Time Series Analysis
|
|
- Sharleen Pearson
- 6 years ago
- Views:
Transcription
1 Time Series Analysis Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby 1
2 Outline of the lecture Chapter 9 Multivariate time series 2
3 Transfer function models with ARMA input Y t = ω(b) δ(b) Bb X t + θ ε(b) ϕ ε (B) ε t X t = θ η(b) ϕ η (B) η t we require {ε t } and {η t } to be mutually uncorrelated. 3
4 Transfer function models with ARMA input From the above we get Y t = ω(b) δ(b) Bb X t + θ ε(b) ϕ ε (B) ε t X t = θ η(b) ϕ η (B) η t δ(b)ϕ ε (B)Y t ϕ η (B)X t = ϕ ε (B)ω(B)B b X t + δ(b)θ ε (B)ε t = θ η (B)η t The term including X t on the RHS is moved to the LHS 3
5 Transfer function models with ARMA input Y t = ω(b) δ(b) Bb X t + θ ε(b) ϕ ε (B) ε t X t = θ η(b) ϕ η (B) η t δ(b)ϕ ε (B)Y t ϕ ε (B)ω(B)B b X t ϕ η (B)X t = δ(b)θ ε (B)ε t = θ η (B)η t Which can be written in matrix notation 3
6 Transfer function models with ARMA input Y t = ω(b) δ(b) Bb X t + θ ε(b) ϕ ε (B) ε t X t = θ η(b) ϕ η (B) η t [ δ(b)ϕε (B) ϕ ε (B)ω(B)B b 0 ϕ η (B) [ Yt X t = [ δ(b)θε (B) 0 0 θ η (B) [ εt η t 3
7 Transfer function models with ARMA input Y t = ω(b) δ(b) Bb X t + θ ε(b) ϕ ε (B) ε t X t = θ η(b) ϕ η (B) η t [ δ(b)ϕε (B) ϕ ε (B)ω(B)B b 0 ϕ η (B) [ Yt X t = [ δ(b)θε (B) 0 0 θ η (B) [ εt η t For multivariate ARMA-models we replace the zeroes by polynomials in B, allow non-zero correlation between ε t and η t, and generalize to more dimensions 3
8 Multivariate ARMA models The model can be written φ(b)(y t c) = θ(b)ǫ t The individual time series may have been transformed and differenced The variance-covariance matrix of the multivariate white noise process {ǫ t } is denoted Σ. The matrices φ(b) and θ(b) has elements which are polynomials in the backshift operator The diagonal elements has leading terms of unity The off-diagonal elements have leading terms of zero (i.e. they normally start in B) 4
9 Air pollution in cities NO and NO 2 [ X1,t X 2,t = [ [ X1,t 1 X 2,t 1 + [ ξ1,t ξ 2,t Σ = [ Or X 1,t 0.9X 1,t + 0.1X 2,t 1 = ξ 1,t 0.4X 1,t 1 + X 2,t 0.8X 2,t 1 = ξ 2,t the LHS can be written using a matrix for which the elements are polynomials i B 5
10 Air pollution in cities NO and NO 2 [ X1,t X 2,t = [ [ X1,t 1 X 2,t 1 + [ ξ1,t ξ 2,t Σ = [ Formulation using the backshift operator: [ 1 0.9B 0.1B 0.4B 1 0.8B X t = ξ t or φ(b)x t = ξ t 5
11 Air pollution in cities NO and NO 2 [ X1,t X 2,t = [ [ X1,t 1 X 2,t 1 + [ ξ1,t ξ 2,t Σ = [ Formulation using the backshift operator: [ 1 0.9B 0.1B 0.4B 1 0.8B X t = ξ t or φ(b)x t = ξ t Alternative formulation: [ X t X t 1 = ξ t or X t φ 1 X t 1 = ξ t 5
12 Stationarity and Invertability The multivariate ARMA process φ(b)(y t c) = θ(b)ǫ t is stationary if is invertible if det(φ(z 1 )) = 0 z < 1 det(θ(z 1 )) = 0 z < 1 6
13 Two formulations (centered data) Matrices with polynomials in B as elements: φ(b)y t = θ(b)ǫ t Without B, but with matrices as coefficients: Y t φ 1 Y t 1... φ p Y t p = ǫ t θ 1 ǫ t 1... θ q ǫ t q 7
14 Auto Covariance Matrix Functions Γ k = E[(Y t k µ Y )(Y t µ Y ) T = Γ T k Example for bivariate case Y t = (Y 1,t Y 2,t ) T : Γ k = [ γ11 (k) γ 12 (k) γ 21 (k) γ 22 (k) = [ γ11 (k) γ 12 (k) γ 12 ( k) γ 22 (k) And therefore we will plot autocovariance or autocorrelation functions for k = 0,1,2,... and one of each pair of cross-covariance or cross-correlation functions for k = 0, ±1, ±2,... 8
15 The Theoretical Autocovariance Matrix Functions Using the matrix coefficients φ 1,...,φ p and θ 1,...,θ q, together with Σ, the theoretical Γ k can be calculated: Pure Autoregressive Models: Γ k is found from a multivariate version of Theorem 5.10 in the book, which leads to the Yule-Walker equations Pure Moving Average Models: Γ k is found from a multivariate version of (5.65) in the book Autoregressive Moving Average Models: Γ k is found multivariate versions of (5.100) and (5.101) in the book Examples can be found in the book note the VAR(1)! 9
