Time Series 4. Robert Almgren. Oct. 5, 2009
|
|
- Edwina Wilkinson
- 6 years ago
- Views:
Transcription
1 Time Series 4 Robert Almgren Oct. 5, Nonstationarity How should you model a process that has drift? ARMA models are intrinsically stationary, that is, they are mean-reverting: when the value of x t is above its long-term mean, its next motions will likely be in a downward direction; when it is below its long-term mean the next motion will likely be up. These models also have decaying autocorrelation: the present value is completely forgotten if you go long enough into the future. In practice a lot of models have some sort of steady drift. For example, if you look at the daily volume of some stock, there is likely to be a substantial upward trend as overall trading volumes have increased. Now we discuss three possible ways that such drift can be included in a model. 1.1 Unit roots When we discussed AR models, we argued that solutions to the model exploded exponentially if P(z) had any roots strictly inside the unit disk. We therefore assumed that all roots of P(z) were strictly outside the unit disk, giving exponential decay. We deliberately ignored the boundary case where P(z) has a root on the unit circle. We shall consider only the case of a root at z = 1 (other cases can be reduced to this by suitable complex rotations). Example Recall the AR(1) model x t = c + φx t 1 + w t. Then P(z) = 1 φz has a single root at z = 1/φ and the root condition requires φ < 1. If φ = 1, then the model is the random walk x t = c + 1
2 Robert Almgren: Time Series 4 Oct. 5, x t 1 + w t. If we define the differenced series y by y = (I B)x, so y t = x t x t 1, then y satisfies y t = c + w t which is MA(1) and hence stationary. This example illustrates the general case. Suppose that P(z) has a zero at z = 1 of order r > 0, and all other roots are strictly outside the unit circle. Then we may write P(z) = P(z) (1 z) r where P(z) is a polynomial of degree p r, with P(0) = 1 and all roots strictly outside the unit circle. Then the original model can be written as P(z) (I B) r x = c + w and if we define the r th difference series y = (I B) r x then y satisfies the AR(p r ) model P(B) y = c + w. An ARIMA(p, r, q) model is one whose r th difference series follows an ARMA(p,q) model. The I is for integrated. An ARIMA(p, r, q) model has solutions that grow not exponentially, as they would if P had a root strictly inside the unit circle, but algebraically, with an order equal to r : If r = 1, then x t x t 1 is stationary, and x t O(t); If r = 2, then x t 2x t 1 + x t 2 is stationary, and x t O(t 2 ); etc. These asymptotic orders assume that c 0, so that y t has a nonzero mean. If c = 0, then cancellation will reduce the growth and we would need a more nuanced probabilisitic statement. For example, for r = 1 we get the classic random walk with E ( ) x t = 0 but E( x t ) O ( t ). The most common case is r = 1. Long-memory processes So far we have always assumed that autocorrelations decayed exponentially in time. This is very convenient theoretically, but not always justified by real data, for which
3 Robert Almgren: Time Series 4 Oct. 5, autocorrelations often decay with power-law behavior: γ l O( l β ). We can do this by interpolating between ARMA and ARIMA models. ARMA(p,q): P(B) x = c + Q(B) w exponential decay ARIMA(p,1,q): P(B)(1 B) x = c + Q(B) w no decay. It is then natural to consider models of the form P(B)(1 B) α x = c + Q(B) w decay O ( l 2α 1) where 0.5 < α < 0.5. In practice, this is equivalent to an ARMA(p,q) model with q =, determined by series expansion of (1 z) α. 1.2 Drift The other way to get nonstationary behavior is to add it explicitly, by making the constant term be explicitly time dependent. Most commonly we just put linear dependence µ t = µ 0 + µ 1 t or c t = c 0 + c 1 t. If the model variable x t is the log of something, then these linear terms correspond to exponential behavior. We would then typically assume that the associated ARMA model has roots strictly outside the unit circle, so that it reverts to a moving mean value. These models have E ( x t ) = a + bt but Var ( x t ) finite. Thus they generally stay within a finite distance of a trending mean value, in contrast to a unit-root model with continually growing variance. Although this distinction is clear in theory, on a particular sample of data it can be difficult to tell the difference, and often you have to make the choice based on theoretical preferences. One might summarize the modeling hierarchy as follows: First, do you think that the distribution should be stationary? If yes then look for an ARMA model with no drift. The autocorrelation should decay rapidly to zero. If no, then ask what causes the nonstationarity. If you think it is a general linear or exponential growth (e.g, the company you are studying grows with the general economy), then look for an ARMA model with explicit drift.
