1
Lecture 1: Introduction to main concepts
1.1
Introduction to time series analysis
The focus of this course is on the analysis, modeling and prediction of
data observed in a sequential order. Standard inferential methods no
longer work in this setup with
Stat 929: Time Series
Lecture 4
University of Waterloo
May 9
In this lecture, we aim to prove the following Theorem:
Theorem
If xt is a white noise sequence with variance 2 and Ext4 < , then
D
T (h) N(0, 4 ).
Review of Math-Stats prerequisites:
1. OP nota
Stat 929: Time Series
Lecture 19
July 11
Topics for today:
1. Diagnostic tests for GARCH modeling
2. Asymmetric GARCH models: Exponential GARCH (EGARCH)
3. Multivariate Time Series/ Or something else.
Last time we discussed:
1. Portmanteau Test for nonlin
Stat 929: Time Series
Lecture 7
University of Waterloo
May 25
Definition
Given a white noise sequence wt , we say that xt is an
Autoregressive Moving Average process of orders p and q
(ARMA(p, q) if
(B)xt = (B)wt ,
where
(x) = 1 1 x p x p ,
and
(x) = 1 +
Stat 929: Time Series
Lecture 16
June 29
Topics for today:
1. A note on Bandwidth Selection in Smoothed Periodogram
estimation
2. ARMA Spectral Density Estimation
3. Cross Spectra and Coherency
There is a great deal of theory (over 100 papers!) discussing
Stat 929: Time Series
Lecture 11
June 8
Topics for today:
1. ARIMA Modeling
Definition
A time series xt is said to be integrated of order d if d xt is
stationary, and j xt is nonstationary for 1 j < d. This is
abbreviated as
xt I (d)
The terminology arise
Stat 929: Time Series
Lecture 8
May 30
Topics for today:
1. Computing the ACF of an ARMA(p,q) model
2. Partial Autocorrelation Function
3. Forecasting
3.1 Projection Theorem
3.2 Best Linear Prediction
Last time we showed that for an ARMA(p, q) process:
(B
Stat 929: Time Series
Greg Rice
Lecture 3
University of Waterloo
May 6
Last time we talked about:
1. Detrending/Di erencing Time Series
2. The space L2
3. White noise, Moving Average, and Autoregressive 1 models
Goals for today:
1. Solution to the AR(1) p
Stat 929: Time Series
Lecture 15
June 27
Topics for today:
1. Confidence intervals for the Periodogram
2. Smoothed periodogram estimates of the spectral density
Let
1/2 X
n
2
xt cos(2j t),
(j ) =
n
t=1
and
1/2 X
n
2
(j ) =
xt sin(2j t),
n
t=1
Then
I (j
STAT 790
SAMPLE MIDTERM EXAM-II
11/29/2015
(Total time : 110min)
1. Let cfw_Yt be a zero mean SOS with ACvFs Y () and spectral density f (). Let wjn
2jn /n, n 1 be a sequence of Fourier frequencies, with jn [1, n 1/2] for all
P
n 1. Suppose that wjn w (
Time Series 1
1
Challenge Problems
Please complete and hand in solutions to at least 4 of the below problems.
1. Let (, F, P ) denote a probability space on which a random variable Z is defined. Let M (Z) be defined by
Y M (Z) Y = (Z) for some Borel funct
Stat 929: Time Series
Lecture 14
June 22
Topics for today:
1. More on the Spectral Density
For a time series x1 , ., xn , Schuster (1898) proposed the (scaled)
periodogram
n
P(j/n) =
2X
xt sin(2j t)
n
t=1
!2
+
!2
n
2X
xt cos(2j t)
n
t=1
where j = j/n, j =
Homework 5, Due Wednesday, July 13th
Problems from Shumway and Stoffer, Chapter 4: 4.7, 4.17, 4.23a
Additional Problems:
(1) Show that if xt is a GARCH(p,q) process, then x2t follows an ARMA process.
(2) In the case of a GARCH(1,1) process, give an explic
Stat 929: Time Series
Lecture 5
University of Waterloo
May 16
Conclusion of the proof of:
Theorem
If xt is a white noise sequence with variance 2 and Ext4 < , then
D
T (h) N(0, 4 ).
