AR(p) + I(d) + MA(q) = ARIMA(p, d, q)
|
|
- Kenneth Stephens
- 5 years ago
- Views:
Transcription
1 AR(p) + I(d) + MA(q) = ARIMA(p, d, q)
2 Outline 1 4.1: Nonstationarity in the Mean 2 ARIMA Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 2/ 19
3 Deterministic Trend Models Polynomial Trend Consider the time series The mean function of this process is So this process is not stationary. More generally, the time series Z t = α 0 + α 1 t + a t µ t = α + α 1 t Z t = α 0 + α 1 t + + α k t k + a t has a kth-order polynomial mean function µ t = α 0 + α 1 t + + α k t k Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 3/ 19
4 Deterministic Trend Models Polynomial Trend Consider the time series The mean function of this process is So this process is not stationary. More generally, the time series Z t = α 0 + α 1 t + a t µ t = α + α 1 t Z t = α 0 + α 1 t + + α k t k + a t has a kth-order polynomial mean function µ t = α 0 + α 1 t + + α k t k Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 3/ 19
5 Deterministic Trend Models Polynomial Trend Consider the time series The mean function of this process is So this process is not stationary. More generally, the time series Z t = α 0 + α 1 t + a t µ t = α + α 1 t Z t = α 0 + α 1 t + + α k t k + a t has a kth-order polynomial mean function µ t = α 0 + α 1 t + + α k t k Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 3/ 19
6 Deterministic Trend Models Polynomial Trend Consider the time series The mean function of this process is So this process is not stationary. More generally, the time series Z t = α 0 + α 1 t + a t µ t = α + α 1 t Z t = α 0 + α 1 t + + α k t k + a t has a kth-order polynomial mean function µ t = α 0 + α 1 t + + α k t k Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 3/ 19
7 Deterministic Trend Models Sinusoidal Frequency Consider the time series Z t = ν 0 + ν cos(ωt + θ) + a t = ν 0 + α cos(ωt) + β sin(ωt) + a t where α = α 2 + β 2 cos(θ) and β = α 2 + β 2 sin(θ) More generally, m Z t = ν 0 + (α j cos(ω j t) + β j sin(ω j t)) + a t j=1 is the model of hidden periodicities". Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 4/ 19
8 Deterministic Trend Models Sinusoidal Frequency Consider the time series Z t = ν 0 + ν cos(ωt + θ) + a t = ν 0 + α cos(ωt) + β sin(ωt) + a t where α = α 2 + β 2 cos(θ) and β = α 2 + β 2 sin(θ) More generally, m Z t = ν 0 + (α j cos(ω j t) + β j sin(ω j t)) + a t j=1 is the model of hidden periodicities". Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 4/ 19
9 Outline 1 4.1: Nonstationarity in the Mean 2 ARIMA Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 5/ 19
10 ARIMA Definition Recall the difference operator Z t = (1 B)Z t = Z t Z t 1 And more generally d Z t = (1 B) d Z t Definition (ARIMA) A process is said to be ARIMA(p,d,q) if d Z t = (1 B) d Z t is ARMA(p, q). Therefore an ARIMA(p,d,q) model (with mean zero) can be written as φ(b)(1 B) d Z t = θ(b)a t Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 6/ 19
11 ARIMA Definition Recall the difference operator Z t = (1 B)Z t = Z t Z t 1 And more generally d Z t = (1 B) d Z t Definition (ARIMA) A process is said to be ARIMA(p,d,q) if d Z t = (1 B) d Z t is ARMA(p, q). Therefore an ARIMA(p,d,q) model (with mean zero) can be written as φ(b)(1 B) d Z t = θ(b)a t Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 6/ 19
12 ARIMA Definition Recall the difference operator Z t = (1 B)Z t = Z t Z t 1 And more generally d Z t = (1 B) d Z t Definition (ARIMA) A process is said to be ARIMA(p,d,q) if d Z t = (1 B) d Z t is ARMA(p, q). Therefore an ARIMA(p,d,q) model (with mean zero) can be written as φ(b)(1 B) d Z t = θ(b)a t Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 6/ 19
13 ARIMA Definition Recall the difference operator Z t = (1 B)Z t = Z t Z t 1 And more generally d Z t = (1 B) d Z t Definition (ARIMA) A process is said to be ARIMA(p,d,q) if d Z t = (1 B) d Z t is ARMA(p, q). Therefore an ARIMA(p,d,q) model (with mean zero) can be written as φ(b)(1 B) d Z t = θ(b)a t Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 6/ 19
14 Return Rate Suppose Z t is the value of an investment at time t and p t is the percentage changes from t 1 to t (which may be negative). Therefore we have Z t = (1 + p t )Z t 1 Taking logs produces or equivalently log(z t ) = log(1 + p t ) + log(z t 1 ) log(z t ) = log(z t ) log(z t 1 ) = log(1 + p t ) p t where the approximation holds when p t is close to zero. Another representation of log(z t ) is log(z t ) = log(z t ) log(z t 1 ) = log ( Zt Z t 1 ). Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 7/ 19
