distributed approximately according to white noise. Likewise, for general ARMA(p,q), the residuals can be expressed as
|
|
- Magdalene Wilkins
- 5 years ago
- Views:
Transcription
1 library(forecast) log_ap <- log(airpassengers) fit <- auto.arima(log_ap, ic="aicc") 7 Model diagnostics The model diagnostics final step in the three-step procedure for time series model building suggested by (and attributed to) the Bo and Jenkins (1970): Identification where we look at the data (with ACF, PACF, differencing, lag plots, periodogram...), and also any subject-specific information about the data, to suggest subclasses of parsimonious models we might consider. Estimation where we fit the chosen model, or models of interest, to the data. Diagnostic checking where we study how the model fits the data, and look for any signs of an inadequate fit using formal hypothesis tests. Thestepsoverlap,asisthecasewithinformationcriteriawhichcanonlybefound after estimation of the parameters. Please bear in mind that this procedure was suggested when computing was epensive, and even then the procedure was meant to be iterative; the most adequate model may not be found in one iteration. 7.1 Residuals Let us net take a closer look at the residuals of the ARMA models. Notice that in the time series contet, there is no natural decomposition of the data to fitted values and the residuals. Please keep this mind when using the R functions fitted and resid with time series models; see Figure 32. Consider first an AR(p). If the data is really from AR(p), and if the estimated parameters are close to their true values, we should have the residuals e i = ( i ˆµ) ˆφ j ( i j ˆµ), i = p+1,...,n distributed approimately according to white noise. Likewise, for general ARMA(p,q), the residuals can be epressed as e i = i E[X i X 1 = 1,...,X i 1 = i 1 ], where the conditional epectation is with respect to the process (X i ) following the ARMA with the estimated parameters (ˆφ,ˆθ,ˆσ 2,ˆµ). The first step in the residual analysis is to look at the ACF and PACF of the residuals, whether they appear similar to those calculated from white noise. 51
2 Figure 32: Residuals of a linear model (top) and residuals of an AR(1) with φ 1 = 3/4 (bottom). 7.2 Residual tests Definition 7.1 (Bo-Pierce test). The Bo-Pierce statistic is calculated for some p+q < K n, K Q = n rj, 2 wherer j isthesampleautocorrelationoftheresidualseries.ifthemodeliscorrect, then Q is approimately distributed as χ 2 K p q.13 Definition 7.2 (Ljung-Bo test). The Ljung-Bo test is eactly as Bo-Pierce, but with a modified statistic Q = n K n+2 n j r2 j, which has been found empirically to be often a more accurate approimation of χ 2 K p q. Eample 7.3. Ljung-Bo with MA(3) fitted to simulated AR(2). 13. That is, the null (that the model is correct) is rejected if Q is greater than the 1 α quantile of χ 2 K p q. 52
3 Standardized Residuals p values for Ljung Bo statistic Figure 33: You should not trust the Ljung-Bo statistic reported by R function tsdiag(fit)... n <- 80; q <- 3; p <- 0 <- arima.sim(model=list(ar=c(1/2, 1/3)), n) fit <- arima(, order=c(p,0,q)); e <- resid(fit); pval <- rep(na,10) for(lag in (p+q+1):10) { pval[lag] <- Bo.test(e, lag=lag, fitdf=p+q, type="ljung")$p.value } Remark 7.4. The R function tsdiag calculates the Ljung-Bo statistics with wrong degrees of freedom, not taking the number of parameters into account, leading into overestimated p-values! Remark 7.5. The Bo-Pierce and Ljung-Bo tests generally may fail to disqualify poorly fitting models with smaller data sets(cf. also Brockwell and Davis, p. 312). Thismeansthatfailingtorejectthenullshouldnotbetakenasastrongindication that the model is necessarily the most adequate one. (Eample 7.3 with n = 200 often leads into clear rejection of the null.) 53
