Definition
A time series cfw_Z t is said to be an Auto-Regressive Moving Average
process of orders p and q , denoted by ARMA( p, q) , if cfw_Z t is stationary and
Z t 1Z t 1 . p Z t p = t + 1t 1 + .q t q for all t ,
where 1 ,., p , 1 ,., q are numbers an

Example
Consider the regression model Yt = X t + t t = 1,., n .
(a)
Find the least squares estimate of using differentiation.
(b)
Find using the general formula obtained from the normal
equations.
Solution:
(a)
Find by minimizing
2
(Y X ) .
i
i
d (Yi X i

Consider the regression model
Yt 0 1 X
t1
. q X
q 1 X
tq
tq 1
. k X
tk
t
under the standard assumptions [i.e. the matrix X is not stochastic and is multivariate normal
with mean 0 and covariance matrix equal to 2 I .
Suppose we want to test
H
0
: q 1

Y1
.
Definition
A random vector Y
is said to have a multivariate normal distribution if for
.
.
Yn
every a 0 ,., a n , the distribution of the random variable a 0
a 1Y1
.
a nY n
is normal.
Note that it follows directly from the definition that all of the

Example
Consider the regression model Yt
Xt
t
t 1,., n .
(a)
Find the least squares estimate of
using differentiation.
(b)
Find using the general formula obtained from the normal
equations.
Solution:
(a)
Find by minimizing
d
X i )2
(Yi
d
0,
In this case X

Forecasting
Definition
Let cfw_Z t be a stationary time series. A function h( Z n , Z n 1 ,.) is said to
be the minimum mean squared error forecast of Z n l , or the best forecast of Z n l ,
based on the set of observations cfw_Z n , Z n 1 ,. if for any

Actuarial Science 430/830
Fall 2012
1.
Homework for Quiz 3
You are given:
(i) cfw_Z t is an MA(1) process satisfying Z t
t 1.
t
(ii) The lag 1 autocorrelation coefficient,
1
for cfw_Z t is equal to 0.4.
(iii) | | 1
Calculate .
2.
You are given the follo

Example
Let
(0 )
cfw_ Z t be
1
(1 )
1
(k
1)
2
1
(k
1)
2
an AR
2
(2)
2
time series with an
(2)
MA (
)
representation. Show that
,
and
(k )
(k
2)
for
k
0
(k
2)
for
k
0
and
(k )
.
Solution: We have
Zt
1
Zt
1
2
Zt
2
t
Zt
1
Zt
1
2
Zt
2
t
,
.
Multiply both sid

Actuarial Science 430/830
Fall 2012
1.
Practice Questions 2
Based on a data set of 26 points, two regression models are estimated. For a two
variable model (the restricted model) Y t 1 2 X t 2 t , we get R 2 . 75 , and a
four variable model (the unrestric

Introduction
Types of forecasts
1. Qualitative forecasts
These are educated guesses that are not
based on a mathematical model.
2. Quantitative forecasts:
These are based on a mathematical or
statistical model, which uses the past
data.
The models are eit

Consider the model Y t 0 1 X t 1 . k X tk t .
We should that the least squares estimator is BLUE if E ( ) 0 and cov( ) 2 I .
When E ( ) 0 , let t E ( t ) . Then
If 1 ,., n are known, let Y t ' Y t t , and t, t t . Then the model
Yt 0 1 X
'
t1
. k X
t
,

Actuarial Science 430/830
Fall 2012
Practice Questions 1
Consider the regression model, Yt X t t where t is a random error term with
1.
mean 0 and variance 2 . Suppose five pairs of values ( X t , Yt ) have produced.
10
X t 21
t 1
10
Yt 42
t 1
10
X tYt

Actuarial Science 430/830
11-22-2011
Exam 2
Name-
Solutions should be written clearly and should
contain sufficient detail to show that you know
how to handle the important aspects of the problem.
No credit will be given to answers that are not supported

Consider the regression model Y X . Suppose X is not stochastic and has mean
Fact
and cov( ) 2 I .
0
SSE
Let S 2
(i)
S
2
n the number
. Then
of coefficien
ts
is an unbiased estimator of 2 .
(ii) When is multivariate normal ( 0 , 2 I ) , S 2 and are indep

