ECON/FIN 250: Forecasting in Finance and Economics: Section 7: Unit Roots & Dickey-Fuller Tests
|
|
|
- Collin Griffin
- 5 years ago
- Views:
Transcription
1 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 250: Spring / 27
2 Course Overview 1 Key Objectives 2 ARIMA Processes 3 Testing for Unit Roots Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
3 Key Objectives 1 Key Objectives 2 ARIMA Processes 3 Testing for Unit Roots Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
4 Key Objectives ARIMA Processes Dickey-Fuller Tests Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
5 Stochastic Trends Random Walk y t = y t 1 + ɛ t, ɛ t WN(0, σ 2 ) (1) Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
6 Stochastic Trends Random Walk y t = y t 1 + ɛ t, ɛ t WN(0, σ 2 ) (1) Random Walk with Drift y t = δ + y t 1 + ɛ t, ɛ t WN(0, σ 2 ) (2) Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
7 Stochastic Trends Geometric Random Walk log(y t ) = log(y t 1 ) + ɛ t, ɛ t WN(0, σ 2 ) (3) Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
8 Stochastic Trends Geometric Random Walk log(y t ) = log(y t 1 ) + ɛ t, ɛ t WN(0, σ 2 ) (3) Geometric Random Walk with Drift log(y t ) = δ + log(y t 1 ) + ɛ t, ɛ t WN(0, σ 2 ) (4) Better for series where percentage changes are constant magnitude over time: stock prices, GDP... Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
9 Basic Random Walk Properties y 0 + y t = y t 1 + ɛ t, ɛ t WN(0, σ 2 ) (5) y t = t ɛ i (6) Starting at y 0 i=1 Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
10 Basic Random Walk Properties y 0 + y t = y t 1 + ɛ t, ɛ t WN(0, σ 2 ) (5) y t = t ɛ i (6) Starting at y 0 i=1 E[y t ] = y 0 (7) Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
11 Basic Random Walk Properties y 0 + y t = y t 1 + ɛ t, ɛ t WN(0, σ 2 ) (5) y t = t ɛ i (6) Starting at y 0 i=1 E[y t ] = y 0 (7) T Var[y t ] = E[ɛ 2 i ] = tσɛ 2 (8) i=1 Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
12 Basic Random Walk Properties y 0 + y t = y t 1 + ɛ t, ɛ t WN(0, σ 2 ) (5) y t = t ɛ i (6) Starting at y 0 i=1 E[y t ] = y 0 (7) T Var[y t ] = E[ɛ 2 i ] = tσɛ 2 (8) i=1 lim Var[y t] = (9) t Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
13 Basic Random Walk with Drift Properties tδ y t = δ + y t 1 + ɛ t, ɛ t WN(0, σ 2 ) (10) yt = y0 + T + ɛ i (11) Starting at y 0 i=1 Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
14 Basic Random Walk with Drift Properties tδ y t = δ + y t 1 + ɛ t, ɛ t WN(0, σ 2 ) (10) yt = y0 + T + ɛ i (11) Starting at y 0 i=1 E[y t ] = y 0 + tδ (12) Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
15 Basic Random Walk with Drift Properties tδ y t = δ + y t 1 + ɛ t, ɛ t WN(0, σ 2 ) (10) yt = y0 + T + ɛ i (11) Starting at y 0 i=1 E[y t ] = y 0 + tδ (12) T Var[y t ] = E[ɛ 2 i ] = tσɛ 2 (13) i=1 Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
16 Basic Random Walk with Drift Properties tδ y t = δ + y t 1 + ɛ t, ɛ t WN(0, σ 2 ) (10) yt = y0 + T + ɛ i (11) Starting at y 0 i=1 E[y t ] = y 0 + tδ (12) T Var[y t ] = E[ɛ 2 i ] = tσɛ 2 (13) i=1 lim Var[y t] = (14) t Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
17 Random Walk Forecasts y t = y t 1 + ɛ t, ɛ t WN(0, σ 2 ) (15) Forecast h steps ahead starting at time T h y T +h = y T + ɛ T +i (16) i=1 Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
18 Random Walk Forecasts y t = y t 1 + ɛ t, ɛ t WN(0, σ 2 ) (15) Forecast h steps ahead starting at time T h y T +h = y T + ɛ T +i (16) i=1 E[y T +h Ω T ] = y T, h (17) Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
19 Random Walk Forecasts y t = y t 1 + ɛ t, ɛ t WN(0, σ 2 ) (15) Forecast h steps ahead starting at time T h y T +h = y T + ɛ T +i (16) i=1 E[y T +h Ω T ] = y T, h (17) Var[y T +h Ω] = E[(y T +h E[y T +h Ω T ]) 2 ] (18) Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
20 Random Walk Forecasts y t = y t 1 + ɛ t, ɛ t WN(0, σ 2 ) (15) Forecast h steps ahead starting at time T h y T +h = y T + ɛ T +i (16) i=1 E[y T +h Ω T ] = y T, h (17) Var[y T +h Ω] = E[(y T +h E[y T +h Ω T ]) 2 ] (18) h Var[y T +h Ω T ] = E[(y T +h y T ) 2 ] = ɛ 2 T +i = hσɛ 2 (19) i=1 Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