4 Figure 34: The incorrect statistics calculated by tsdiag (), and the correct Bo- Ljung ( ) and Bo-Pierce (o) statistics. 7.3 Overfitting Sometimes, it can be instructive to fit higher order model to reassure that the chosen model should, in fact, be sufficient. If the preliminary model is, say, AR(2), we may try to fit AR(3), and inspect the coefficients of the AR(3). If the first two coefficients of the fitted do not significantly differ from those of the AR(2), and the third does not significantly differ from zero, this overfitting procedure can given further support to our choice of the AR(2). Eample 7.6. Suppose we have fitted an AR(1) to the data, and both residual analysis and information criteria support our choice. We fit AR(2) and compare the coefficients. ˆφ 1 ±s.d. ˆφ2 ±s.d. ˆσ 2 What would you conclude? AR(1) ± AR(2) ± ± Forecasting Forecasting in time-series models relies on calculating forecasts from the model with the estimated parameters, in the mean square sense. This means calculating the conditional epectations ˆ i+h 1:i := E[X i+h X 1 = 1,...,X i = i ], h > 1, where the conditional epectation is with respect to the process (X i ) following the ARMA with the estimated parameters. In order to have confidence invervals of the prediction, we should consider the conditional distribution of X i+h given X 1 = 1,...,X i = i. Under the 54
5 assumption of Gaussian white noise, we only need to calculate only the predictive variance v i+h 1:i = Var(X i+h X 1 = 1,...,X i = i ). 14 Remark 8.1. Note that these confidence intervals may be optimistic because of this Gaussian assumption heavier tailed noise might well imply wider confidence intervals. Note also that the parameter uncertainty is not taken into account, so the prediction confidence intervals may be optimistic. 8.1 Autoregressive process In the case of AR(p), we already discussed noted in Section 7.1 that the one-step predictors come directly from the definition for i > p ˆ i+1 1:i := E[X i+1 X 1 = 1,...,X i = i ] = ˆµ+ ˆφ j ( i j+1 ˆµ). The conditional variance is just the variance of W i+1, that is, ˆσ 2. For the rest of this section, we assume ˆµ = 0 to simplify epressions if (X i) was the non-centred AR(p) process, we consider ˆX i = X i ˆµ and so forth. The two-step predictor can be calculated as [ ˆ i+2 1:i = E W i+2 + ] ˆφ j X i+2 j X 1 = 1,...,X i = i = ˆφ 1 E[X i+1 X 1 = 1,...,X i = i ]+ = ˆφ 1ˆ i+1 1:i + ˆφ j i+2 j, j=2 ˆφ j i+2 j where the latter two sums equal zero if p = 1. If we denote ˆ j 1:i = j if 1 j i, we have the general result for i p and h 1 j=2 ˆ i+h 1:i = ˆφ jˆ i+h j 1:i. This just means that we calculate ˆ i+h 1:i from the previous values and previous predictions via the AR(p) definition, ignoring the noise. Remark 8.2. For any stationary AR(p), the predictors ˆ i+h 1:i converge to ˆµ as h increases (at an eponential rate). 14. If the prediction (ˆ i+1 1:i,...,ˆ i+h 1:i ) is considered simultaneously, then one could consider also the conditional covariance matri. 55
6 Eample 8.3. The variance of the prediction of AR(1) satisfies for i 2 and h 2, v i+h 1:i = Var(X 2 i+h X 1 = 1,...,X i = i ) = ˆσ 2 + ˆφ 2 1Var(X i+h 1 X 1 = 1,...,X i = 1 ) h 1 = = ˆσ 2 (ˆφ 2 1) k. k=0 We observe that v i+h 1:i ˆσ2 as h increases. 1 ˆρ 2 1 Any stationary AR(p) behaves similarly, that is, the variance of the predictor stabilises to the stationary variance (at eponential rate). 