It therefore follows by Slutskys Theorem that
D
T (h) N(0, 1).
This resul
Stat 929: Time Series
Greg Rice
Lecture 2
University of Waterloo
May 4
Announcements
Goals for today:
1. Discuss basic time series models, detrending, and differencing
2. Mathematical foundations: L2 theory for time series
Suppose xt , t = 1, ., T is a ti
Stat 929: Time Series
Lecture 18
July 6
Topics for today:
1. General conditions for the existence of a stationary solution to
the GARCH(p,q)
2. Distribution of the ACF of a GARCH sequence.
3. Estimating the GARCH parameters
Definition
Let wt be a unit var
Stat 929: Time Series
Lecture 13
June 15
Topics for today:
1. Spectral Density/ Spectral Distribution
Last time we showed that if
xt = U1 cos(2t) + U2 sin(2t),
where U1 and U2 are independent, mean zero normal random
variables with variance 2 , then xt is
6
Lecture 6: ARIMA Models
We have already discussed the importance of the class of ARMA models for representing stationary series. A
generalization of this class, which incorporates a wide range of non-stationary series, is provided by the ARIMA
processes
2
Lecture 2: Linear Filters
2.1
Introduction
Suppose there are two processes X and Y related by
Yt =
cr Xtr ,
c2 <
r
< t < , where
r=
(2.1)
r=
and suppose their spectral densities are fX () and fY ().
We have
Cov (Yt , Yt+k ) = Cov
cr Xtr ,
r=
=
cr cs k
5
Lecture 5: Forecasting with ARMA models
We consider forecasting using an ARMA(p, q) model dened as
(B)Xt = (B) t ,
t
W N (0, 2 ).
(5.1)
For forecasting, it is convenient to assume that the moving average operator (B) is invertible,
i.e. the zeros of th
3
Lecture 3: Modelling and Forecasting with ARMA processes
3.1
Introduction
The process of tting an ARMA model, as it was made explicit by Box and Jenkins, may be
divided into three components,
Identication
Estimation
Verication
which are iterated unti
4
Lecture 4: Verication diagnostics of ARMA models
The third step of the Box-Jenkins cycle is to conrm that the model in fact ts the data. There
are two basic techniques here:
Overtting: i.e. add extra parameters to the model and use likelihood ratio or
7
Lecture 7: Modelling seasonal ARMA processes. SARIMA
17.1 Example
If a time series contains a seasonal component St then the ordinary dierencing
methods of the previous section will not work. We will illustrate this with an
example.
Example. Monthly pro
9
Lecture 9: State Space Models and the Kalman Filter
9.1
Introduction
Many time-series models used in econometrics are special cases of the class of linear state space models developed by engineers to describe physical systems. The
Kalman lter, an ecient
10
Lecture 10: ARCH and GARCH models
In contrast to the traditional time series analysis which focuses on modeling
the conditional rst moment, ARCH and GARCH models specically take the
dependency of the conditional second moment into modeling consideratio
8
Lecture 8: Estimating Spectral Densities
8.1
Regression on sinusoidal components
The simplest form of spectral analysis consists of regression on a periodic component:
Yt = A cos t + B sin t + C + t ,
1 t T,
(8.1)
where t W N (0, 2 ). Without loss of ge
11
Lecture 11: Estimation and testing of ARCH and GARCH models
We have observations X1 , . . . , XT and would like to estimate the parameters of ARCH/GARCH model.
11.1
Conditional MLE
For the ARCH(p) model
Xt = t t ,
2
2
2
t = c0 + b1 Xt1 + . . . + bp Xtp
12
Lecture 12: Introduction to ARFIMA models
12.1
Long range dependence
The phenomenon of long memory had been known well before suitable stochastic models were developed. Scientists in
diverse elds of statistical applications observed empirically that co
Stat 929: Time Series
Lecture 12
June 13
Topics for today:
1. Wold Decomposition
2. Introduction to Spectral Analysis of Time Series
Suppose that xt is a stationary process, and let
Mn = sp(x
t , < t n).
For a closed linear subspace M, let
PM Y = Project