21 Random Walk Forecast Intuitive Properties Best forecast is current value Longer forecasts don t converge to the mean Forecast variance expand linearly in h as hσ 2 ɛ Forecast std s expand as h or hσ ɛ Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
22 ARIMA Processes 1 Key Objectives 2 ARIMA Processes 3 Testing for Unit Roots Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
23 Integrated Processes ARIMA(p,d,q) An AR(p) process is a unit root process if one of the roots of the lag operator polynomial is equal to one Unit roots result in nonstationary behavior Differencing a random walk process integrates or undoes the unit root. Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
24 Integrated Processes ARIMA(p,d,q) An AR(p) process is a unit root process if one of the roots of the lag operator polynomial is equal to one Unit roots result in nonstationary behavior Differencing a random walk process integrates or undoes the unit root. y t y t 1 = ɛ t (20) The difference is a white noise process z t = ɛ t, ɛ t WN(0, σ 2 ) (21) Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
25 More Examples Consider an ARIMA(1,1,0) Difference is AR(1) y t y t 1 = φ(y t 1 y t 2 ) + ɛ t (22) z t = φz t 1 + ɛ t (23) Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
26 More Examples Consider an ARIMA(1,1,1) Difference is ARMA(1,1) y t y t 1 = φ(y t 1 y t 2 ) + θɛ t 1 + ɛ t (24) z t = φz t 1 + θɛ t 1 + ɛ t (25) Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
27 Differencing vs. Detrending Differencing a Stochastic Trend Model y t = a 0 + y t 1 + ɛ t (26) t y t = y 0 + a 0 t + ɛ i (27) i=1 t 1 y t 1 = y 0 + a 0 (t 1) + ɛ i (28) i=1 y t = a 0 + ɛ t (29) The difference is stationary A series with a unit root can be transformed into a stationary series by differencing Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
28 Differencing vs. Detrending Differencing a Deterministic Trend Model y t = y 0 + a 1 t + ɛ t (30) y t 1 = y 0 + a 1 (t 1) + ɛ t 1 (31) y t = a 1 + ɛ t + ɛ t 1 (32) The difference is not stationary (not invertible) Need to detrend data with deterministic time trend, not (necessarily) difference A trend-stationary series can be transformed into a stationary series by removing the deterministic trend Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
29 ARIMA Notation and Methods ARIMA(p,d,q) p = AR q = MA d = differencing level What is differencing level? d = 1, y t y t 1 is ARMA(p,q) d = 2, (y t y t 1 ) (y t 1 y t 2 ) is ARMA(p,q) d = 3, keep going. Higher order differencing is rare in economics data Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
30 What do you do? If you know d Difference y t d times Estimate ARMA components Generate forecasts of ẑ t Add back together ŷ t = ŷ t 1 + ẑ t Problem: You often don t know d In economics data, the question is often between d = 1 or d = 0 Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
31 In Lag Notation ARMA(p,q) Φ(L)y t = c + Θ(L)ɛ t (33) ARIMA(p,1,q) ARIMA(p,d,q) Φ(L) = 1 φ 1 L φ 2 L φ p L p (34) Θ(L) = 1 θ 1 L θ 2 L θ q L q (35) Φ(L)(1 L)y t = c + Θ(L)ɛ t (36) Φ(L)(1 L) d y t = c + Θ(L)ɛ t (37) Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
32 Testing for Unit Roots 1 Key Objectives 2 ARIMA Processes 3 Testing for Unit Roots Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
33 Testing d = 0 or d = 1 What about regressing y t = φy t 1 + ɛ t? Test for φ = 1 Many people did this a while ago It turns out the distribution of φ is not t-distribution Proper tests have Dickey-Fuller-distribution Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
34 Dickey-Fuller Tests y t = φy t 1 + ɛ t (38) y t y t 1 = (φ 1)y t 1 + ɛ t (39) y t = γy t 1 + ɛ t (40) Regress y t y t 1 on y t 1 Testing is γ = 0 is equivalent to testing φ = 1 The null hypothesis is γ = 0, φ = 1, or {y t } has a unit root the alternative is (φ 1) < 0 Failing to reject unit root Rejecting the null no unit root Alternative would be φ < 1 stationary AR(1) Alternatives are important Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
