Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Chapter 1 - Introduction
Y. Chen S. Tavakoli
A Cambridge Part III Course, Lent 2015
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Outline
1
2
3
4
5
6
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Administration and examination
24 lectures in total, 12 on Time Series and 12 on Monte Carlo
Inference
4 example sheets and 4 example classes
6 exam questions in total, 3 on Time Series and 3 on Monte
Carlo Inference
Need to pick 4 out of 6 exam questions
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Course website
My email address is Y.Chen@statslab.cam.ac.uk
Course website:
http://www.statslab.cam.ac.uk/~yc319/ts.html
Feedback form on the course website
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Books - I
(by P. J.
Brockwell and R. A. Davis) serves as a good introduction,
especially for those completely new to time series analysis.
Introduction to Time Series and Forecasting
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Books - II
Time Series: Applications to Finance with R and S-Plus
(by N. H. Chan) covers large parts of this course, presented in
a less mathematical and very concise style.
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Books - III
Time Series Analysis and its Applications: with R
Examples
(by R. H. Shummway and D. S. Stoer) is another
great book on time series analysis aimed at roughly the same
level as our course.
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Books - more advanced
(by P. J. Brockwell and R. A.
Davis) covers the rst part of our course (linear time series models)
in much greater depth than we do. It also contains rigorous proofs
of many theoretical results that we state but do not prove in the
lectures.
Time Series: Theory and Methods
GARCH Models: Structure, Statistical Inference and Financial
Applications (by C. Francq and J-M. Zakoian) is an excellent read
for those who are keen to know more about non-linear time series
models. It provides a comprehensive and systematic approach from
a mathematical (theoretical) perspective.
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Outline
1
2
3
4
5
6
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
What is a time series
A time series is a set of data collected at successive (discrete)
time-points
In this course we will conne ourselves to observations made
at regularly spaced time-points, without loss of generality
taking the interval between such points to be 1
We shall write the series as {Xt , t ∈ Z}
Other time series: continuous time series, discrete observed
values, multivariate time series etc.
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Applications
economics e.g., monthly data for unemployment, hospital
admissions
nance e.g., daily exchange rate, a share price
environmental e.g., daily rainfall, air quality readings
medicine e.g., brain wave activity
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Outline
1
2
3
4
5
6
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
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Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
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Chapter 1 - Introduction
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Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
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Chapter 1 - Introduction
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Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
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Y. Chen, S. Tavakoli
Chapter 1 - Introduction
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Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Outline
1
2
3
4
5
6
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Classical decomposition I
One simple method of describing a series is that of classical
decomposition.
The notion is that the series can be decomposed into three
elements:
Trend (Tt ): long term movements in the mean
Seasonal eects (St ): cyclical uctuations related to the
calendar or business cycles
Microscopic part (Mt ): other random or systematic
uctuations
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Classical decomposition II
The idea is to create separate models for these three elements and
then combine them
either additively Xt = Tt + St + Mt
or multiplicatively Xt = Tt St Mt
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Classical decomposition - example
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Estimation of trends and seasonal cycles I
Least squares method for linear trend:
suppose that the seasonal part is absent
a simple linear time trend structure Tt = a + bt
estimate a and b by minimising ∑(Xt − Tt )
Major drawbacks:
cannot deal with trend changing over time
linear form might be too restrictive
2
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Estimation of trends and seasonal cycles II
Smoothing or ltering:
s
Tt = ∑ ar Xt+r
r =−q
where the weights {ar } of the lter are usually assumed to be
symmetric and normalized, so ar = a r and ∑ ar = 1, This lter is
also known as the moving average lter.
−
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Estimation of trends and seasonal cycles III
Dierencing:
First order dierencing: Yt = Xt − Xt
Higher order dierencing: e.g. second order,
Yt = (Xt − Xt ) − (Xt − Xt )
Seasonal dierencing: Yt = Xt − Xt d , if the cycle is of length
−1
−1
−1
d
−2
−
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Estimation of trends and seasonal cycles IV
Reasons behind dierencing:
if Xt = Tt + Mt with Tt = a + bt , then Tt is eliminated in the
new seriesYt = Xt − Xt
same argument works for order-k dierencing, where Tt is a
k -degree polynomial of t
−1
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Outline
1
2
3
4
5
6
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Dierent approaches
In this course, we mainly focus on modelling the microscopic part
Mt :
Time domain
linear models: autoregressive (AR) models, moving average
(MA) models, ARMA models, ARIMA models
non-linear models: generalised autogressive heteroskedasticity
(GARCH) models, threshold models
Frequency domain
Spectral analysis
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Statistical software
Most of the common time series data analysis can be done in
R
Some useful packages include: ARIMA, fGarch, stochvol
Most frequently used functions include: lter, acf, pacf, arima,
garch, spectrum, etc
More details can be found here:
cran.r-project.org/web/views/TimeSeries.html
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Outline
1
2
3
4
5
6
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
General framework
Data generating mechanism
Probability structure
Estimation procedure
Model selection
Model diagnostics
Y. Chen, S. Tavakoli
Chapter 1 - Introduction
Admin
Basic description
Some examples
Trend and seasonal cycles
Course overview
On statistical modelling
Some quotes
All models are wrong, but some are useful. (George Box)
There are no routine statistical questions, only questionable
statistical routines. (David Cox)
If you torture the data enough, nature will always confess.
(Ronald Coase)
In theory there is no dierence between theory and practice.
But, in practice, there is. (Attributed to multiple people)
Y. Chen, S. Tavakoli
Chapter 1 - Introduction