8.2 General ARIMA For general ARIMA process, closed form epressions are not available, but both prediction and the variance of the prediction (under Gaussian assumption) can be calculated numerically. In case of a regular stationary ARMA (Condition 4.26), we could write, in principle, X i = β j X i 1 +W i, where the constants β j converge to zero eponentially fast. Therefore, it comes by no surprise that the long-horizon predictions behave similarly as in the AR(p) case, that is, ˆ i+h 1:i ˆµ and v i+h 1:i γ 0 (at an eponential rate). When the model involves differencing, we can consider the model as an non-stationary ARMA, and do the prediction and calculate the variance of the prediction with the same numerical tools as for stationary ARMA. However, the model is in this case non-stationary, and the predictive variances increase towards infinity as h increases. Eample 8.4. Prediction from ARIMA(1,0,0)(1,0,0) 12 fitted to NY births data (from 1948) with a linear trend regressor. h <- 48; t <- time(b) f1 <- arima(b, reg=t, order=c(1,0,0), seasonal=list(order=c(1,0,0), season=12)) p1 <- predict(f1, h, newreg=t[n]+(1:h)/12) m1 <- p1$pred; s1 <- p1$se ts.plot(b, m1, m1+1.96*s1, m1-1.96*s1, col=c(1,2,2,2), lty=c(1,1,2,2)) 56
7 Figure 35: Predictions of ARIMA(1,0,0)(1,0,0) 12 of Eample 8.4 (top) and a similarly fitted non-stationary ARIMA(1,1,0)(1,0,0) 12 (bottom). 9 Spectrum of a stationary process The periodogram was a transform calculated from a finite length vector. We net consider a slightly more abstract concept of the spectrum of a stationary process. There are two key differences: the process (X i ) i Z is of infinite length, and the process can take multiple realisations. 9.1 Spectral density The spectral density of a stationary process is, in fact, a discrete-time Fourier transform (DTFT) of the autocovariance function. It is analogous to DFT, but with infinite number of frequencies. Definition 9.1 (Spectral density). Suppose that (X t ) is a stationary process with autocovariance sequence satisfying γ k <. k=0 The spectral density of the process (X t ) (or equivalently of the autocovariance (γ k )) is the function f(λ) = 1 2π k= γ k e ikλ, for λ ( π,π]. 57
Figure 29: AR model fit into speech sample ah (top), the residual, and the random sample of the model (bottom).
Original 0.4 0.0 0.4 ACF 0.5 0.0 0.5 1.0 0 500 1000 1500 2000 0 50 100 150 200 Residual 0.05 0.05 ACF 0 500 1000 1500 2000 0 50 100 150 200 Generated 0.4 0.0 0.4 ACF 0.5 0.0 0.5 1.0 0 500 1000 1500 2000
More informationChapter 8: Model Diagnostics
Chapter 8: Model Diagnostics Model diagnostics involve checking how well the model fits. If the model fits poorly, we consider changing the specification of the model. A major tool of model diagnostics
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 informationat least 50 and preferably 100 observations should be available to build a proper model
III Box-Jenkins Methods 1. Pros and Cons of ARIMA Forecasting a) need for data at least 50 and preferably 100 observations should be available to build a proper model used most frequently for hourly or
More informationMinitab Project Report - Assignment 6
.. Sunspot data Minitab Project Report - Assignment Time Series Plot of y Time Series Plot of X y X 7 9 7 9 The data have a wavy pattern. However, they do not show any seasonality. There seem to be an
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 informationAdvanced Econometrics
Advanced Econometrics Marco Sunder Nov 04 2010 Marco Sunder Advanced Econometrics 1/ 25 Contents 1 2 3 Marco Sunder Advanced Econometrics 2/ 25 Music Marco Sunder Advanced Econometrics 3/ 25 Music Marco
More informationModelling using ARMA processes
Modelling using ARMA processes Step 1. ARMA model identification; Step 2. ARMA parameter estimation Step 3. ARMA model selection ; Step 4. ARMA model checking; Step 5. forecasting from ARMA models. 33
More informationSTAT 520 FORECASTING AND TIME SERIES 2013 FALL Homework 05
STAT 520 FORECASTING AND TIME SERIES 2013 FALL Homework 05 1. ibm data: The random walk model of first differences is chosen to be the suggest model of ibm data. That is (1 B)Y t = e t where e t is a mean
More informationComment about AR spectral estimation Usually an estimate is produced by computing the AR theoretical spectrum at (ˆφ, ˆσ 2 ). With our Monte Carlo
