STAT 4005
ASSIGNMENT 5
Due date: December 6, 2012
1. The following data were generated by the process Zt = at + at1 .
t
Zt
1
0
2
2
3
1
0 ()
Set an initial estimate = 0. Let a0 = 0 and dad = 0. By using the
Gauss-Newton algorithm, nd the next new estimate
STAT 4005
ASSIGNMENT 1
Due date: October 4, 2012
1
1. Suppose E(X) = 4, V ar(X) = 4, E(Y ) = 9, V ar(Y ) = 9, and Corr(X, Y ) = 4 .
Find:
(a) V ar(X + Y ), (b) Cov(X, X + Y ), (c) Corr(X + Y, X Y ).
2. Suppose Zt = 8 + 3t + Xt , where cfw_Xt is a station
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Practise Numerical Reasoning: Question 1 of8
Which mine can produce the greatest amount of units of electricity
before it runs out ofcoal?
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Numerical Reasoning: Question 9 of 18
A trader buys Rupees with £15,000 Sterling at "Today's High“ rate.
[.hannﬁ IWhat is the maximum amount pren the trader cpuld tiu‘lur with
Em p.31} -. ; these Rupees
STAT4005 Time Series (2013-2014)
Tutorial 10
(Continue) Estimation Methods-Maximum Likelihood Estimation
Example
Consider the AR(1) model
Z t 0.6 Z t 1 = 2 + at
with observations Z1 = 2 , Z 2 = 3 , Z 3 = 4 , Z 4 = 4 .
(a) Derive the likelihood function of
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Verbal Reasoning test: Question 28 of 30
In the past, to fly was not as yiable a means oftrayel as it is
today.
The airline industry has always acknowledged ﬁrst-
STAT4005 Time Series
Tutorial 1
Properties of Covariance
Suppose X, Y and Z are random variables, and a, b, c and d are constant (or deterministic).
i) Cov(X, Y ) = E[(X E[X])(Y E[Y ])] = E[XY ] E[X]E[Y ]
ii) Cov(a + bX, c + dY ) = bdCov(X, Y )
Cov(X + Y,
STAT4005 Time Series 2013-2014
Tutorial 9
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 specification. For
STAT4005 Time Series (2013-2014)
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 (2013-2014)
Tutorial 5
Example 1:
2
Consider the stationary process Zt = + Zt1 + at , at W N (0, a ), and | < 1. Find E(Zt ) and
Cov(Zt , Zt+k ) for k = 0, 1, 2, . . .
Example 2:
Consider the process Zt = 0.5Zt1 0.06Zt2 +at 0.8at1 +0
STAT4005 Time Series (2014-2015)
Tutorial 2
Theorem. Suppose cfw_Zt is a stationary time series and Zn =
Var(Z) =
0
1+2
n
n1
1
k=1
k
n
k =
0
n
1
n
n
t=1 Zt ,
then
n1
1
k=n+1
|k|
n
k
Examples
1. Is it possible to have a series cfw_Zt with a constant mean
STAT 4005 Tutorial 4 solution
1. 1 1 x = 0 x =
1
1
Hence stationary condition is
2. 1 1 x 2 x2 = 0 x =
1
r1
Hence stationary condition is
1
1
>1
or x =
1
r1
1
r2
> 1 and
1
r2
>1
3. (a) AR characteristic function f (x) = 1 x x2 with roots x1 and
x2 .
For >
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 + . . .
=
STAT 4005 Tutorial 2 solution
1. Yes. Consider Zt = A + tXt where A is a constant, Xt is a stationary
process with mean zero and autocovariance function k .
Cov(Zt , Zt+k ) = t(t + k)k
V ar(Zt ) = t2 0
V ar(Zt+k ) = (t + k)2 0
Corr(Zt , Zt+k ) = t(t+k)k
STAT4005 Time Series (2013-2014)
Serie
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 )
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
STAT 4005 Mid-term Examination Solution
1. A stochastic process cfw_Zt is said to be strictly stationary if the joint distribution of
Zt1 , Zt2 , . . . , Ztn is the same as the joint distribution of Zt1 k , Zt2 k , . . . , Ztn k for all
choices of time p
STAT 4005
ASSIGNMENT 4
Due date: April 11, 2016
1. Identify Zt = 0.25Zt2 + at 0.3at1 . as a specific ARIMA model
2. From a series of 144 observations, we calculate r1 = 0.96, r2 = 0.84, r3 =
0.7, r4 = 0.01, and |rk | 0.01 for k > 4. On the basis of this i
Setup
Step 1: Save the excel file as .csv format.
Step 2: Open the R program and change directory
File Change dir location of the file
Step 3: Import the data and assign to data2. Add header=F if the raw data columns do not have
names. In R, they will be
STAT4005 Assignment 1 Solution
p
p
1. (a) Cov(X, Y ) = Corr(X, Y ) V ar(X) V ar(Y ) = 3
V ar(2X + Y ) = 4V ar(X) + 4Cov(X, Y ) + V ar(Y ) = 64
(b) Cov(2X, X + Y ) = 2V ar(X) + 2Cov(X, Y ) = 24
(c) Cov(2X + Y, X 3Y ) = 2V ar(X) 6Cov(X, Y ) + Cov(X, Y )
3V