15 Return Rate Suppose Z t is the value of an investment at time t and p t is the percentage changes from t 1 to t (which may be negative). Therefore we have Z t = (1 + p t )Z t 1 Taking logs produces or equivalently log(z t ) = log(1 + p t ) + log(z t 1 ) log(z t ) = log(z t ) log(z t 1 ) = log(1 + p t ) p t where the approximation holds when p t is close to zero. Another representation of log(z t ) is log(z t ) = log(z t ) log(z t 1 ) = log ( Zt Z t 1 ). Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 7/ 19
16 Return Rate Suppose Z t is the value of an investment at time t and p t is the percentage changes from t 1 to t (which may be negative). Therefore we have Z t = (1 + p t )Z t 1 Taking logs produces or equivalently log(z t ) = log(1 + p t ) + log(z t 1 ) log(z t ) = log(z t ) log(z t 1 ) = log(1 + p t ) p t where the approximation holds when p t is close to zero. Another representation of log(z t ) is log(z t ) = log(z t ) log(z t 1 ) = log ( Zt Z t 1 ). Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 7/ 19
17 Return Rate Suppose Z t is the value of an investment at time t and p t is the percentage changes from t 1 to t (which may be negative). Therefore we have Z t = (1 + p t )Z t 1 Taking logs produces or equivalently log(z t ) = log(1 + p t ) + log(z t 1 ) log(z t ) = log(z t ) log(z t 1 ) = log(1 + p t ) p t where the approximation holds when p t is close to zero. Another representation of log(z t ) is log(z t ) = log(z t ) log(z t 1 ) = log ( Zt Z t 1 ). Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 7/ 19
18 US Gross National Product We consider the seasonally adjusted quarterly US GNP from 1947(1) to 2003(3) giving a total of n = 223 observations. (Economic Data FREDR Gross Domestic Product (GDP) and Components GDP/GNP GNP) > gnp96 = read.table("mydata/gnp96.dat") > gnp = ts(gnp96[,2], start=1947, frequency=4) > plot(gnp,lwd=3) Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 8/ 19
19 GDP and Housing Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 9/ 19
20 US Gross National Product (cont) Just for kicks, lets look at the acf. > acf(gnp, 50) Simple differencing may not be the answer. > plot(diff(gnp)) Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 10/ 19
21 Percentage Quarterly Growth of US GNP Instead, we consider the growth rate Z t = log(z t ). > gnpgr = diff(log(gnp)) # growth rate > plot.ts(gnpgr) Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 11/ 19
22 Modeling Percentage Quarterly Growth of US GNP The plots of the ACF and PACF of the GNP growth rate indicates two potential models for the log GNP series: ARIMA(0,1,2) ARIMA(1,1,0) We fit AR(1) to log(gnp). > (gnpgr.ar = arima(gnpgr, order = c(1, 0, 0))) Call: arima(x = gnpgr, order = c(1, 0, 0)) Coefficients: ar1 intercept s.e sigma^2 estimated as 9.03e-05: log likelihood = , aic = Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 12/ 19
23 Modeling in R R says intercept but means mean. Therefore the fitted model is Z t.0083 =.347(Z t ) + a t or equivalently Z t = Z t 1 + a t i.e. if α is the intercept and µ is the mean, then α = µ(1 φ) Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 13/ 19
24 Modeling in R From the expression α = µ(1 φ), we see σ α = σ m u(1 φ). Therefore we can write down the fitted model which incorporates the standard errors of the estimators Z t =.005 (.0006) (.063) Z t 1 + a t and σ = Also R has an issue with the I part of ARIMA fits where there is an AR component, so first difference the data then fit an ARMA model. Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 14/ 19
25 Modeling Percentage Quarterly Growth of US GNP We fit MA(2) to log(gnp). > (gnpgr.ma = arima(gnpgr, order = c(0, 0, 2))) Call: arima(x = gnpgr, order = c(0, 0, 2)) Coefficients: ma1 ma2 intercept s.e sigma^2 estimated as 8.92e-05: log likelihood = , aic = The R output indicates the model with σ = Z t =.0083 (.001) (.065) a t (.064) a t 2 + w t Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 15/ 19
26 The Two Models Aren t That Different The first 10 terms of the MA( ) representation of the AR(1) model is computed in R as > ARMAtoMA(ar=.35, ma=0, 10) # prints psi-weights [1] e e e e-02 [5] e e e e-04 [9] e e-05 So one (rather crude) approximation to the model Z t =.35Z t 1 + a t is Z t =.35a t a t 2 + a t which is close to the fitted MA(2) model Z t =.0083 (.001) (.065) a t (.064) a t 2 + w t. Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 16/ 19