Comment aout AR spectral estimation Usually an estimate is produced y computing the AR theoretical spectrum at (ˆφ, ˆσ 2 ). With our Monte Carlo simulation approach, for every draw (φ,σ 2 ), we can compute
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 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 information5 Autoregressive-Moving-Average Modeling
5 Autoregressive-Moving-Average Modeling 5. Purpose. Autoregressive-moving-average (ARMA models are mathematical models of the persistence, or autocorrelation, in a time series. ARMA models are widely
More informationTime Series Analysis -- An Introduction -- AMS 586
Time Series Analysis -- An Introduction -- AMS 586 1 Objectives of time series analysis Data description Data interpretation Modeling Control Prediction & Forecasting 2 Time-Series Data Numerical data
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 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 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 informationARIMA Modelling and Forecasting
ARIMA Modelling and Forecasting Economic time series often appear nonstationary, because of trends, seasonal patterns, cycles, etc. However, the differences may appear stationary. Δx t x t x t 1 (first
More informationAPPLIED ECONOMETRIC TIME SERIES 4TH EDITION
APPLIED ECONOMETRIC TIME SERIES 4TH EDITION Chapter 2: STATIONARY TIME-SERIES MODELS WALTER ENDERS, UNIVERSITY OF ALABAMA Copyright 2015 John Wiley & Sons, Inc. Section 1 STOCHASTIC DIFFERENCE EQUATION
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 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 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 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 informationTIME SERIES ANALYSIS AND FORECASTING USING THE STATISTICAL MODEL ARIMA
CHAPTER 6 TIME SERIES ANALYSIS AND FORECASTING USING THE STATISTICAL MODEL ARIMA 6.1. Introduction A time series is a sequence of observations ordered in time. A basic assumption in the time series analysis
More informationReview Session: Econometrics - CLEFIN (20192)
Review Session: Econometrics - CLEFIN (20192) Part II: Univariate time series analysis Daniele Bianchi March 20, 2013 Fundamentals Stationarity A time series is a sequence of random variables x t, t =
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 informationCHAPTER 8 MODEL DIAGNOSTICS. 8.1 Residual Analysis
CHAPTER 8 MODEL DIAGNOSTICS We have now discussed methods for specifying models and for efficiently estimating the parameters in those models. Model diagnostics, or model criticism, is concerned with testing
More informationIntroduction to Time Series Analysis. Lecture 11.
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
More informationLesson 14: Model Checking
Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@univaq.it Model checking Given the time series {x t ; t = 1,..., T } suppose that we have estimated
More informationCh 6. Model Specification. Time Series Analysis
We start to build ARIMA(p,d,q) models. The subjects include: 1 how to determine p, d, q for a given series (Chapter 6); 2 how to estimate the parameters (φ s and θ s) of a specific ARIMA(p,d,q) model (Chapter
More informationLab: Box-Jenkins Methodology - US Wholesale Price Indicator
Lab: Box-Jenkins Methodology - US Wholesale Price Indicator In this lab we explore the Box-Jenkins methodology by applying it to a time-series data set comprising quarterly observations of the US Wholesale
More informationMidterm Suggested Solutions
CUHK Dept. of Economics Spring 2011 ECON 4120 Sung Y. Park Midterm Suggested Solutions Q1 (a) In time series, autocorrelation measures the correlation between y t and its lag y t τ. It is defined as. ρ(τ)
More informationITSM-R Reference Manual
ITSM-R Reference Manual George Weigt February 11, 2018 1 Contents 1 Introduction 3 1.1 Time series analysis in a nutshell............................... 3 1.2 White Noise Variance.....................................