16 Identification using Autocovariance Matrix Functions Sample Correlation Matrix Function; R k near zero for pure moving average processes of order q when k > q Sample Partial Correlation Matrix Function; S k near zero for pure autoregressive processes of order p when k > p Sample q-conditioned Partial Correlation Matrix Function; S k (q) near zero for autoregressive moving average processes of order (p,q) when k > p can be used for univariate processes also. 10
17 Identification using (multivariate) prewhitening Fit univariate models to each individual series Investigate the residuals as a multivariate time series The cross correlations can then be compared with ±2/ N This is not the same form of prewhitening as in Chapter 8 The multivariate model φ(b)y t = θ(b)ǫ t is equivalent to diag(det(φ(b)))y t = adj(φ(b))θ(b)ǫ t Therefore the corresponding univariate models will have much higher order, so although this approach is often used in the literature: Don t use this approach! 11
18 Example Muskrat and Mink skins traded 12
19 Raw data (maybe not exactly as in the paper) Skins traded Muskrat Mink
20 Stationary and Gaussian data Skins traded log transformed and muskrat data differenced Muskrat Mink
21 w1 SACF w1 and w2 15 ACF w2 and w1 w2 ACF Lag Lag w1 SPACF w1 and w Partial ACF w2 and w1 w Lag Lag Partial ACF
22 Yule-Walker Estimates > fit.ar3yw <- ar(mmdata.tr, order=3) > fit.ar3yw$ar[1,, # lag 1 [,1 [,2 w1 [1, [2, > fit.ar3yw$ar[2,, # lag 2 [,1 [,2 [1, w2 and w1 [2, > fit.ar3yw$ar[3,, # lag 3 [,1 [,2 [1, [2, Lag > acf(fit.ar3yw$resid) ACF ACF Multivariate Series : fit.ar3yw$resid w1 and w w Lag 16
23 Maximum likelihood estimates > ## Load module > module(finmetrics) S+FinMetrics Version for Linux : 2005 > ## Means: > colmeans(mmdata.tr) w1 w > ## Center non-differences data: > tmp.dat <- mmdata.tr > tmp.dat[,2 <- tmp.dat[,2 - mean(tmp.dat[,2) > colmeans(tmp.dat) w1 w e-15 17
24 Maximum likelihood estimates (cont nd) > mgarch( -1 + ar(3), dvec(0,0), series=tmp.dat, armatype="full")...[deleted... Convergence reached. Call: mgarch(formula.mean = -1 + ar(3), formula.var = dvec(0, 0), series = tmp.dat, armatype = "full") Mean Equation: -1 + ar(3) Conditional Variance Equation: dvec(0, 0) Coefficients: AR(1; 1, 1)
25 Maximum likelihood estimates (cont nd) Coefficients: AR(1; 1, 1) AR(1; 2, 1) AR(1; 1, 2) AR(1; 2, 2) AR(2; 1, 1) AR(2; 2, 1) AR(2; 1, 2) AR(2; 2, 2) AR(3; 1, 1) AR(3; 2, 1) AR(3; 1, 2) AR(3; 2, 2) A(1, 1) A(2, 1) A(2, 2)
26 Model Validation For the individual residual series; all the methods from Chapter 6 in the book with the extension for the cross correlation as mentioned in Chapter 8 in the book 20
27 Forecasting The model: Y t+l = φ 1 Y t+l φ p Y t+l p +ǫ t+l θ 1 ǫ t+l 1... θ q ǫ t+l q 1-step: Ŷ t+1 t = φ 1 Y t φ p Y t+1 p +0 θ 1 ǫ t θ q ǫ t+1 q 2-step: Ŷ t+2 t = φ 1 Ŷ t+2 1 t φ p Y t+2 p + 0 θ θ q ǫ t+2 q and so on... in S-PLUS: > predict(fit.ml, 10) # fit.ml from mgarch() above However, this does not calculate the variance-covariance matrix of the forecast errors use the hint given in the text book. 21
Time Series Analysis
Time Series Analysis hm@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby 1 Outline of the lecture Input-Output systems The z-transform important issues
More informationRoss Bettinger, Analytical Consultant, Seattle, WA
ABSTRACT DYNAMIC REGRESSION IN ARIMA MODELING Ross Bettinger, Analytical Consultant, Seattle, WA Box-Jenkins time series models that contain exogenous predictor variables are called dynamic regression
More information3 Theory of stationary random processes
3 Theory of stationary random processes 3.1 Linear filters and the General linear process A filter is a transformation of one random sequence {U t } into another, {Y t }. A linear filter is a transformation
More informationTime Series I Time Domain Methods
Astrostatistics Summer School Penn State University University Park, PA 16802 May 21, 2007 Overview Filtering and the Likelihood Function Time series is the study of data consisting of a sequence of DEPENDENT