27 Diagnostic Checking Investigate the residuals Z t x t t 1 or standardized residuals e t = Z t x t t 1 P t 1 t If the model fits well, the residuals should behave like an iid sequence with mean zero and variance one. Diagnostic Checks Check the plot of Standardized residuals for patterns and outliers. Check the ACF, ˆρ, for significance lags. Use the Ljung-Box-Pierce Q-statistic to measure collective autocorralative (not just significance at a single lag). The Ljung-Box-Pierce Q-statistic is given as H ρ 2 Q = n(n + 2) e(h) n h Under the null of model adequacy, Q as the asymptotic distribution Q χ 2 H p q. Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 17/ 19 h=1
28 Diagnostic Checking of gnpgr.ma > tsdiag(gnpgr.ma, gof.lag=20) Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 18/ 19
29 Model Selection in US GNP Series n = length(gnpgr) kma = length(gnpgr.ma$coef) sma=gnpgr.ma$sigma2 kar = length(gnpgr.ar$coef) sar=gnpgr.ar$sigma2 # AIC Returned Value log(sma) + (n+2*kma)/n # MA log(sar) + (n+2*kar)/n # AR # AICc log(sma) + (n+kma)/(n-kma-2) # MA log(sar) + (n+kar)/(n-kar-2) # AR # BIC log(sma) + kma*log(n)/n # MA log(sar) + kar*log(n)/n # AR # sample size # number of parameters in ma model # mle of sigma^2 # number of parameters in ar model # mle of sigma^2 Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 19/ 19
STA 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 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 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 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 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 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 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 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 informationSTAT 436 / Lecture 16: Key
STAT 436 / 536 - Lecture 16: Key Modeling Non-Stationary Time Series Many time series models are non-stationary. Recall a time series is stationary if the mean and variance are constant in time and the
More informationForecasting using R. Rob J Hyndman. 2.5 Seasonal ARIMA models. Forecasting using R 1
Forecasting using R Rob J Hyndman 2.5 Seasonal ARIMA models Forecasting using R 1 Outline 1 Backshift notation reviewed 2 Seasonal ARIMA models 3 ARIMA vs ETS 4 Lab session 12 Forecasting using R Backshift
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 informationPart 1. Multiple Choice (50 questions, 1 point each) Part 2. Problems/Short Answer (10 questions, 5 points each)
GROUND RULES: This exam contains two parts: Part 1. Multiple Choice (50 questions, 1 point each) Part 2. Problems/Short Answer (10 questions, 5 points each) The maximum number of points on this exam is
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 informationForecasting using R. Rob J Hyndman. 3.2 Dynamic regression. Forecasting using R 1
Forecasting using R Rob J Hyndman 3.2 Dynamic regression Forecasting using R 1 Outline 1 Regression with ARIMA errors 2 Stochastic and deterministic trends 3 Periodic seasonality 4 Lab session 14 5 Dynamic
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 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 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 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 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 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 informationLecture Notes of Bus (Spring 2017) Analysis of Financial Time Series Ruey S. Tsay
Lecture Notes of Bus 41202 (Spring 2017) Analysis of Financial Time Series Ruey S. Tsay Simple AR models: (Regression with lagged variables.) Motivating example: The growth rate of U.S. quarterly real
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 informationStat 565. (S)Arima & Forecasting. Charlotte Wickham. stat565.cwick.co.nz. Feb
Stat 565 (S)Arima & Forecasting Feb 2 2016 Charlotte Wickham stat565.cwick.co.nz Today A note from HW #3 Pick up with ARIMA processes Introduction to forecasting HW #3 The sample autocorrelation coefficients
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 informationUsing Analysis of Time Series to Forecast numbers of The Patients with Malignant Tumors in Anbar Provinc
Using Analysis of Time Series to Forecast numbers of The Patients with Malignant Tumors in Anbar Provinc /. ) ( ) / (Box & Jenkins).(.(2010-2006) ARIMA(2,1,0). Abstract: The aim of this research is to
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 informationThe log transformation produces a time series whose variance can be treated as constant over time.
