Homework 2 solutions
Joe Neeman
September 22, 2010
1. (a) We compute three cases: since the Wt are uncorrelated, we can ignore
any cross-terms of the form EWs Wt when s = t. Then
9
19
25
EWt2 1 + EWt2 2 =
4
4
2
15
5
5
(1) = EWt2 + EWt2 1 =
2
4
4
3
3
(2
Introduction to Time Series Analysis. Lecture 11.
Peter Bartlett 1. Review: Time series modelling and forecasting 2. Parameter estimation 3. Maximum likelihood estimator 4. Yule-Walker estimation 5. Yule-Walker estimation: example
1
Review (Lecture 1): Ti
Introduction to Time Series Analysis. Lecture 5.
Peter Bartlett
www.stat.berkeley.edu/bartlett/courses/153-fall2010
Last lecture: 1. ACF, sample ACF 2. Properties of the sample ACF 3. Convergence in mean square
1
Introduction to Time Series Analysis. Lect
Introduction to Time Series Analysis. Lecture 1.
Peter Bartlett 1. Organizational issues. 2. Objectives of time series analysis. Examples. 3. Overview of the course. 4. Time series models. 5. Time series modelling: Chasing stationarity.
1
Organizational I
STAT 758, Spring 2012
Key solution for Home Work 4
Prepared by Tracy Backes
MA(q )
Below we assume that Zt W N (0, 2 ).
4.1 Consider MA(1) process Xt = a Zt + b Zt1 . Find the white noise Wt such that
the process Xt is presented as Xt = Wt + Wt1 with Wt W
Homework 1 solutions
Joe Neeman
September 10, 2010
1. To check that cfw_Xt is white noise, we need to compute its means and
covariances. For the means, EXt = EWt (1 Wt1 )Zt = (EWt )(1
EWt1 )(EZt ) = 0. For the covariances,
(s, t) = E Ws (1 Ws1 )Zs Wt (
Statistics 758, Fall 2014
University of Nevada Reno
Homework 7 Solutions
Problem 1: The rvs Y and X are related as Y 10 20 X , ~ N 0, 42
a) Find the conditional distribution of Y given X x
Y | X x ~ N 10 20x, 42
b) Find the conditional expectation of Y g
STAT 758, Fall 2014
Home Work 9
Spectral analysis
2
We assume below that Zt is a white noise with mean 0 and variance Z .
6.1 [similar to Chateld, Ex. 6.2] Find the power spectrum (spectral density) of the
following processes:
a) Xt = 0.5 Zt ;
b) Xt = Zt
STAT 758, Fall 2014
Home Work 8
Second-order forecasting, Prediction operator
2
We assume below that Zt is a white noise with mean 0 and variance Z .
Problem 1 Construct 1, 2, and 3-step forecasts for AR(2) process Xt = 1 Xt1 +
2 Xt2 + Zt , calculate the
We assume below that Zt W N (0, 2 ), B is a backshift operator.
1. Construct 1, 2, and 3-step forecasts for AR(2) process Xt = 1 Xt1 + 2 Xt2 + Zt and calculate the
forecast errors. Find the values of 1 , 2 that minimize the forecast errors. Discuss.
h =
Stat 153: Homework 1
Due Fri 9/18
September 9, 2015
1. [5pts] (Problem 1.8) Consider the model
Xt = + Xt1 + Wt ,
for t = 1, 2 . . . with X0 = 0, and where Wt is a white noise with variance 2 .
(a) Plot samples of Xt by simulation, by picking = 0 and = 0.
Stat 153: Homework 2
Due Mon 10/5
September 27, 2015
1. [5pts] (Problem 1.29)
(a) Suppose Xt is a weakly stationary process with zero mean and
Show that if
h= RX (h) = 0 then
h |RX (h)|
< .
p
nX0,
where X is the sample mean.
