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 cfw_Z1,t , cfw_Z2,t are independent purely random pro
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 > 0. Dene
n
Sn =
i=0
i.
Show that Sn =
1 n+1 . 1
Let S =
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 statement scenarios
Conditional expected value, E(Y|X) f (
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 : frequency, radians/unit time : phase t = 0,1,2,. t = 0,1
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 frequencies p = 2p/N, p = 0,.,N-1 Fourier components t=
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 smoothed averages) n = 508, 1970-1979
acf ccf
Correlation is
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 century movement of planets and sun
Symbol graph. Good f
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 are observations of some process made sequentially in ti
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 stochastic processes Mixed models Integrated models
2 / 16
3.1
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. identication of the order (p ,d ,q ), 2. estimation of the mod
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 an observed time series x1 , . . . , xN . We would like
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 periodic components can be modelled as
k
(6.1)
Xt =
j =1
Rj
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 error term. We will consider the following model: Xt =
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 bivariate stochastic process (Xt , Yt). Denition 8.1.1: T
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 to output 1 y1 (t) + 2 y2 (t) where y1 (t) and y2 (t) a
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 (10.3) (10.2) (10.1)
Equation (10.3) says that the uncond
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 should be aware of from the rst year. Section 0.9 is a sma
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 ) = bdCov (X, Y ) (c) Cov (X, Y ) = E (XY ) X Y 2. Find th
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 ) = bdCov (X, Y ) (c) Cov (X, Y ) = E (XY ) X Y 2. Find th
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 second order MA process given by Xt =
+ 0.7
t1
0.2
t2
2
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 rst order AR process dened by Xt = 0.7(Xt1 ) + t . Plot
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 second order MA process given by Xt =
+ 0.7
t1
0.2
t2
2. S
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 has r1 = 0.31, r2 = 0.37, r3 = 0.05, r4 = 0.06, r5 = 0.
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: (* Recommended) Chateld, C. The analysis of time series
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 supported by evidence. Your solutions to this sheet COUNT T
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 supported by evidence. Your solutions to this sheet COUNT TOW
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 supported by evidence. You will also need to at least read t