More informationEASTERN MEDITERRANEAN UNIVERSITY ECON 604, FALL 2007 DEPARTMENT OF ECONOMICS MEHMET BALCILAR ARIMA MODELS: IDENTIFICATION
ARIMA MODELS: IDENTIFICATION A. Autocorrelations and Partial Autocorrelations 1. Summary of What We Know So Far: a) Series y t is to be modeled by Box-Jenkins methods. The first step was to convert y t
More informationLecture 4a: ARMA Model
Lecture 4a: ARMA Model 1 2 Big Picture Most often our goal is to find a statistical model to describe real time series (estimation), and then predict the future (forecasting) One particularly popular model
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 informationUniversity of Oxford. Statistical Methods Autocorrelation. Identification and Estimation
University of Oxford Statistical Methods Autocorrelation Identification and Estimation Dr. Órlaith Burke Michaelmas Term, 2011 Department of Statistics, 1 South Parks Road, Oxford OX1 3TG Contents 1 Model
More informationModeling and forecasting global mean temperature time series
Modeling and forecasting global mean temperature time series April 22, 2018 Abstract: An ARIMA time series model was developed to analyze the yearly records of the change in global annual mean surface
More informationProblem Set 2: Box-Jenkins methodology
Problem Set : Box-Jenkins methodology 1) For an AR1) process we have: γ0) = σ ε 1 φ σ ε γ0) = 1 φ Hence, For a MA1) process, p lim R = φ γ0) = 1 + θ )σ ε σ ε 1 = γ0) 1 + θ Therefore, p lim R = 1 1 1 +
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 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 informationEconometrics I: Univariate Time Series Econometrics (1)
Econometrics I: Dipartimento di Economia Politica e Metodi Quantitativi University of Pavia Overview of the Lecture 1 st EViews Session VI: Some Theoretical Premises 2 Overview of the Lecture 1 st EViews
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 informationAkaike criterion: Kullback-Leibler discrepancy
Model choice. Akaike s criterion Akaike criterion: Kullback-Leibler discrepancy Given a family of probability densities {f ( ; ψ), ψ Ψ}, Kullback-Leibler s index of f ( ; ψ) relative to f ( ; θ) is (ψ
More informationAR, MA and ARMA models
AR, MA and AR by Hedibert Lopes P Based on Tsay s Analysis of Financial Time Series (3rd edition) P 1 Stationarity 2 3 4 5 6 7 P 8 9 10 11 Outline P Linear Time Series Analysis and Its Applications For
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 informationSuan Sunandha Rajabhat University
Forecasting Exchange Rate between Thai Baht and the US Dollar Using Time Series Analysis Kunya Bowornchockchai Suan Sunandha Rajabhat University INTRODUCTION The objective of this research is to forecast
More informationEconometric Forecasting
Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna October 1, 2014 Outline Introduction Model-free extrapolation Univariate time-series models Trend
More informationSTA 6857 ARIMA and SARIMA Models ( 3.8 and 3.9)
STA 6857 ARIMA and SARIMA Models ( 3.8 and 3.9) Outline 1 Building ARIMA Models 2 SARIMA 3 Homework 4c Arthur Berg STA 6857 ARIMA and SARIMA Models ( 3.8 and 3.9) 2/ 34 Outline 1 Building ARIMA Models
More informationWhite Noise Processes (Section 6.2)
White Noise Processes (Section 6.) Recall that covariance stationary processes are time series, y t, such. E(y t ) = µ for all t. Var(y t ) = σ for all t, σ < 3. Cov(y t,y t-τ ) = γ(τ) for all t and τ
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 informationSTA 6857 ARIMA and SARIMA Models ( 3.8 and 3.9) Outline. Return Rate. US Gross National Product
STA 6857 ARIMA and SARIMA Models ( 3.8 and 3.9) Outline 1 Building ARIMA Models 2 SARIMA 3 Homework 4c Arthur Berg STA 6857 ARIMA and SARIMA Models ( 3.8 and 3.9) 2/ 34 Return Rate Suppose x t is the value
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 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 informationEconometrics for Policy Analysis A Train The Trainer Workshop Oct 22-28, 2016 Organized by African Heritage Institution