35 Dickey-Fuller Tests: Expanding the Alternative Random Walk + Drift y t = a 0 + γy t 1 + ɛ t (41) Random Walk + Drift + Deterministic Time Trend y t = a 0 + γy t 1 + a 2 t + ɛ t (42) Test if γ = 0 to determine if process has unit root Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
36 Augmented Dickey-Fuller Null: Random Walk + AR(p); Alternative: AR(p) + No Mean p y t = γy t 1 + β i y t i+1 + ɛ t (43) i=2 Null: Random Walk + Drift + AR(p); Alternative: AR(p) + Mean p y t = a 0 + γy t 1 + β i y t i+1 + ɛ t (44) i=2 Null: Random Walk + Drift + Deterministic Time Trend + AR(p); Alternative: AR(p)+Trend p y t = a 0 + γy t 1 + a 2 t + β i y t i+1 + ɛ t (45) i=2 Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
37 Dickey-Fuller Instructions Don t worry about critical values and tests, they are performed by all good software. Do worry about interpretation and alternatives. No software does this Basic Steps Is your data trending over time? If yes, use form 3 If no obvious trend, then use form 1 or 2 depending on mean Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
38 Stata Code: Forms dfuller lgdp, noconstant reg lags(k) 2 dfuller lgdp, reg lags(k) 3 dfuller lgdp, trend reg lags(k) The reg option prints out regression coefficients Skipping the drift option Try this with U.S. GDP data Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
39 Size and Power: Important Unit root testing can be difficult The Power of the Dickey-Fuller test can be low Higher chance of Type II error Failing to reject a false null hypothesis Failing to reject a unit root when there is no unit root Often accept random walk null when time trend might be true model Think about your data Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN 250: Spring / 27
ECON/FIN 250: Forecasting in Finance and Economics: Section 6: Standard Univariate Models
ECON/FIN 250: Forecasting in Finance and Economics: Section 6: Standard Univariate Models Patrick Herb Brandeis University Spring 2016 Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN
ECON/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
Forecasting with ARMA
Forecasting with ARMA Eduardo Rossi University of Pavia October 2013 Rossi Forecasting Financial Econometrics - 2013 1 / 32 Mean Squared Error Linear Projection Forecast of Y t+1 based on a set of variables
Univariate, 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,
Univariate 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
Ch 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
10) Time series econometrics
30C00200 Econometrics 10) Time series econometrics Timo Kuosmanen Professor, Ph.D. 1 Topics today Static vs. dynamic time series model Suprious regression Stationary and nonstationary time series Unit
FE570 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
Ch 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
Chapter 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
Chapter 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
Non-Stationary Time Series and Unit Root Testing
Econometrics II Non-Stationary Time Series and Unit Root Testing Morten Nyboe Tabor Course Outline: Non-Stationary Time Series and Unit Root Testing 1 Stationarity and Deviation from Stationarity Trend-Stationarity
Empirical 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
9) Time series econometrics
30C00200 Econometrics 9) Time series econometrics Timo Kuosmanen Professor Management Science http://nomepre.net/index.php/timokuosmanen 1 Macroeconomic data: GDP Inflation rate Examples of time series
1 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
Non-Stationary Time Series and Unit Root Testing
Econometrics II Non-Stationary Time Series and Unit Root Testing Morten Nyboe Tabor Course Outline: Non-Stationary Time Series and Unit Root Testing 1 Stationarity and Deviation from Stationarity Trend-Stationarity
Lecture 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
BCT Lecture 3. Lukas Vacha.
