5) State, with explanation, which of the following time series is second order stationary,
which are strictly stationary and which are both.
(i) cfw_t are iid random variables with mean zero and variance one.
cfw_t is strictly s
Solutions 1 (STAT673)
(1.1)
(i) Import the yearly temperature data (le global mean temp.txt) into R and t
the a linear trend to the data (use the R command lsfit).
(ii) Suppose the errors in model are correlated. Under the correlated assumption,
explain w
Solutions 7 (STAT673)
(6.1) Under the assumption that cfw_Xt are iid random variables show that cn (1) is asymp
totically normal.
n1
k=1
Hint: Let m = n/(b + 1) and partition the sum
n1
(b+1)+b
b
Xt Xt+1 =
t=1
Xt Xt+1 as follows
Xt Xt+1 + Xb+1 Xb+2 +
t=1
1.
X
810
(a) With a sample proportion of = n = 2500 = 0.324, the estimate of standard error is given
(0.324)(0.676)
= 0.00936.
by s.e. =
2500
Hence
95% margin of error (z.025)( s.e.) = (1.96)(0.00936) = 0.0183456, so that
95% Confidence Interval for = (0.
Time Series Analysis (Final Exam) 2 hours (Statistics Majors)
Short Questions
(1) Suppose that cfw_Xt is a second order stationary time series with autocovariance function
cfw_c(r), with r |rc(r)| < . Directly verify that the DFT of the covariances (you
Chapter 4
Estimation for Linear models
Prerequisites
Linear regression.
The Gaussian likelihood.
Some idea of what a cumulant is.
Objectives
To derive the sample autocovariance of a time series, and show that this is a positive
denite sequence.
To sh
STAT 673 Homework 1
(1) Consider the MA(2) process
1
1
Xt = t t 1 t 2 ,
2
4
where cfw_t are iid random variables with mean zero and variance one. Derive the
autocovariance function of cfw_Xt .
(2) Show for the AR(2) model Xt = 1 Xt1 + 2 Xt2 + t to have a
Chapter 1
Introduction
A time series is a series of observations xt , each observed at the time t. Typically the observations
can be over an entire interval, randomly sampled on an interval or at xed time points. Dierent
types of time sampling require die
Chapter 2
Linear time series
Prerequisites
Familarity with linear models.
Solve polynomial equations.
Be familiar with complex numbers.
Understand under what conditions the sequences have well dened limits, with particular application to the innite su
Chapter 3
Prediction
Prerequisites
The best linear predictor.
Some idea of what a basis of a vector space is.
Objectives
Understand that prediction using a long past can be dicult because a large matrix
has to be inverted, thus alternative, recursive m
Chapter 6
Spectral Analysis
Prerequisites
The Gaussian likelihood.
The approximation of a Toeplitz by a Circulant (covered in previous chapters).
Objectives
The DFTs are close to uncorrelated but have a frequency dependent variance (under
stationarity)
Chapter 5
Spectral Representations
Prerequisites
Knowledge of complex numbers.
Have some idea of what the covariance of a complex random variable (we do dene it
below).
Some idea of a Fourier transform.
Objectives
Know the denition of the spectral den