Econometrics for Policy Analysis A Train The Trainer Workshop Oct 22-28, 2016 Organized by African Heritage Institution Delivered by Dr. Nathaniel E. Urama Department of Economics, University of Nigeria,
More informationCHAPTER 8 FORECASTING PRACTICE I
CHAPTER 8 FORECASTING PRACTICE I Sometimes we find time series with mixed AR and MA properties (ACF and PACF) We then can use mixed models: ARMA(p,q) These slides are based on: González-Rivera: Forecasting
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 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 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 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 informationStat 153 Time Series. Problem Set 4
Stat Time Series Problem Set Problem I generated 000 observations from the MA() model with parameter 0.7 using the following R function and then fitted the ARMA(, ) model to the data. order = c(, 0, )),
More informationDecision 411: Class 9. HW#3 issues
Decision 411: Class 9 Presentation/discussion of HW#3 Introduction to ARIMA models Rules for fitting nonseasonal models Differencing and stationarity Reading the tea leaves : : ACF and PACF plots Unit
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 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 informationECONOMETRIA II. CURSO 2009/2010 LAB # 3
ECONOMETRIA II. CURSO 2009/2010 LAB # 3 BOX-JENKINS METHODOLOGY The Box Jenkins approach combines the moving average and the autorregresive models. Although both models were already known, the contribution
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 informationSTAT Financial Time Series
STAT 6104 - Financial Time Series Chapter 4 - Estimation in the time Domain Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 1 / 46 Agenda 1 Introduction 2 Moment Estimates 3 Autoregressive Models (AR
More informationStochastic Modelling Solutions to Exercises on Time Series
Stochastic Modelling Solutions to Exercises on Time Series Dr. Iqbal Owadally March 3, 2003 Solutions to Elementary Problems Q1. (i) (1 0.5B)X t = Z t. The characteristic equation 1 0.5z = 0 does not have
More informationAnalysis. Components of a Time Series
Module 8: Time Series Analysis 8.2 Components of a Time Series, Detection of Change Points and Trends, Time Series Models Components of a Time Series There can be several things happening simultaneously
More informationThe Identification of ARIMA Models
APPENDIX 4 The Identification of ARIMA Models As we have established in a previous lecture, there is a one-to-one correspondence between the parameters of an ARMA(p, q) model, including the variance of
More informationFinal Examination 7/6/2011
The Islamic University of Gaza Faculty of Commerce Department of Economics & Applied Statistics Time Series Analysis - Dr. Samir Safi Spring Semester 211 Final Examination 7/6/211 Name: ID: INSTRUCTIONS:
More informationECON/FIN 250: Forecasting in Finance and Economics: Section 8: Forecast Examples: Part 1
ECON/FIN 250: Forecasting in Finance and Economics: Section 8: Forecast Examples: Part 1 Patrick Herb Brandeis University Spring 2016 Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN
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 informationThe autocorrelation and autocovariance functions - helpful tools in the modelling problem
The autocorrelation and autocovariance functions - helpful tools in the modelling problem J. Nowicka-Zagrajek A. Wy lomańska Institute of Mathematics and Computer Science Wroc law University of Technology,
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 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 informationAkaike criterion: Kullback-Leibler discrepancy
Model choice. Akaike s criterion Akaike criterion: Kullback-Leibler discrepancy Given a family of probability densities {f ( ; ), 2 }, Kullback-Leibler s index of f ( ; ) relativetof ( ; ) is Z ( ) =E
More informationFirstly, the dataset is cleaned and the years and months are separated to provide better distinction (sample below).