4 Robert Almgren: Time Series 4 Oct. 5, t t t Mean reverting Mean reverting with drift Random walk Figure 1: Sample paths for a mean-reverting AR(1) model, AR(1) with drift, and ARIMA(1,1,0) random walk. In each we have emphasized one realization to emphasize that in practice that is all you see. From just this path, you would have trouble distinguishing the mean reverting model with drift (middle) from the random walk (right). If you think the value drifts randomly, then look for an ARMA model for differences. Of course, if the quality of your data permits, you will always formulate a model that contains all possible dependencies and see which coefficients come out zero. 1.3 Seasonality Some series may have periodic components: for example, daily market data may have weekly effect, corresponding to a signal in the data of period 5 (with occasional disruptions when one or more days in a week are a holiday). Special techniques should be used to remove this component. 1.4 Variable volatility Even if a series has no drift, integrated effects, or seasonality, the magnitude of changes may vary, for example varying volatility of asset price returns. These problems are well covered by ARCH/GARCH methodology which we will talk about in a few weeks.
5 Robert Almgren: Time Series 4 Oct. 5, Breaks The entire data series may change at some specific point, for example, because of exogeneous effects such as decimalization in equity prices, or Reg NMS. Or it might be intrinsic regime shifts when the entire model shifts from one configuration to another. Again, specialized techniques exists for identifying these changes. 2 Fitting strategy Now suppose you have a data sequence x 1,..., x n (equally spaced in time) and you want to determine a model that fits it to some acceptable extent. We advocate the following sequence of steps: 1. Remove obvious trends and seasonality. At a minimum, this will probably include subtracting an average drift. You could determine the average drift by doing a linear regression x j µ 0 + µ 1 j + ξ j where the residuals ξ j are your new series. Or you could simply take the drift from the endpoints (x n x 1 )/n. For seasonality, there are standard techniques built into most software packages, or you can use sophisticated Kalman filter methods, or you can construct some approximation that makes sense to you. This step is inevitably rather messy and there is no easy formula. 2. Look for the best AR model. Carry out a sequence of linear regressions of each term on its predecessors, at increasing order: x t x t c + β 1 x t 1 + w t c + β 1 x t 1 + β 2 x t 2 + w t. At each stage, the last coefficient β k is the partial auto-correlation coefficient. Hopefully, these coefficients will have nontrivial magnitude for the first few orders, and then will suddenly drop to a negligible magnitude at a particular order, telling you exactly where to stop.
6 Robert Almgren: Time Series 4 Oct. 5, More precisely, if x is truly described by an AR(p) process, then β k will be close to zero for k > p. If the noise terms have finite variance, and are not too far from Gaussian, then the variance in the estimate of β k is 1/n for k > p (note that the coefficients β j are nondimensional so the variance does not enter into this estimate). Tsay describes a more sophisticated approach that chooses the order to maximise an information criterion, balancing precision of fit against number of parameters. By whatever means, choose the optimal AR order. 3. Fit an MA model. After fitting the best AR model, the residuals may not yet be white noise. Examine their empirical autocorrelation, and see whether it plausibly drops to zero beyond a particular finite order q. Then the residuals can be fit by an MA(q) model, and you have constructed an ARMA(p,q) model. 4. Test the final residuals as white noise. If the model constructed in the previous steps is adequate, then the final residuals should have no identifiable structure. They should pass empirical test for white noise: zero mean, constant variance, and zero correlation. If you can convince yourself that they pass those tests then you are done. Unit root tests To identify ARIMA models from the AR fit, we need to identify whether a polynomial of order p has a root at z = 1, which sounds complicated. In fact, by rewriting the problem it becomes equally easy. The main observation is that we can write the polynomial as P(z) = 1 φ 1 z φ p z p = 1 β z z(1 z) P(z) where P(z) = α 0 + α 1 z + + α p 2 z p 2 has degree p 2. (To convince yourself that you can rewrite P(z) this way, write out the conditions that determine β, α 0,..., α p 2 in terms of φ 1,..., φ p and convince yourself they can always be solved.) In this form, P(1) = 0 if and only if β = 1.
7 Robert Almgren: Time Series 4 Oct. 5, Then the AR(p) model P(B)x = c + w can be written as x t = c t + β x t 1 + p 1 l=1 α l 1 ( x) t l + w t where = I B is the difference operator, and we have allowed for potential time dependence in c t. Thus all we have to do is regress x t on ( 1, t, x t 1, x t 1 x t 2,..., x t p+1 x t p ) which is almost as easy as the original regression. Standard packages give us the estimate of β and its uncertainty, and we evaluate it as above.