More informationEconometría 2: Análisis de series de Tiempo
Econometría 2: Análisis de series de Tiempo Karoll GOMEZ kgomezp@unal.edu.co http://karollgomez.wordpress.com Segundo semestre 2016 III. Stationary models 1 Purely random process 2 Random walk (non-stationary)
More informationLecture 3: Autoregressive Moving Average (ARMA) Models and their Practical Applications
Lecture 3: Autoregressive Moving Average (ARMA) Models and their Practical Applications Prof. Massimo Guidolin 20192 Financial Econometrics Winter/Spring 2018 Overview Moving average processes Autoregressive
More informationTime Series Analysis
Time Series Analysis hm@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby 1 Outline of the lecture Regression based methods, 1st part: Introduction (Sec.
More informationBooth School of Business, University of Chicago Business 41914, Spring Quarter 2017, Mr. Ruey S. Tsay Midterm
Booth School of Business, University of Chicago Business 41914, Spring Quarter 2017, Mr. Ruey S. Tsay Midterm Chicago Booth Honor Code: I pledge my honor that I have not violated the Honor Code during
More information9. Multivariate Linear Time Series (II). MA6622, Ernesto Mordecki, CityU, HK, 2006.
9. Multivariate Linear Time Series (II). MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture: Introduction to Time Series and Forecasting. P.J. Brockwell and R. A. Davis, Springer Texts
More information{ } Stochastic processes. Models for time series. Specification of a process. Specification of a process. , X t3. ,...X tn }
Stochastic processes Time series are an example of a stochastic or random process Models for time series A stochastic process is 'a statistical phenomenon that evolves in time according to probabilistic
More informationMultivariate ARMA Processes
LECTURE 8 Multivariate ARMA Processes A vector y(t) of n elements is said to follow an n-variate ARMA process of orders p and q if it satisfies the equation (1) A 0 y(t) + A 1 y(t 1) + + A p y(t p) = M
More informationBooth School of Business, University of Chicago Business 41914, Spring Quarter 2017, Mr. Ruey S. Tsay. Solutions to Midterm
Booth School of Business, University of Chicago Business 41914, Spring Quarter 017, Mr Ruey S Tsay Solutions to Midterm Problem A: (51 points; 3 points per question) Answer briefly the following questions
More informationTime Series Outlier Detection
Time Series Outlier Detection Tingyi Zhu July 28, 2016 Tingyi Zhu Time Series Outlier Detection July 28, 2016 1 / 42 Outline Time Series Basics Outliers Detection in Single Time Series Outlier Series Detection
More informationMarcel Dettling. Applied Time Series Analysis SS 2013 Week 05. ETH Zürich, March 18, Institute for Data Analysis and Process Design
Marcel Dettling Institute for Data Analysis and Process Design Zurich University of Applied Sciences marcel.dettling@zhaw.ch http://stat.ethz.ch/~dettling ETH Zürich, March 18, 2013 1 Basics of Modeling
More informationBasics: Definitions and Notation. Stationarity. A More Formal Definition
Basics: Definitions and Notation A Univariate is a sequence of measurements of the same variable collected over (usually regular intervals of) time. Usual assumption in many time series techniques is that
More informationBooth School of Business, University of Chicago Business 41914, Spring Quarter 2013, Mr. Ruey S. Tsay. Midterm
Booth School of Business, University of Chicago Business 41914, Spring Quarter 2013, Mr. Ruey S. Tsay Midterm Chicago Booth Honor Code: I pledge my honor that I have not violated the Honor Code during
More informationSome Time-Series Models
Some Time-Series Models Outline 1. Stochastic processes and their properties 2. Stationary processes 3. Some properties of the autocorrelation function 4. Some useful models Purely random processes, random
More informationSTAT 443 Final Exam Review. 1 Basic Definitions. 2 Statistical Tests. L A TEXer: W. Kong