Project: Forecasting Sales Step 1: Plan Your Analysis Answer the following questions to help you plan out your analysis: 1. Does the dataset meet the criteria of a time series dataset? Make sure to explore
More informationPart II. Time Series
Part II Time Series 12 Introduction This Part is mainly a summary of the book of Brockwell and Davis (2002). Additionally the textbook Shumway and Stoffer (2010) can be recommended. 1 Our purpose is to
More information6. The econometrics of Financial Markets: Empirical Analysis of Financial Time Series. MA6622, Ernesto Mordecki, CityU, HK, 2006.
6. The econometrics of Financial Markets: Empirical Analysis of Financial Time Series MA6622, Ernesto Mordecki, CityU, HK, 2006. References for Lecture 5: Quantitative Risk Management. A. McNeil, R. Frey,
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 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 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 informationExercises - Time series analysis
Descriptive analysis of a time series (1) Estimate the trend of the series of gasoline consumption in Spain using a straight line in the period from 1945 to 1995 and generate forecasts for 24 months. Compare
More informationARMA MODELS Herman J. Bierens Pennsylvania State University February 23, 2009
1. Introduction Given a covariance stationary process µ ' E[ ], the Wold decomposition states that where U t ARMA MODELS Herman J. Bierens Pennsylvania State University February 23, 2009 with vanishing
More informationUnivariate linear models
Univariate linear models The specification process of an univariate ARIMA model is based on the theoretical properties of the different processes and it is also important the observation and interpretation
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 informationChapter 6: Model Specification for Time Series
Chapter 6: Model Specification for Time Series The ARIMA(p, d, q) class of models as a broad class can describe many real time series. Model specification for ARIMA(p, d, q) models involves 1. Choosing
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 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 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 informationRead Section 1.1, Examples of time series, on pages 1-8. These example introduce the book; you are not tested on them.
TS Module 1 Time series overview (The attached PDF file has better formatting.)! Model building! Time series plots Read Section 1.1, Examples of time series, on pages 1-8. These example introduce the book;
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 informationMultivariate Time Series
Multivariate Time Series Notation: I do not use boldface (or anything else) to distinguish vectors from scalars. Tsay (and many other writers) do. I denote a multivariate stochastic process in the form
More informationKernel-based portmanteau diagnostic test for ARMA time series models
STATISTICS RESEARCH ARTICLE Kernel-based portmanteau diagnostic test for ARMA time series models Esam Mahdi 1 * Received: 01 October 2016 Accepted: 07 February 2017 First Published: 21 February 2017 *Corresponding
More informationThe t-distribution. Patrick Breheny. October 13. z tests The χ 2 -distribution The t-distribution Summary
Patrick Breheny October 13 Patrick Breheny Biostatistical Methods I (BIOS 5710) 1/25 Introduction Introduction What s wrong with z-tests? So far we ve (thoroughly!) discussed how to carry out hypothesis
More informationHomework 4. 1 Data analysis problems
Homework 4 1 Data analysis problems This week we will be analyzing a number of data sets. We are going to build ARIMA models using the steps outlined in class. It is also a good idea to read section 3.8
More informationARIMA Models. Richard G. Pierse
ARIMA Models Richard G. Pierse 1 Introduction Time Series Analysis looks at the properties of time series from a purely statistical point of view. No attempt is made to relate variables using a priori
More informationAnalysis of Violent Crime in Los Angeles County
Analysis of Violent Crime in Los Angeles County Xiaohong Huang UID: 004693375 March 20, 2017 Abstract Violent crime can have a negative impact to the victims and the neighborhoods. It can affect people
More informationMAT3379 (Winter 2016)
MAT3379 (Winter 2016) Assignment 4 - SOLUTIONS The following questions will be marked: 1a), 2, 4, 6, 7a Total number of points for Assignment 4: 20 Q1. (Theoretical Question, 2 points). Yule-Walker estimation
More information