Time 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 informationTime Series 3. Robert Almgren. Sept. 28, 2009
Time Series 3 Robert Almgren Sept. 28, 2009 Last time we discussed two main categories of linear models, and their combination. Here w t denotes a white noise: a stationary process with E w t ) = 0, E
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 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 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 informationTopic 4 Unit Roots. Gerald P. Dwyer. February Clemson University
Topic 4 Unit Roots Gerald P. Dwyer Clemson University February 2016 Outline 1 Unit Roots Introduction Trend and Difference Stationary Autocorrelations of Series That Have Deterministic or Stochastic Trends
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 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 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 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 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 informationUnivariate, Nonstationary Processes
Univariate, Nonstationary Processes Jamie Monogan University of Georgia March 20, 2018 Jamie Monogan (UGA) Univariate, Nonstationary Processes March 20, 2018 1 / 14 Objectives By the end of this meeting,
More informationTime Series Models and Inference. James L. Powell Department of Economics University of California, Berkeley
Time Series Models and Inference James L. Powell Department of Economics University of California, Berkeley Overview In contrast to the classical linear regression model, in which the components of the
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 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 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 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 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 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 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 information11. Further Issues in Using OLS with TS Data
11. Further Issues in Using OLS with TS Data With TS, including lags of the dependent variable often allow us to fit much better the variation in y Exact distribution theory is rarely available in TS applications,
More informationStochastic Processes
Stochastic Processes Stochastic Process Non Formal Definition: Non formal: A stochastic process (random process) is the opposite of a deterministic process such as one defined by a differential equation.
More informationLecture 5: Unit Roots, Cointegration and Error Correction Models The Spurious Regression Problem
Lecture 5: Unit Roots, Cointegration and Error Correction Models The Spurious Regression Problem Prof. Massimo Guidolin 20192 Financial Econometrics Winter/Spring 2018 Overview Stochastic vs. deterministic
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 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 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 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 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 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 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 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 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 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 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 informationVolatility. Gerald P. Dwyer. February Clemson University
Volatility Gerald P. Dwyer Clemson University February 2016 Outline 1 Volatility Characteristics of Time Series Heteroskedasticity Simpler Estimation Strategies Exponentially Weighted Moving Average Use
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 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 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 information1 Teaching notes on structural VARs.
Bent E. Sørensen February 22, 2007 1 Teaching notes on structural VARs. 1.1 Vector MA models: 1.1.1 Probability theory The simplest (to analyze, estimation is a different matter) time series models are
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 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 informationCh 9. FORECASTING. Time Series Analysis
In this chapter, we assume the model is known exactly, and consider the calculation of forecasts and their properties for both deterministic trend models and ARIMA models. 9.1 Minimum Mean Square Error
More informationHeteroskedasticity in Time Series
Heteroskedasticity in Time Series Figure: Time Series of Daily NYSE Returns. 206 / 285 Key Fact 1: Stock Returns are Approximately Serially Uncorrelated Figure: Correlogram of Daily Stock Market Returns.
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 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 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 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 informationLecture 6a: Unit Root and ARIMA Models
Lecture 6a: Unit Root and ARIMA Models 1 2 Big Picture A time series is non-stationary if it contains a unit root unit root nonstationary The reverse is not true. For example, y t = cos(t) + u t has no
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 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 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 informationdata lam=36.9 lam=6.69 lam=4.18 lam=2.92 lam=2.21 time max wavelength modulus of max wavelength cycle
AUTOREGRESSIVE LINEAR MODELS AR(1) MODELS The zero-mean AR(1) model x t = x t,1 + t is a linear regression of the current value of the time series on the previous value. For > 0 it generates positively
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 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 informationA nonparametric test for seasonal unit roots
Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna To be presented in Innsbruck November 7, 2007 Abstract We consider a nonparametric test for the
More information7. Integrated Processes
7. Integrated Processes Up to now: Analysis of stationary processes (stationary ARMA(p, q) processes) Problem: Many economic time series exhibit non-stationary patterns over time 226 Example: We consider