STAT 443 Final Exam Review L A TEXer: W Kong 1 Basic Definitions Definition 11 The time series {X t } with E[X 2 t ] < is said to be weakly stationary if: 1 µ X (t) = E[X t ] is independent of t 2 γ X
More informationFE570 Financial Markets and Trading. Stevens Institute of Technology
FE570 Financial Markets and Trading Lecture 5. Linear Time Series Analysis and Its Applications (Ref. Joel Hasbrouck - Empirical Market Microstructure ) Steve Yang Stevens Institute of Technology 9/25/2012
More informationA time series is called strictly stationary if the joint distribution of every collection (Y t
5 Time series A time series is a set of observations recorded over time. You can think for example at the GDP of a country over the years (or quarters) or the hourly measurements of temperature over a
More informationStat 5100 Handout #12.e Notes: ARIMA Models (Unit 7) Key here: after stationary, identify dependence structure (and use for forecasting)
Stat 5100 Handout #12.e Notes: ARIMA Models (Unit 7) Key here: after stationary, identify dependence structure (and use for forecasting) (overshort example) White noise H 0 : Let Z t be the stationary
More informationUnivariate Time Series Analysis; ARIMA Models
Econometrics 2 Fall 24 Univariate Time Series Analysis; ARIMA Models Heino Bohn Nielsen of4 Outline of the Lecture () Introduction to univariate time series analysis. (2) Stationarity. (3) Characterizing
More informationγ 0 = Var(X i ) = Var(φ 1 X i 1 +W i ) = φ 2 1γ 0 +σ 2, which implies that we must have φ 1 < 1, and γ 0 = σ2 . 1 φ 2 1 We may also calculate for j 1
4.2 Autoregressive (AR) Moving average models are causal linear processes by definition. There is another class of models, based on a recursive formulation similar to the exponentially weighted moving
More informationLecture 1: Fundamental concepts in Time Series Analysis (part 2)
Lecture 1: Fundamental concepts in Time Series Analysis (part 2) Florian Pelgrin University of Lausanne, École des HEC Department of mathematics (IMEA-Nice) Sept. 2011 - Jan. 2012 Florian Pelgrin (HEC)
More informationSTA 6857 VAR, VARMA, VARMAX ( 5.7)
STA 6857 VAR, VARMA, VARMAX ( 5.7) Outline 1 Multivariate Time Series Modeling 2 VAR 3 VARIMA/VARMAX Arthur Berg STA 6857 VAR, VARMA, VARMAX ( 5.7) 2/ 16 Outline 1 Multivariate Time Series Modeling 2 VAR
More informationModule 3. Descriptive Time Series Statistics and Introduction to Time Series Models
Module 3 Descriptive Time Series Statistics and Introduction to Time Series Models Class notes for Statistics 451: Applied Time Series Iowa State University Copyright 2015 W Q Meeker November 11, 2015
More informationEconometrics II Heij et al. Chapter 7.1
Chapter 7.1 p. 1/2 Econometrics II Heij et al. Chapter 7.1 Linear Time Series Models for Stationary data Marius Ooms Tinbergen Institute Amsterdam Chapter 7.1 p. 2/2 Program Introduction Modelling philosophy
More informationNew Introduction to Multiple Time Series Analysis
Helmut Lütkepohl New Introduction to Multiple Time Series Analysis With 49 Figures and 36 Tables Springer Contents 1 Introduction 1 1.1 Objectives of Analyzing Multiple Time Series 1 1.2 Some Basics 2
More informationLecture # 37. Prof. John W. Sutherland. Nov. 28, 2005
Lecture # 37 Prof. John W. Sutherland Nov. 28, 2005 Linear Regression 8 y 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 x Modeling To describe the data above, propose the model: y = B 0 + B 1 x + ε Fitted model will
More informationCovariance Stationary Time Series. Example: Independent White Noise (IWN(0,σ 2 )) Y t = ε t, ε t iid N(0,σ 2 )
Covariance Stationary Time Series Stochastic Process: sequence of rv s ordered by time {Y t } {...,Y 1,Y 0,Y 1,...} Defn: {Y t } is covariance stationary if E[Y t ]μ for all t cov(y t,y t j )E[(Y t μ)(y
More informationIdentifiability, Invertibility
Identifiability, Invertibility Defn: If {ǫ t } is a white noise series and µ and b 0,..., b p are constants then X t = µ + b 0 ǫ t + b ǫ t + + b p ǫ t p is a moving average of order p; write MA(p). Q:
