STAT 4005
ASSIGNMENT 4
Due date: November 17, 2011
1. Identify the following as a specic ARIMA model
Zt = 0.8Zt1 + at 0.5at1 0.3at2 .
2. Consider the IMA(1,1) model
(1 B )Zt = 0.6 + (1 + 0.2B )at
where Zt = 0 for t 20. Find Corr(Z20 , Z10 ).
3. Rewrite th
STA 4005A
Mid-term Examination (Total Marks 35)
Answer ALL questions:
October 25, 2007
Time allowed: 1 1/2 hours.
2
Let at N ID(0, a ).
1. Consider a MA(2) model:
Zt = at 1.3at1 0.4at2 ,
2
a = 5.
(a) (2 marks) Find E (Zt ) and V ar(Zt ).
(b) (3 marks) Fin
STA4005B Time Series
2007-2008
Mid-Term Solution
1a) A stochastic process cfw_Z t is said to be weakly stationary if t is constant for all
time point t and Cov ( Z t , Z t + k ) = t ,t + k = 0 ,k = k for all time point t and lag k.
1b) A stochastic proces
STAT4005 Course Outline
1. Course title and description
Course title:
STAT4005 Time Series
September 2011
Instructor: Professor WU, Ka Ho (LSB 112)
Teaching Assistant: Mr. CHUNG, Shing Fung (LSB G32)
Time and Venue (Lecture): M5 (LSB LT6), H1-2 (LSB LT6)
An example
Let X1 and X2 be independent random variables, each with
p.d.f.
f ( x) = e x ,
0< x<
Consider Y1 = X1 X2 and Y2 = X1 + X2
find the joint p.d.f. of Y1 and Y2 .
An example
Their joint p.d.f. is
f ( x1 ) f ( x2 ) = e x1 x2
0 < x1 < ,0 < x2 <
Henc
STAT4005 Time Series (2011-2012)
Tutorial 1
Def: Suppose X, Y and Z are random variables and a, b, c and d are constant.
i)
Cov ( X , Y ) = E[ X EX ][Y EY ] = EXY EXEY
ii)
Cov ( a + bX , c + dY ) = bdCov ( X , Y ) ( Cov( aX , bY ) = abCov ( X , Y ) )
Cov
STAT 4005 Time Series (2011 - 2012)
Tutorial 2
Example 1:
Is it possible to have a series cfw_Zt with a constant mean and Corr(Zt , Ztk ) free of t but with
cfw_Zt not stationary? If the answer is yes, give an example. If the answer is no, explain why.
STAT 4005
ASSIGNMENT 5
Due date: December 1, 2011
1. Consider the invertible MA(1) process Zt = 3 + at + at1 . Find the leastsquares estimate of based on the a time series of length n = 4 with Z1 =
3, Z2 = 3, Z3 = 6 and Z4 = 4.
2. Suppose the joint p.d.f.
STAT4005 Time Series (2011-2012)
Tutorial 3
How to analyze a time series using S-plus?
Suppose a time series cfw_Z t : t = 1, 2,3, ,100 , is given in a dataset called dataset1.xls.
STEP 1 Rewrite models into matrix form
Model (a) Z t = 0 + 1 cos(
2 t
2 t
STAT 4005 Time Series (2011 - 2012)
Tutorial 4
Table for Stationary Conditions and Invertible Conditions
cfw_Zt is the following model Is cfw_Zt stationary? Is cfw_Zt invertible?
AR(p) Model
MA(q) Model
ARMA(p,q) Model
Stationary Conditions and Inverti
STAT 4005 Time Series (2011 - 2012)
Tutorial 6 (31/10,3/11)
1. Consider an innite MA process Zt = at + C (at1 + at2 + . . .), where C is a xed constant,
2
E (at ) = 0, and V ar(at ) = a .
(a) Show that cfw_Zt is not second order stationary (weakly statio
NOTES 9
Forecasting:
Mean squares error prediction
Give a random vector (Y, X) , where Y is a scalar and X is a vector. We want to get a
function g (X) which predict Y as closely as possible, in the Mean squares error sense,
i.e
M SE = E [Y g (X)]2
is to
Notes 8
Diagnostics Checking:
Recall
Zt =
j Ztj + at
j =1
Put
at = Z t
j Ztj
j =1
Residual = actual value - estimated value
Residual Analysis
1. Plot residuals at against t (See whether trend exists)
2. Histogram of at (or standardized residuals), normal
NOTES 1
Introduction:
Many Statistics courses focus on independent data. We assume X1 , X2 , X3 , ., Xn are
independent.
For time series, we assume X1 , X2 , X3 , ., XT are are dependent (correlated).
Problems of interest:
1. Estimating t = mean at time t
Time series Examples:
1
Daily closing stock prices
Unemployment rate
Monthly Temperature
Tourists number
Carbon dioxides concentration
Sales
Oil price
Time Series Plot
It appears to vary about a fixed level.
2
Time Series Plot
It does not vary about a fix
NOTES 2
Stochastic Process and Time Series:
Denition: A stochastic process is a family of random variables cfw_Zt , t T . T is
thought of as representing time. If T is an interval, then the process cfw_Zt , t T is said
to be continuous. If T is discrete
NOTES 3
Trends of a time series:
Deterministic versus Stochastic Trends
Examples:
2
yt = at 0.5at1 where at N (0, a );
yt = 0 + 1 t + 2 t2 ;
Purely Stochastic.
Purely Deterministic
2
yt = 0 + 1 t + 2 t2 + at where at N (0, a );
Stochastic + Deterministic
Notes 4
Models for Stationary Time Series:
General Linear Processes
A general linear process cfw_Zt is one that can be represented as a weighted linear combination of the present and past terms of a white noise process:
Zt = at + 1 at1 + 2 at2 + . . .
=
Notes 5
Models for Non-stationary Time Series:
In Notes 4, the time series we studied are all stationary processes. However, in practice,
a lot of useful time series are nonstationary. At present, we introduce a class of nonstationary time series models c
Notes 6
Model Specication:
Overall strategy - Box Jenkins Approach:
1. To decide a reasonable - but tentative - values for p, d, q .
2
2. Estimate , and a , for that model.
3. Check the models adequacy.
4. If the model appears inadequate, consider the nat
Notes 7
Parameter estimation:
In general, for AR(p), MA(q) and ARMA(p,q) models, what we want to estimate are
the series mean , the AR parameters s, the MA parameters s and the error variance
2
a .
Estimation methods:
1. Method of moment
2. Method based o
STAT4005 Time Series (2010-2011)
Tutorial 7 (7/11,10/11)
The partial autocorrelation function of AR(p)
Initially, we may not know which order of autoregressive process (p) to fit to an observed time series.
The partial autocorrelation is a device to deter
STAT4005 Time Series (2010-2011)
Tutorial 8 (14/11, 17/11)
Estimation Methods
Suppose a sample realization cfw_Z t , t = 1,2, K , n is obtained and ARIMA model is proposed to fit the data.
Then we want to estimate the parameters in the model after model s