TAT 520 Homework 6 Fall 2017 Note: Problem 5 is mandatory for graduate students and extra credit for undergraduates. 1) The quarterly earnings per share for 1960-1980 are in the object in the TA package.
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 informationCh 5. Models for Nonstationary Time Series. Time Series Analysis
We have studied some deterministic and some stationary trend models. However, many time series data cannot be modeled in either way. Ex. The data set oil.price displays an increasing variation from the
More informationCircle the single best answer for each multiple choice question. Your choice should be made clearly.
TEST #1 STA 4853 March 6, 2017 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 32 multiple choice
More informationChapter 5: Models for Nonstationary Time Series
Chapter 5: Models for Nonstationary Time Series Recall that any time series that is a stationary process has a constant mean function. So a process that has a mean function that varies over time must be
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 informationForecasting. Simon Shaw 2005/06 Semester II
Forecasting Simon Shaw s.c.shaw@maths.bath.ac.uk 2005/06 Semester II 1 Introduction A critical aspect of managing any business is planning for the future. events is called forecasting. Predicting future
More informationUnit root problem, solution of difference equations Simple deterministic model, question of unit root
Unit root problem, solution of difference equations Simple deterministic model, question of unit root (1 φ 1 L)X t = µ, Solution X t φ 1 X t 1 = µ X t = A + Bz t with unknown z and unknown A (clearly X
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 informationA SEASONAL TIME SERIES MODEL FOR NIGERIAN MONTHLY AIR TRAFFIC DATA
www.arpapress.com/volumes/vol14issue3/ijrras_14_3_14.pdf A SEASONAL TIME SERIES MODEL FOR NIGERIAN MONTHLY AIR TRAFFIC DATA Ette Harrison Etuk Department of Mathematics/Computer Science, Rivers State University
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 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 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 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 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 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 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 informationLecture 7: Model Building Bus 41910, Time Series Analysis, Mr. R. Tsay
Lecture 7: Model Building Bus 41910, Time Series Analysis, Mr R Tsay An effective procedure for building empirical time series models is the Box-Jenkins approach, which consists of three stages: model
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 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 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 informationModelling Monthly Rainfall Data of Port Harcourt, Nigeria by Seasonal Box-Jenkins Methods
International Journal of Sciences Research Article (ISSN 2305-3925) Volume 2, Issue July 2013 http://www.ijsciences.com Modelling Monthly Rainfall Data of Port Harcourt, Nigeria by Seasonal Box-Jenkins
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 informationSTAT STOCHASTIC PROCESSES. Contents
STAT 3911 - STOCHASTIC PROCESSES ANDREW TULLOCH Contents 1. Stochastic Processes 2 2. Classification of states 2 3. Limit theorems for Markov chains 4 4. First step analysis 5 5. Branching processes 5
More informationARIMA Models. Jamie Monogan. January 16, University of Georgia. Jamie Monogan (UGA) ARIMA Models January 16, / 27
ARIMA Models Jamie Monogan University of Georgia January 16, 2018 Jamie Monogan (UGA) ARIMA Models January 16, 2018 1 / 27 Objectives By the end of this meeting, participants should be able to: Argue why
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 informationProblems from Chapter 3 of Shumway and Stoffer s Book
UNIVERSITY OF UTAH GUIDED READING TIME SERIES Problems from Chapter 3 of Shumway and Stoffer s Book Author: Curtis MILLER Supervisor: Prof. Lajos HORVATH November 10, 2015 UNIVERSITY OF UTAH DEPARTMENT
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 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 informationSTA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7)
STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) Outline 1 AR Bootstrap 2 ARIMA 3 Homework 4b Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 2/ 20 Outline 1 AR Bootstrap
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 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 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 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 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 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 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 informationSeasonal Autoregressive Integrated Moving Average Model for Precipitation Time Series
Journal of Mathematics and Statistics 8 (4): 500-505, 2012 ISSN 1549-3644 2012 doi:10.3844/jmssp.2012.500.505 Published Online 8 (4) 2012 (http://www.thescipub.com/jmss.toc) Seasonal Autoregressive Integrated
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 informationQuantitative Finance I
