Time Series Analysis Problem Sheet 6 1. Show that the cross-covariance function of the discrete bivariate process cfw_Xt , Yt where Xt = Z1,t + 11 Z1,t1 + 12 Z2,t1 Yt = Z2,t + 21 Z1,t1 + 22 Z2,t1 and
Time Series Analysis Problem Sheet 1
If you wish hand in solutions to questions 1 and 3 on Thursday 23rd October at the 10.00 lecture. These solutions do not count for credit. 1. (Revision). Suppose >
Stat 153 - 25 Sept 2008 D. R. Brillinger Chapter 5 - Forecasting Data x1 ,., xN What about xN+h, h>0 Forecast x( N , h) or x N (h) No single method universally applicable extrapolation conditional sta
Stat 153 - 7 Oct 2008 D. R. Brillinger Chapter 6 - Stationary Processes in the Frequency Domain One model X t = R cos(t + ) + Z t Another X t = R expcfw_t cos(t + ) + Z t R: amplitude : decay rate : f
Stat 153 - 13 Oct 2008 D. R. Brillinger Chapter 7 - Spectral analysis 7.1 Fourier analysis Xt = + cos t + sin t + Zt Cases known versus unknown, "hidden frequency" Study/fit via least squares
Fourier
Bivariate time series Los Angeles polution mortality study
Shumway at al (1988) Environ. Res. 45, 224-241
Los Angeles County: average daily cardiovascular mortality particulate polution (six day smoot
Stat 153 - 13 Nov 2008 D. R. Brillinger Chapter 14 - Examples and practical advice ". drawing a time plot . arguably the most important step in any times series analysis."
Tufte (1983). 10th or 11th c
Chapter 1: Introduction
Li Chen
Department of Mathematics University of Bristol
1 / 13
Outline
Examples General denitions The objectives of time series analysis (TSA)
2 / 13
Introduction
Time series a
Chapter 3: Probability Models for Time Series
Li Chen
Department of Mathematics University of Bristol
1 / 16
Outline
Introduction Stationary processes The autocorrelation function Some useful stochast
Chapter 4: Estimation in the Time Domain
Li Chen
Department of Mathematics University of Bristol
1 / 10
Outline
Fitting an ARIMA process to an observed time series proceeds in three stages: 1. identic
Chapter 5: Forecasting
Li Chen
Department of Mathematics University of Bristol
1/7
Introduction
Forecasting future values of an observed time series is an important, but dicult task. Suppose we have a
6
6.1
Frequency-based Methods for Time Series
Stationary Processes in the Frequency Domain
If a time series has a periodic component, this can be modelled as Xt = R cos(t + ) + Zt . More than one peri
7
7.1
Spectral Analysis
A Simple Sinusoidal Model
Suppose we suspect that a given time series, with observations made at unit time intervals, contains a sinusoidal component of frequency plus a random
8
8.1
Bivariate Processes
Time-domain Quantities
Suppose we have N observations recorded on two variables, (x1 , y1 ), . . . , (xN , yN ). This can be thought of as a nite realization of a discrete bi
9
9.1
Linear Systems
Introduction
We consider linear systems which have stochastic processes as their input and output. Denition 9.1.1: A system is linear if each input 1 x1 (t) + 2 x2 (t) gives rise
10
10.1
Time series models in Finance and Econometrics
Background
Recall that the AR(1) model is given by Xt = Xt1 + Zt where cfw_Zt is white noise, ie E(Zt ) = 0 and E(Zt Zs ) = 2, t = s 0, t = s (1
Time Series Analysis Essential Concepts
G. P. Nason 1st October 1998 (updated 28th September 2005)
0
Essential concepts for time series analysis
This chapter reiterates important concepts that you sho
Time Series (MATH5/30085) 2004 Exercises 1 1. Properties of covariance. Using the denition Cov (X, Y ) = E [(X X )(Y Y )] Prove the following: (a) Cov (X, Y ) = Cov (Y, X ) (b) Cov (a + bX, c + dY ) =
Time Series (MATH5/30085) 2005 Exercises 1 1. Properties of covariance. Using the denition Cov (X, Y ) = E [(X X )(Y Y )] Prove the following: (a) Cov (X, Y ) = Cov (Y, X ) (b) Cov (a + bX, c + dY ) =
Time Series (MA5/30085) 2005 Exercises 2 In these questions, cfw_ t is a discrete, purely random process, such that E ( t ) = 0, V AR( t ) = 2 , COV ( t ,
t+ )
= 0 for = 0.
t
1. Find the ACF of the s
Time Series (M5/30085) 2005 Exercises 3 In these questions, cfw_ t is a discrete, purely random process, such that E ( t ) = 0, V AR( t ) = 2 , COV ( t ,
t+k )
= 0 for k = 0.
1. Find the ACF of the r
Time Series (M30085) 2002 Exercises 3 In these questions, cfw_ t is a discrete, purely random process, such that E ( t ) = 0, V AR( t ) = 2 , COV ( t ,
t+ )
= 0 for = 0.
t
1. Find the ACF of the seco
Time Series (MA3/50085) 2005 Exercises 4 1. Find the partial ACF of the AR(2) process given by Xt = 29Xt2 + t .
1 3 Xt1
+
2. Suppose that a correlogram of a time series consisting of 100 observations
Time Series Analysis
Dr. Gavin Shaddick [email protected] www.bath.ac.uk/masgs
Department of Mathematical Sciences University of Bath Bath BA2 7AY
2004
1
Books
Most suitable books for the course:
Time Series Analysis Problem Sheet 2
Hand in solutions to questions 3, 4 and 6 on Thursday 6th November at the 10.00 lecture. LATE solutions will NOT be accepted unless you have very good reasons supp
Time Series Analysis Problem Sheet 3
Hand in solutions to questions 2 and 3 on Thursday 20th November at the 10.00 lecture. LATE solutions will NOT be accepted unless you have very good reasons suppor
Time Series Analysis Problem Sheet 4
Hand in solutions to questions 3, 4 and 5 on Tuesday 2nd December at the 12.00 lecture. LATE solutions will NOT be accepted unless you have very good reasons suppo