(b) Give an example of a proc
Stat 153: Homework 3
Due Mon 11/2
October 23, 2015
1. [5pts] Consider a dataset of size n generated according to the zero-mean AR(1) model
with parameter and Gaussian noise. The Yule-Walker estimate of is approxi2 )/n. In this problem, we check this
matel
Statistical Models of Time Series
ARIMA Models
Important Examples
Statistical Measurements
Stationarity
Estimation of Correlation
Sample ACF
Prediction with ACF
More Properties of ACF
Review: Stationarity and Correlation
A time series cfw_Xt has mean fun
Important Examples
Statistical Measurements
Stationarity
Estimation of Correlation
Sample ACF
Prediction with ACF
More Properties of ACF
Statistical Models of Time Series
ARIMA Models
Review: Sample Autocovariance
The sample mean is
n
1X
b=
xi .
n
i=1
The
Statistical Models of Time Series
ARIMA Models
Important Examples
Statistical Measurements
Stationarity
Estimation of Correlation
Sample ACF
Prediction with ACF
More Properties of ACF
Review: Sample Mean
The sample mean is
n
b=
1X
xi .
n
i=1
We saw that
Logistics
Introduction to Time Series
Statistical Models of Time Series
Stat 153: Introduction to Time Series
Joan Bruna
Department of Statistics
UC, Berkeley
September 7, 2015
Joan Bruna
Stat 153: Introduction to Time Series
Logistics
Introduction to Tim
We assume below that Zt W N (0, 2 ), B is a backshift operator.
6.1 For the model (1 B)(1 0.2B)Xt = (1 0.5B)Zt :
a) Classify the model as an ARIMA(p, d, q) process (i.e. nd p, d, q). ARIMA(1,1,1)
b) Determine whether the process is stationary, causal, inv
STAT 758, Fall 2014
Home Work 6
SARIMA
2
We assume below that Zt is a white noise with mean 0 and variance Z .
Problem 1
For the model (1 B)(1 0.2 B)Xt = (1 0.5 B)Zt :
a) Classify the model as an ARIMA(p, d, q) process (i.e. nd p, d, q).
b) Determine whet
Intuitive Application of the Wold Representation Theorem
Suppose we want to specify a covariance stationary time series
cfw_Xi to model actual data from a real time series
cfw_Xt,t=0,1,., T
Consider the following strategy:
a Initialize a parameter p, th
Definitions of Stationarity
Definition: A time series cfw_Xi is Covariance Stationary if
E09) : M
Var(Xt) or;
COV(Xt:Xf+T) W)
(all constant over time t)
The autocorrelation function of cfw_Xi is
p(7) = Cov(Xt?Xt+T)/\/Var(Xt) Var(Xt+T)
7(T)/Y(0)
0 Apply time series methods to the time series of residuals
cfw_pl to specify a moving average model:
Elf) : 2:20 jmi
yielding cfw_1,13, and cfw_tF estimates of parameters and
innovations.
0 Conduct a case analysis diagnosing consistency with model
assump
Representation Theorem
Wold Representation Theorem: Any zeromean covariance
stationary time series cfw_Xi can be decomposed as Xt : Vt + St
where
o cfw_Vt is a linearly deterministic process, i.e., a linear
combination of past values of Vt with constant c
1. Let cfw_1Q be a deubly innite stochastic process that is stationary with autocovariarioe function
Ty. Let
X: = (a + has: + Yr,
where a and b are real numbers and 5; is a deterministic seasonal function with period cfw_1 (Le,
Sg_d = 3; for all t)
(a) Is
0 Estimate the linear projection of Xi on (Xt_1,Xt_2, . . .,Xt_p)
(continued)
)1 1 yo yta *" yLrpl)
h 1 h m * LQ
y : . Z : . . . _ . (p l
yn 1 ynl Yin2 ynp
0 Apply OLS to specify the projection:
9 : Z(ZTZ)1Zy
: P(5/t l Yfla YtQa ' ' ' Viip)
: (P)
0 Comput
STAT 153 Fall 2015, Midterm 2 Exam
Nov 12, 2015
Name:
SID:
Person on left:
Person on right:
If you are stuck in one question of a problem, you can move to the following questions.
Partial credit will be given.
Try to answer short and to the point.
Goo
STAT 758, Spring 2012
Key solution for Home Work 1
Prepared by Tracy Backes
Dierencing, backshift operator
All notations are from lectures.
1.1 Show that the dierence operators and 12 are commutative, that is
12 = 12 .
Solution: Let us apply the operators
STAT 758, Spring 2012
Key solution for Home Work 3
Prepared by Tracy Backes
ACF, iid sequence, white noise, random walk
3.1 Give two examples (specify distributions) of each:
a) iid sequence:
(a.1) Xt iid N (0, 1), t Z,
(a.2) Xt iid Uniform([0, 1]), t Z
b
STAT 758, Fall 2014
Home Work 1
Due date: Sep. 10
Dierencing, backshift operator
All notations are from lectures.
1.1 Show that the dierence operators
and
12
are commutative, that is
12
=
12 .
1.2 Show that the dierence operator
7
eliminates a linear tren