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 estim
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 anal
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(
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 cf
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"
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 variable
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
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
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)
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)| <
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, a
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) Sh
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
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 den
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 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 h
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 trans