More informationSOME BASICS OF TIME-SERIES ANALYSIS
SOME BASICS OF TIME-SERIES ANALYSIS John E. Floyd University of Toronto December 8, 26 An excellent place to learn about time series analysis is from Walter Enders textbook. For a basic understanding of
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 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 informationFinQuiz Notes
Reading 9 A time series is any series of data that varies over time e.g. the quarterly sales for a company during the past five years or daily returns of a security. When assumptions of the regression
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 information7. Integrated Processes
7. Integrated Processes Up to now: Analysis of stationary processes (stationary ARMA(p, q) processes) Problem: Many economic time series exhibit non-stationary patterns over time 226 Example: We consider
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 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 informationLecture 6: Univariate Volatility Modelling: ARCH and GARCH Models
Lecture 6: Univariate Volatility Modelling: ARCH and GARCH Models Prof. Massimo Guidolin 019 Financial Econometrics Winter/Spring 018 Overview ARCH models and their limitations Generalized ARCH models
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 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 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 informationCh. 19 Models of Nonstationary Time Series
Ch. 19 Models of Nonstationary Time Series In time series analysis we do not confine ourselves to the analysis of stationary time series. In fact, most of the time series we encounter are non 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 informationNonstationary Time Series:
Nonstationary Time Series: Unit Roots Egon Zakrajšek Division of Monetary Affairs Federal Reserve Board Summer School in Financial Mathematics Faculty of Mathematics & Physics University of Ljubljana September
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 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 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 informationCh 4. Models For Stationary Time Series. Time Series Analysis
This chapter discusses the basic concept of a broad class of stationary parametric time series models the autoregressive moving average (ARMA) models. Let {Y t } denote the observed time series, and {e
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 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 informationBox-Jenkins ARIMA Advanced Time Series
Box-Jenkins ARIMA Advanced Time Series www.realoptionsvaluation.com ROV Technical Papers Series: Volume 25 Theory In This Issue 1. Learn about Risk Simulator s ARIMA and Auto ARIMA modules. 2. Find out
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 information3 Time Series Regression
3 Time Series Regression 3.1 Modelling Trend Using Regression Random Walk 2 0 2 4 6 8 Random Walk 0 2 4 6 8 0 10 20 30 40 50 60 (a) Time 0 10 20 30 40 50 60 (b) Time Random Walk 8 6 4 2 0 Random Walk 0
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 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 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 informationData Mining Techniques
Data Mining Techniques CS 6220 - Section 3 - Fall 2016 Lecture 18: Time Series Jan-Willem van de Meent (credit: Aggarwal Chapter 14.3) Time Series Data http://www.capitalhubs.com/2012/08/the-correlation-between-apple-product.html
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 informationMultiple Regression Analysis
1 OUTLINE Basic Concept: Multiple Regression MULTICOLLINEARITY AUTOCORRELATION HETEROSCEDASTICITY REASEARCH IN FINANCE 2 BASIC CONCEPTS: Multiple Regression Y i = β 1 + β 2 X 1i + β 3 X 2i + β 4 X 3i +
More informationAlgebra & Trig Review
Algebra & Trig Review 1 Algebra & Trig Review This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The
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 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 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 informationFinancial Econometrics
Financial Econometrics Nonlinear time series analysis Gerald P. Dwyer Trinity College, Dublin January 2016 Outline 1 Nonlinearity Does nonlinearity matter? Nonlinear models Tests for nonlinearity Forecasting
More informationScenario 5: Internet Usage Solution. θ j
Scenario : Internet Usage Solution Some more information would be interesting about the study in order to know if we can generalize possible findings. For example: Does each data point consist of the total
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 information7 Introduction to Time Series
Econ 495 - Econometric Review 1 7 Introduction to Time Series 7.1 Time Series vs. Cross-Sectional Data Time series data has a temporal ordering, unlike cross-section data, we will need to changes some
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 informationMultiscale and multilevel technique for consistent segmentation of nonstationary time series
Multiscale and multilevel technique for consistent segmentation of nonstationary time series Haeran Cho Piotr Fryzlewicz University of Bristol London School of Economics INSPIRE 2009 Imperial College London
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 information7. Forecasting with ARIMA models
7. Forecasting with ARIMA models 309 Outline: Introduction The prediction equation of an ARIMA model Interpreting the predictions Variance of the predictions Forecast updating Measuring predictability
More informationESSE Mid-Term Test 2017 Tuesday 17 October :30-09:45
ESSE 4020 3.0 - Mid-Term Test 207 Tuesday 7 October 207. 08:30-09:45 Symbols have their usual meanings. All questions are worth 0 marks, although some are more difficult than others. Answer as many questions
More informationTrend-Cycle Decompositions
Trend-Cycle Decompositions Eric Zivot April 22, 2005 1 Introduction A convenient way of representing an economic time series y t is through the so-called trend-cycle decomposition y t = TD t + Z t (1)
More information