More informationEmpirical Market Microstructure Analysis (EMMA)
Empirical Market Microstructure Analysis (EMMA) Lecture 3: Statistical Building Blocks and Econometric Basics Prof. Dr. Michael Stein michael.stein@vwl.uni-freiburg.de Albert-Ludwigs-University of Freiburg
More informationClassic Time Series Analysis
Classic Time Series Analysis Concepts and Definitions Let Y be a random number with PDF f Y t ~f,t Define t =E[Y t ] m(t) is known as the trend Define the autocovariance t, s =COV [Y t,y s ] =E[ Y t t
More informationModule 4. Stationary Time Series Models Part 1 MA Models and Their Properties
Module 4 Stationary Time Series Models Part 1 MA Models and Their Properties Class notes for Statistics 451: Applied Time Series Iowa State University Copyright 2015 W. Q. Meeker. February 14, 2016 20h
More informationMultivariate Time Series: VAR(p) Processes and Models
Multivariate Time Series: VAR(p) Processes and Models A VAR(p) model, for p > 0 is X t = φ 0 + Φ 1 X t 1 + + Φ p X t p + A t, where X t, φ 0, and X t i are k-vectors, Φ 1,..., Φ p are k k matrices, with
More informationAdvanced Econometrics
Based on the textbook by Verbeek: A Guide to Modern Econometrics Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna May 2, 2013 Outline Univariate
More informationLecture 2: Univariate Time Series
Lecture 2: Univariate Time Series Analysis: Conditional and Unconditional Densities, Stationarity, ARMA Processes Prof. Massimo Guidolin 20192 Financial Econometrics Spring/Winter 2017 Overview Motivation:
More informationStatistics 349(02) Review Questions
Statistics 349(0) Review Questions I. Suppose that for N = 80 observations on the time series { : t T} the following statistics were calculated: _ x = 10.54 C(0) = 4.99 In addition the sample autocorrelation
More informationLINEAR STOCHASTIC MODELS
LINEAR STOCHASTIC MODELS Let {x τ+1,x τ+2,...,x τ+n } denote n consecutive elements from a stochastic process. If their joint distribution does not depend on τ, regardless of the size of n, then the process
More informationForecasting using R. Rob J Hyndman. 2.4 Non-seasonal ARIMA models. Forecasting using R 1
Forecasting using R Rob J Hyndman 2.4 Non-seasonal ARIMA models Forecasting using R 1 Outline 1 Autoregressive models 2 Moving average models 3 Non-seasonal ARIMA models 4 Partial autocorrelations 5 Estimation
More information18.S096 Problem Set 4 Fall 2013 Time Series Due Date: 10/15/2013
18.S096 Problem Set 4 Fall 2013 Time Series Due Date: 10/15/2013 1. Covariance Stationary AR(2) Processes Suppose the discrete-time stochastic process {X t } follows a secondorder auto-regressive process
More informationDiscrete time processes
Discrete time processes Predictions are difficult. Especially about the future Mark Twain. Florian Herzog 2013 Modeling observed data When we model observed (realized) data, we encounter usually the following
More informationChapter 4: Models for Stationary Time Series
Chapter 4: Models for Stationary Time Series Now we will introduce some useful parametric models for time series that are stationary processes. We begin by defining the General Linear Process. Let {Y t
More informationTime Series Analysis
Time Series Analysis hm@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby 1 Outline of the lecture State space models, 1st part: Model: Sec. 10.1 The
More informationINDIAN INSTITUTE OF SCIENCE STOCHASTIC HYDROLOGY. Lecture -33 Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc.
INDIAN INSTITUTE OF SCIENCE STOCHASTIC HYDROLOGY Lecture -33 Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc. Summary of the previous lecture Regression on Principal components
More informationUnivariate ARIMA Models
Univariate ARIMA Models ARIMA Model Building Steps: Identification: Using graphs, statistics, ACFs and PACFs, transformations, etc. to achieve stationary and tentatively identify patterns and model components.
More informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY & Contents PREFACE xiii 1 1.1. 1.2. Difference Equations First-Order Difference Equations 1 /?th-order Difference
More informationECON/FIN 250: Forecasting in Finance and Economics: Section 6: Standard Univariate Models
ECON/FIN 250: Forecasting in Finance and Economics: Section 6: Standard Univariate Models Patrick Herb Brandeis University Spring 2016 Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN
More informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY PREFACE xiii 1 Difference Equations 1.1. First-Order Difference Equations 1 1.2. pth-order Difference Equations 7
More information1 Linear Difference Equations
ARMA Handout Jialin Yu 1 Linear Difference Equations First order systems Let {ε t } t=1 denote an input sequence and {y t} t=1 sequence generated by denote an output y t = φy t 1 + ε t t = 1, 2,... with
More information2. Multivariate ARMA
2. Multivariate ARMA JEM 140: Quantitative Multivariate Finance IES, Charles University, Prague Summer 2018 JEM 140 () 2. Multivariate ARMA Summer 2018 1 / 19 Multivariate AR I Let r t = (r 1t,..., r kt
More information1 Class Organization. 2 Introduction
Time Series Analysis, Lecture 1, 2018 1 1 Class Organization Course Description Prerequisite Homework and Grading Readings and Lecture Notes Course Website: http://www.nanlifinance.org/teaching.html wechat
More informationFORECASTING SUGARCANE PRODUCTION IN INDIA WITH ARIMA MODEL
FORECASTING SUGARCANE PRODUCTION IN INDIA WITH ARIMA MODEL B. N. MANDAL Abstract: Yearly sugarcane production data for the period of - to - of India were analyzed by time-series methods. Autocorrelation
More informationCh 8. MODEL DIAGNOSTICS. Time Series Analysis
Model diagnostics is concerned with testing the goodness of fit of a model and, if the fit is poor, suggesting appropriate modifications. We shall present two complementary approaches: analysis of residuals
More informationTime Series Examples Sheet
Lent Term 2001 Richard Weber Time Series Examples Sheet This is the examples sheet for the M. Phil. course in Time Series. A copy can be found at: http://www.statslab.cam.ac.uk/~rrw1/timeseries/ Throughout,
More informationDynamic Time Series Regression: A Panacea for Spurious Correlations
International Journal of Scientific and Research Publications, Volume 6, Issue 10, October 2016 337 Dynamic Time Series Regression: A Panacea for Spurious Correlations Emmanuel Alphonsus Akpan *, Imoh
More informationChapter 2: Unit Roots
Chapter 2: Unit Roots 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and undeconometrics II. Unit Roots... 3 II.1 Integration Level... 3 II.2 Nonstationarity
More informationProf. Dr. Roland Füss Lecture Series in Applied Econometrics Summer Term Introduction to Time Series Analysis
Introduction to Time Series Analysis 1 Contents: I. Basics of Time Series Analysis... 4 I.1 Stationarity... 5 I.2 Autocorrelation Function... 9 I.3 Partial Autocorrelation Function (PACF)... 14 I.4 Transformation
More informationTime Series Analysis
Time Series Analysis Christopher Ting http://mysmu.edu.sg/faculty/christophert/ christopherting@smu.edu.sg Quantitative Finance Singapore Management University March 3, 2017 Christopher Ting Week 9 March
More informationInternational Journal of Advancement in Physical Sciences, Volume 4, Number 2, 2012
International Journal of Advancement in Physical Sciences, Volume, Number, RELIABILIY IN HE ESIMAES AND COMPLIANCE O INVERIBILIY CONDIION OF SAIONARY AND NONSAIONARY IME SERIES MODELS Usoro, A. E. and
More informationNANYANG TECHNOLOGICAL UNIVERSITY SEMESTER II EXAMINATION MAS451/MTH451 Time Series Analysis TIME ALLOWED: 2 HOURS
NANYANG TECHNOLOGICAL UNIVERSITY SEMESTER II EXAMINATION 2012-2013 MAS451/MTH451 Time Series Analysis May 2013 TIME ALLOWED: 2 HOURS INSTRUCTIONS TO CANDIDATES 1. This examination paper contains FOUR (4)
More informationTime Series: Theory and Methods
Peter J. Brockwell Richard A. Davis Time Series: Theory and Methods Second Edition With 124 Illustrations Springer Contents Preface to the Second Edition Preface to the First Edition vn ix CHAPTER 1 Stationary
More informationVector autoregressions, VAR
1 / 45 Vector autoregressions, VAR Chapter 2 Financial Econometrics Michael Hauser WS17/18 2 / 45 Content Cross-correlations VAR model in standard/reduced form Properties of VAR(1), VAR(p) Structural VAR,
More informationVector Auto-Regressive Models
Vector Auto-Regressive Models Laurent Ferrara 1 1 University of Paris Nanterre M2 Oct. 2018 Overview of the presentation 1. Vector Auto-Regressions Definition Estimation Testing 2. Impulse responses functions
More informationVector autoregressive Moving Average Process. Presented by Muhammad Iqbal, Amjad Naveed and Muhammad Nadeem