BCT Lecture 3 Lukas Vacha vachal@utia.cas.cz Stationarity and Unit Root Testing Why do we need to test for Non-Stationarity? The stationarity or otherwise of a series can strongly influence its behaviour
ECON 616: Lecture 1: Time Series Basics
ECON 616: Lecture 1: Time Series Basics ED HERBST August 30, 2017 References Overview: Chapters 1-3 from Hamilton (1994). Technical Details: Chapters 2-3 from Brockwell and Davis (1987). Intuition: Chapters
7. 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
Financial Time Series Analysis: Part II
Department of Mathematics and Statistics, University of Vaasa, Finland Spring 2017 1 Unit root Deterministic trend Stochastic trend Testing for unit root ADF-test (Augmented Dickey-Fuller test) Testing
7. 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
Non-Stationary Time Series and Unit Root Testing
Econometrics II Non-Stationary Time Series and Unit Root Testing Morten Nyboe Tabor Course Outline: Non-Stationary Time Series and Unit Root Testing 1 Stationarity and Deviation from Stationarity Trend-Stationarity
Nonstationary 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
CHAPTER 21: TIME SERIES ECONOMETRICS: SOME BASIC CONCEPTS
CHAPTER 21: TIME SERIES ECONOMETRICS: SOME BASIC CONCEPTS 21.1 A stochastic process is said to be weakly stationary if its mean and variance are constant over time and if the value of the covariance between
Lecture 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
Problem Set 2 Solution Sketches Time Series Analysis Spring 2010
Problem Set 2 Solution Sketches Time Series Analysis Spring 2010 Forecasting 1. Let X and Y be two random variables such that E(X 2 ) < and E(Y 2 )
Chapter 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
Prof. 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
Ch 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
EASTERN 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
Univariate Nonstationary Time Series 1
Univariate Nonstationary Time Series 1 Sebastian Fossati University of Alberta 1 These slides are based on Eric Zivot s time series notes available at: http://faculty.washington.edu/ezivot Introduction
Topic 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
Review 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 =
Covariance Stationary Time Series. Example: Independent White Noise (IWN(0,σ 2 )) Y t = ε t, ε t iid N(0,σ 2 )
Covariance Stationary Time Series Stochastic Process: sequence of rv s ordered by time {Y t } {...,Y 1,Y 0,Y 1,...} Defn: {Y t } is covariance stationary if E[Y t ]μ for all t cov(y t,y t j )E[(Y t μ)(y
Advanced 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
ECON/FIN 250: Forecasting in Finance and Economics: Section 4.1 Forecasting Fundamentals
ECON/FIN 250: Forecasting in Finance and Economics: Section 4.1 Forecasting Fundamentals Patrick Herb Brandeis University Spring 2016 Patrick Herb (Brandeis University) Forecasting Fundamentals ECON/FIN
Exercises - 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
Econometrics. Week 11. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 11 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 30 Recommended Reading For the today Advanced Time Series Topics Selected topics
Autoregressive and Moving-Average Models
Chapter 3 Autoregressive and Moving-Average Models 3.1 Introduction Let y be a random variable. We consider the elements of an observed time series {y 0,y 1,y2,...,y t } as being realizations of this randoms
at 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
ECON 616: Lecture Two: Deterministic Trends, Nonstationary Processes
ECON 616: Lecture Two: Deterministic Trends, Nonstationary Processes ED HERBST September 11, 2017 Background Hamilton, chapters 15-16 Trends vs Cycles A commond decomposition of macroeconomic time series
Classic 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
ARIMA 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
SOME 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
5 Transfer function modelling