Recall:
cov( i , j ) E[i E ( i )][ i E ( j )]. Since we have assumed E[ i ] 0 for all i ,
E[ i j ] cov( i , j ) in our regression models.
Note that covariance is a measure of linearity of the relation between two random variables and
zero covariance does

Actuarial Science 430/830
10-04-2011
Exam 1
Name-
Solutions should be written clearly and should
contain sufficient detail to show that you know
how to handle the important aspects of the problem.
1. You fit the following model to four observations:
Yt 1

Time Series
Definition
A time series is a set of random variables cfw_ X t : t T ordered in time.
The set T is the index set, and the index t denotes time. When T is countable, the time
series is called a discrete time series. When T is an interval, the

Nov 2004: 39
Nov. 2002: 9
Nov. 2001: 32
Nov. 2000: 30
May 2000: 5
And the following two problems.
1.
You are using an AR(1) model to represent a time series of 100 observations,
Yt , t 1,2,.,100 . You have determined: 1 0.83
2 = 3.0
Calculate the length

Example Show that for the regression model Y t X t t ,
2
2
SSR ( X i X )
.
Solution:
We have
Yi X
i
and
Y X
Therefore
Yi Y ( X i X )
SSR
i
(0, I ) ,
.
For a regression model Y t X t t , where is multivariate normal
Example
2
2
2
Y ) ( X i X )
(Y
2
,
wi

Actuarial Science 430/830
Homework for Quiz 2
Fall 2012
You are given the regression model Y i X i i , for i 1,., 20 , where i s are
multivariate normal with mean zero and covariance matrix equal to 2 I.
You have determined:
138 . 561 , 1 . 104 ,
X
i
X

Serial correlation
Consider the regression model
Yi 0 1 X i1 . ik i .
We say the errors in the above regression are serially correlated they do not satisfy the
assumption that cov( i , j ) 0 when i j .
A case when the errors are serially correlated is whe

Actuarial Science 430/830
Fall 2012
Homework for Quiz 4
1.
Calculate 1 for the ARMA (1,1) model
Yt .5Yt 1 t .25 t 1
where the cfw_ t is a WN (0, 2 ) .
2.
For the ARMA (1,1) model
Yt .8Yt 1 t . 1 t 1
you are given 1 .6. Determine 2
3.
Calculate 1 for the

ACTS 430/830:
Actuarial Applications of Applied Statistics
Instructor: Yanxin (Graham) Liu
Department of Finance, University of Nebraska-Lincoln
September 12, 2016
Instructor: Yanxin (Graham) Liu
Actuarial Applications of Applied Statistics
September 12,

Definition
A time series cfw_Z t is called an auto-regressive process of order p ,
denoted by AR ( p) , if
Z t 1Z t 1 . p Z t p = t , for every t ,
where 1 ,., p are numbers and cfw_t is a WN (0, 2 ) .
The process cfw_Z t is said to be AR ( p) with mea

Time Series
Definition
A time series is a set of random variables cfw_ X t : t T ordered in time.
The set T is the index set, and the index t denotes time. When T is countable, the time
series is called a discrete time series. When T is a continuum of po

When nothing is known about 1 ,., n , the number of unknown parameters is more than the
number of observations and we cannot have a reasonable statistical inference when the number of
unknown parameters is more than the number of observations.
Generalized

Recall:
cov( i , j ) = E[i E ( i )][ i E ( j )]. Since we have assumed E[ i ] = 0 for all i ,
E[ i j ] = cov( i , j ) in our regression models.
Note that covariance is a measure of linearity of the relation between two random variables and
zero covariance

&v1 #
$. !
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Let v = $. ! be a k -dimensional random vector. The covariance matrix of v ,
$!
$. !
$v k !
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denoted by cov(v) , is the matrix
&cov(v1 , v1 ) . . . cov(v1 , v k ) #
$.
!
$
!
$.
!.
$
!
$.
!
$cov(v k , v1 ) . . . cov(v k , v k )!
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