Quantitative Finance I Linear AR and MA Models (Lecture 4) Winter Semester 01/013 by Lukas Vacha * If viewed in.pdf format - for full functionality use Mathematica 7 (or higher) notebook (.nb) version
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 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 informationTransformations for variance stabilization
FORECASTING USING R Transformations for variance stabilization Rob Hyndman Author, forecast Variance stabilization If the data show increasing variation as the level of the series increases, then a transformation
More informationibm: daily closing IBM stock prices (dates not given) internet: number of users logged on to an Internet server each minute (dates/times not given)
Remark: Problem 1 is the most important problem on this assignment (it will prepare you for your project). Problem 2 was taken largely from last year s final exam. Problem 3 consists of a bunch of rambling
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 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 informationEconomics 618B: Time Series Analysis Department of Economics State University of New York at Binghamton
Problem Set #1 1. Generate n =500random numbers from both the uniform 1 (U [0, 1], uniformbetween zero and one) and exponential λ exp ( λx) (set λ =2and let x U [0, 1]) b a distributions. Plot the histograms
More informationThe ARIMA Procedure: The ARIMA Procedure
Page 1 of 120 Overview: ARIMA Procedure Getting Started: ARIMA Procedure The Three Stages of ARIMA Modeling Identification Stage Estimation and Diagnostic Checking Stage Forecasting Stage Using ARIMA Procedure
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 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 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 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 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 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 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 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 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 informationSeasonality. Matthieu Stigler January 8, Version 1.1
Seasonality Matthieu Stigler Matthieu.Stigler@gmail.com January 8, 2009 Version 1.1 This document is released under the Creative Commons Attribution-Noncommercial 2.5 India license. Matthieu Stigler Matthieu.Stigler@gmail.com
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 informationPart 1. Multiple Choice (40 questions, 1 point each) Part 2. Problems/Short Answer (10 questions, 6 points each)
GROUND RULES: This exam contains two parts: Part 1. Multiple Choice (40 questions, 1 point each) Part 2. Problems/Short Answer (10 questions, 6 points each) The maximum number of points on this exam is
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 informationEconometrics for Policy Analysis A Train The Trainer Workshop Oct 22-28, 2016
Econometrics for Policy Analysis A Train The Trainer Workshop Delivered by Dr. Nathaniel E. Urama Department of Economics, University of Nigeria, Nsukka Loading Time Series data in E-views: Review For
More informationEcon 623 Econometrics II Topic 2: Stationary Time Series
1 Introduction Econ 623 Econometrics II Topic 2: Stationary Time Series In the regression model we can model the error term as an autoregression AR(1) process. That is, we can use the past value of the
More informationAutomatic seasonal auto regressive moving average models and unit root test detection
ISSN 1750-9653, England, UK International Journal of Management Science and Engineering Management Vol. 3 (2008) No. 4, pp. 266-274 Automatic seasonal auto regressive moving average models and unit root
More informationARIMA Models. Jamie Monogan. January 25, University of Georgia. Jamie Monogan (UGA) ARIMA Models January 25, / 38
ARIMA Models Jamie Monogan University of Georgia January 25, 2012 Jamie Monogan (UGA) ARIMA Models January 25, 2012 1 / 38 Objectives By the end of this meeting, participants should be able to: Describe
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 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 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 information4. MA(2) +drift: y t = µ + ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t 2. Mean: where θ(l) = 1 + θ 1 L + θ 2 L 2. Therefore,
61 4. MA(2) +drift: y t = µ + ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t 2 Mean: y t = µ + θ(l)ɛ t, where θ(l) = 1 + θ 1 L + θ 2 L 2. Therefore, E(y t ) = µ + θ(l)e(ɛ t ) = µ 62 Example: MA(q) Model: y t = ɛ t + θ 1 ɛ
More informationA Data-Driven Model for Software Reliability Prediction
A Data-Driven Model for Software Reliability Prediction Author: Jung-Hua Lo IEEE International Conference on Granular Computing (2012) Young Taek Kim KAIST SE Lab. 9/4/2013 Contents Introduction Background
More informationECON/FIN 250: Forecasting in Finance and Economics: Section 7: Unit Roots & Dickey-Fuller Tests
ECON/FIN 250: Forecasting in Finance and Economics: Section 7: Unit Roots & Dickey-Fuller Tests Patrick Herb Brandeis University Spring 2016 Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN
More information