Vector autoregressive Moving Average Process Presented by Muhammad Iqbal, Amjad Naveed and Muhammad Nadeem Road Map 1. Introduction 2. Properties of MA Finite Process 3. Stationarity of MA Process 4. VARMA
More informationVAR Models and Applications
VAR Models and Applications Laurent Ferrara 1 1 University of Paris West M2 EIPMC Oct. 2016 Overview of the presentation 1. Vector Auto-Regressions Definition Estimation Testing 2. Impulse responses functions
More informationExamination paper for TMA4285 Time Series Models
Department of Mathematical Sciences Examination paper for TMA4285 Series Models Academic contact during examination: Professor Jarle Tufto Phone: 99 70 55 19 Examination date: December 8, 2016 Examination
More informationTime Series Solutions HT 2009
Time Series Solutions HT 2009 1. Let {X t } be the ARMA(1, 1) process, X t φx t 1 = ɛ t + θɛ t 1, {ɛ t } WN(0, σ 2 ), where φ < 1 and θ < 1. Show that the autocorrelation function of {X t } is given by
More informationLesson 13: Box-Jenkins Modeling Strategy for building ARMA models
Lesson 13: Box-Jenkins Modeling Strategy for building ARMA models Facoltà di Economia Università dell Aquila umberto.triacca@gmail.com Introduction In this lesson we present a method to construct an ARMA(p,
More informationMultivariate Time Series Analysis and Its Applications [Tsay (2005), chapter 8]
1 Multivariate Time Series Analysis and Its Applications [Tsay (2005), chapter 8] Insights: Price movements in one market can spread easily and instantly to another market [economic globalization and internet
More information2. An Introduction to Moving Average Models and ARMA Models
. An Introduction to Moving Average Models and ARMA Models.1 White Noise. The MA(1) model.3 The MA(q) model..4 Estimation and forecasting of MA models..5 ARMA(p,q) models. The Moving Average (MA) models
More informationApplied time-series analysis
Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna October 18, 2011 Outline Introduction and overview Econometric Time-Series Analysis In principle,
More informationGeneralised AR and MA Models and Applications
Chapter 3 Generalised AR and MA Models and Applications 3.1 Generalised Autoregressive Processes Consider an AR1) process given by 1 αb)x t = Z t ; α < 1. In this case, the acf is, ρ k = α k for k 0 and
More informationIDENTIFICATION OF ARMA MODELS
IDENTIFICATION OF ARMA MODELS A stationary stochastic process can be characterised, equivalently, by its autocovariance function or its partial autocovariance function. It can also be characterised by
More informationChapter 12: An introduction to Time Series Analysis. Chapter 12: An introduction to Time Series Analysis
Chapter 12: An introduction to Time Series Analysis Introduction In this chapter, we will discuss forecasting with single-series (univariate) Box-Jenkins models. The common name of the models is Auto-Regressive
More informationEconometría 2: Análisis de series de Tiempo
Econometría 2: Análisis de series de Tiempo Karoll GOMEZ kgomezp@unal.edu.co http://karollgomez.wordpress.com Segundo semestre 2016 IX. Vector Time Series Models VARMA Models A. 1. Motivation: The vector
More informationNotes on Time Series Modeling
Notes on Time Series Modeling Garey Ramey University of California, San Diego January 17 1 Stationary processes De nition A stochastic process is any set of random variables y t indexed by t T : fy t g
More informationIntroduction to ARMA and GARCH processes
Introduction to ARMA and GARCH processes Fulvio Corsi SNS Pisa 3 March 2010 Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March 2010 1 / 24 Stationarity Strict stationarity: (X 1,
More informationTMA4285 December 2015 Time series models, solution.
Norwegian University of Science and Technology Department of Mathematical Sciences Page of 5 TMA4285 December 205 Time series models, solution. Problem a) (i) The slow decay of the ACF of z t suggest that
More informationCircle a single answer for each multiple choice question. Your choice should be made clearly.
TEST #1 STA 4853 March 4, 215 Name: Please read the following directions. DO NOT TURN THE PAGE UNTIL INSTRUCTED TO DO SO Directions This exam is closed book and closed notes. There are 31 questions. Circle
More informationAutoregressive Moving Average (ARMA) Models and their Practical Applications
Autoregressive Moving Average (ARMA) Models and their Practical Applications Massimo Guidolin February 2018 1 Essential Concepts in Time Series Analysis 1.1 Time Series and Their Properties Time series:
More informationMAT 3379 (Winter 2016) FINAL EXAM (SOLUTIONS)
MAT 3379 (Winter 2016) FINAL EXAM (SOLUTIONS) 15 April 2016 (180 minutes) Professor: R. Kulik Student Number: Name: This is closed book exam. You are allowed to use one double-sided A4 sheet of notes.
More informationTIME SERIES ANALYSIS. Forecasting and Control. Wiley. Fifth Edition GWILYM M. JENKINS GEORGE E. P. BOX GREGORY C. REINSEL GRETA M.