MSc Further Time Series Analysis 5 Transfer function modelling 5.1 The model Consider the construction of a model for a time series (Y t ) whose values are influenced by the earlier values of a series
Lecture 8: ARIMA Forecasting Please read Chapters 7 and 8 of MWH Book
Lecture 8: ARIMA Forecasting Please read Chapters 7 and 8 of MWH Book 1 Predicting Error 1. y denotes a random variable (stock price, weather, etc) 2. Sometimes we want to do prediction (guessing). Let
Midterm 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)
Testing for non-stationarity
20 November, 2009 Overview The tests for investigating the non-stationary of a time series falls into four types: 1 Check the null that there is a unit root against stationarity. Within these, there are
Forecasting 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
Introduction 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,
ECON 4160, Spring term Lecture 12
ECON 4160, Spring term 2013. Lecture 12 Non-stationarity and co-integration 2/2 Ragnar Nymoen Department of Economics 13 Nov 2013 1 / 53 Introduction I So far we have considered: Stationary VAR, with deterministic
Moreover, the second term is derived from: 1 T ) 2 1
170 Moreover, the second term is derived from: 1 T T ɛt 2 σ 2 ɛ. Therefore, 1 σ 2 ɛt T y t 1 ɛ t = 1 2 ( yt σ T ) 2 1 2σ 2 ɛ 1 T T ɛt 2 1 2 (χ2 (1) 1). (b) Next, consider y 2 t 1. T E y 2 t 1 T T = E(y
Ch. 15 Forecasting. 1.1 Forecasts Based on Conditional Expectations
Ch 15 Forecasting Having considered in Chapter 14 some of the properties of ARMA models, we now show how they may be used to forecast future values of an observed time series For the present we proceed
Basic concepts and terminology: AR, MA and ARMA processes
ECON 5101 ADVANCED ECONOMETRICS TIME SERIES Lecture note no. 1 (EB) Erik Biørn, Department of Economics Version of February 1, 2011 Basic concepts and terminology: AR, MA and ARMA processes This lecture
7 Introduction to Time Series Time Series vs. Cross-Sectional Data Detrending Time Series... 15
Econ 495 - Econometric Review 1 Contents 7 Introduction to Time Series 3 7.1 Time Series vs. Cross-Sectional Data............ 3 7.2 Detrending Time Series................... 15 7.3 Types of Stochastic
Trend-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)
Multivariate Time Series
Multivariate Time Series Fall 2008 Environmental Econometrics (GR03) TSII Fall 2008 1 / 16 More on AR(1) In AR(1) model (Y t = µ + ρy t 1 + u t ) with ρ = 1, the series is said to have a unit root or a
Econometrics 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
Time Series 4. Robert Almgren. Oct. 5, 2009
Time Series 4 Robert Almgren Oct. 5, 2009 1 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
Università di Pavia. Forecasting. Eduardo Rossi
Università di Pavia Forecasting Eduardo Rossi Mean Squared Error Forecast of Y t+1 based on a set of variables observed at date t, X t : Yt+1 t. The loss function MSE(Y t+1 t ) = E[Y t+1 Y t+1 t ]2 The
Econ 424 Time Series Concepts
Econ 424 Time Series Concepts Eric Zivot January 20 2015 Time Series Processes Stochastic (Random) Process { 1 2 +1 } = { } = sequence of random variables indexed by time Observed time series of length
Consider the trend-cycle decomposition of a time series y t
1 Unit Root Tests Consider the trend-cycle decomposition of a time series y t y t = TD t + TS t + C t = TD t + Z t The basic issue in unit root testing is to determine if TS t = 0. Two classes of tests,
Time Series Methods. Sanjaya Desilva
Time Series Methods Sanjaya Desilva 1 Dynamic Models In estimating time series models, sometimes we need to explicitly model the temporal relationships between variables, i.e. does X affect Y in the same
E 4101/5101 Lecture 9: Non-stationarity
E 4101/5101 Lecture 9: Non-stationarity Ragnar Nymoen 30 March 2011 Introduction I Main references: Hamilton Ch 15,16 and 17. Davidson and MacKinnon Ch 14.3 and 14.4 Also read Ch 2.4 and Ch 2.5 in Davidson
Univariate 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.