TIME SERIES ANALYSIS Forecasting and Control Fifth Edition GEORGE E. P. BOX GWILYM M. JENKINS GREGORY C. REINSEL GRETA M. LJUNG Wiley CONTENTS PREFACE TO THE FIFTH EDITION PREFACE TO THE FOURTH EDITION
More informationCointegrated VARIMA models: specification and. simulation
Cointegrated VARIMA models: specification and simulation José L. Gallego and Carlos Díaz Universidad de Cantabria. Abstract In this note we show how specify cointegrated vector autoregressive-moving average
More informationEcon 424 Time Series Concepts
Econ 424 Time Series Concepts Eric Zivot January 20 2015 Time Series Processes Stochastic (Random) Process { 1 2 +1 } = { } = sequence of random variables indexed by time Observed time series of length
More informationActa Universitatis Carolinae. Mathematica et Physica
Acta Universitatis Carolinae. Mathematica et Physica Jitka Zichová Some applications of time series models to financial data Acta Universitatis Carolinae. Mathematica et Physica, Vol. 52 (2011), No. 1,
More information5: MULTIVARATE STATIONARY PROCESSES
5: MULTIVARATE STATIONARY PROCESSES 1 1 Some Preliminary Definitions and Concepts Random Vector: A vector X = (X 1,..., X n ) whose components are scalarvalued random variables on the same probability
More informationEstimation and application of best ARIMA model for forecasting the uranium price.
Estimation and application of best ARIMA model for forecasting the uranium price. Medeu Amangeldi May 13, 2018 Capstone Project Superviser: Dongming Wei Second reader: Zhenisbek Assylbekov Abstract This
More informationARMA (and ARIMA) models are often expressed in backshift notation.
Backshift Notation ARMA (and ARIMA) models are often expressed in backshift notation. B is the backshift operator (also called the lag operator ). It operates on time series, and means back up by one time
More informationChapter 9: Forecasting
Chapter 9: Forecasting One of the critical goals of time series analysis is to forecast (predict) the values of the time series at times in the future. When forecasting, we ideally should evaluate the
More informationWe will only present the general ideas on how to obtain. follow closely the AR(1) and AR(2) cases presented before.
ACF and PACF of an AR(p) We will only present the general ideas on how to obtain the ACF and PACF of an AR(p) model since the details follow closely the AR(1) and AR(2) cases presented before. Recall that
More informationTime Series 2. Robert Almgren. Sept. 21, 2009
Time Series 2 Robert Almgren Sept. 21, 2009 This week we will talk about linear time series models: AR, MA, ARMA, ARIMA, etc. First we will talk about theory and after we will talk about fitting the models
More informationMODELING INFLATION RATES IN NIGERIA: BOX-JENKINS APPROACH. I. U. Moffat and A. E. David Department of Mathematics & Statistics, University of Uyo, Uyo
Vol.4, No.2, pp.2-27, April 216 MODELING INFLATION RATES IN NIGERIA: BOX-JENKINS APPROACH I. U. Moffat and A. E. David Department of Mathematics & Statistics, University of Uyo, Uyo ABSTRACT: This study
More informationELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process
Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Definition of stochastic process (random
More informationPart III Example Sheet 1 - Solutions YC/Lent 2015 Comments and corrections should be ed to
TIME SERIES Part III Example Sheet 1 - Solutions YC/Lent 2015 Comments and corrections should be emailed to Y.Chen@statslab.cam.ac.uk. 1. Let {X t } be a weakly stationary process with mean zero and let
More informationMCMC analysis of classical time series algorithms.
MCMC analysis of classical time series algorithms. mbalawata@yahoo.com Lappeenranta University of Technology Lappeenranta, 19.03.2009 Outline Introduction 1 Introduction 2 3 Series generation Box-Jenkins
More informationA SARIMAX coupled modelling applied to individual load curves intraday forecasting
A SARIMAX coupled modelling applied to individual load curves intraday forecasting Frédéric Proïa Workshop EDF Institut Henri Poincaré - Paris 05 avril 2012 INRIA Bordeaux Sud-Ouest Institut de Mathématiques
More informationSTAT 443 (Winter ) Forecasting
Winter 2014 TABLE OF CONTENTS STAT 443 (Winter 2014-1141) Forecasting Prof R Ramezan University of Waterloo L A TEXer: W KONG http://wwkonggithubio Last Revision: September 3, 2014 Table of Contents 1
More informationElements of Multivariate Time Series Analysis
Gregory C. Reinsel Elements of Multivariate Time Series Analysis Second Edition With 14 Figures Springer Contents Preface to the Second Edition Preface to the First Edition vii ix 1. Vector Time Series
More informationARMA models with time-varying coefficients. Periodic case.
ARMA models with time-varying coefficients. Periodic case. Agnieszka Wy lomańska Hugo Steinhaus Center Wroc law University of Technology ARMA models with time-varying coefficients. Periodic case. 1 Some
More information