Nonstationary time series models
13 November, 2009 Goals Trends in economic data. Alternative models of time series trends: deterministic trend, and stochastic trend. Comparison of deterministic and stochastic trend models The statistical
Romanian Economic and Business Review Vol. 3, No. 3 THE EVOLUTION OF SNP PETROM STOCK LIST - STUDY THROUGH AUTOREGRESSIVE MODELS
THE EVOLUTION OF SNP PETROM STOCK LIST - STUDY THROUGH AUTOREGRESSIVE MODELS Marian Zaharia, Ioana Zaheu, and Elena Roxana Stan Abstract Stock exchange market is one of the most dynamic and unpredictable
Autoregressive 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:
Class 1: Stationary Time Series Analysis
Class 1: Stationary Time Series Analysis Macroeconometrics - Fall 2009 Jacek Suda, BdF and PSE February 28, 2011 Outline Outline: 1 Covariance-Stationary Processes 2 Wold Decomposition Theorem 3 ARMA Models
Time Series Econometrics 4 Vijayamohanan Pillai N
Time Series Econometrics 4 Vijayamohanan Pillai N Vijayamohan: CDS MPhil: Time Series 5 1 Autoregressive Moving Average Process: ARMA(p, q) Vijayamohan: CDS MPhil: Time Series 5 2 1 Autoregressive Moving
7 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
MA Advanced Econometrics: Applying Least Squares to Time Series
MA Advanced Econometrics: Applying Least Squares to Time Series Karl Whelan School of Economics, UCD February 15, 2011 Karl Whelan (UCD) Time Series February 15, 2011 1 / 24 Part I Time Series: Standard
Permanent Income Hypothesis (PIH) Instructor: Dmytro Hryshko
Permanent Income Hypothesis (PIH) Instructor: Dmytro Hryshko 1 / 36 The PIH Utility function is quadratic, u(c t ) = 1 2 (c t c) 2 ; borrowing/saving is allowed using only the risk-free bond; β(1 + r)
3. ARMA Modeling. Now: Important class of stationary processes
3. ARMA Modeling Now: Important class of stationary processes Definition 3.1: (ARMA(p, q) process) Let {ɛ t } t Z WN(0, σ 2 ) be a white noise process. The process {X t } t Z is called AutoRegressive-Moving-Average
Estimation 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
Time series: Cointegration
Time series: Cointegration May 29, 2018 1 Unit Roots and Integration Univariate time series unit roots, trends, and stationarity Have so far glossed over the question of stationarity, except for my stating
4. 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
Questions and Answers on Unit Roots, Cointegration, VARs and VECMs
Questions and Answers on Unit Roots, Cointegration, VARs and VECMs L. Magee Winter, 2012 1. Let ɛ t, t = 1,..., T be a series of independent draws from a N[0,1] distribution. Let w t, t = 1,..., T, be
Multivariate 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
Lecture 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:
Ch. 14 Stationary ARMA Process
Ch. 14 Stationary ARMA Process A general linear stochastic model is described that suppose a time series to be generated by a linear aggregation of random shock. For practical representation it is desirable
Lecture 1: Fundamental concepts in Time Series Analysis (part 2)
Lecture 1: Fundamental concepts in Time Series Analysis (part 2) Florian Pelgrin University of Lausanne, École des HEC Department of mathematics (IMEA-Nice) Sept. 2011 - Jan. 2012 Florian Pelgrin (HEC)
Econometrics and Structural
Introduction to Time Series Econometrics and Structural Breaks Ziyodullo Parpiev, PhD Outline 1. Stochastic processes 2. Stationary processes 3. Purely random processes 4. Nonstationary processes 5. Integrated
13. Time Series Analysis: Asymptotics Weakly Dependent and Random Walk Process. Strict Exogeneity
Outline: Further Issues in Using OLS with Time Series Data 13. Time Series Analysis: Asymptotics Weakly Dependent and Random Walk Process I. Stationary and Weakly Dependent Time Series III. Highly Persistent
MEI Exam Review. June 7, 2002
MEI Exam Review June 7, 2002 1 Final Exam Revision Notes 1.1 Random Rules and Formulas Linear transformations of random variables. f y (Y ) = f x (X) dx. dg Inverse Proof. (AB)(AB) 1 = I. (B 1 A 1 )(AB)(AB)
University 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
Week 5 Quantitative Analysis of Financial Markets Characterizing Cycles
Week 5 Quantitative Analysis of Financial Markets Characterizing Cycles Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036
Problem 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 +
E 4160 Autumn term Lecture 9: Deterministic trends vs integrated series; Spurious regression; Dickey-Fuller distribution and test
E 4160 Autumn term 2016. Lecture 9: Deterministic trends vs integrated series; Spurious regression; Dickey-Fuller distribution and test Ragnar Nymoen Department of Economics, University of Oslo 24 October
A lecture on time series
A lecture on time series Bernt Arne Ødegaard 15 November 2018 Contents 1 Survey of time series issues 1 1.1 Definition.............................. 2 1.2 Examples.............................. 2 1.3 Typical
11/18/2008. So run regression in first differences to examine association. 18 November November November 2008
Time Series Econometrics 7 Vijayamohanan Pillai N Unit Root Tests Vijayamohan: CDS M Phil: Time Series 7 1 Vijayamohan: CDS M Phil: Time Series 7 2 R 2 > DW Spurious/Nonsense Regression. Integrated but
Econometrics of Panel Data
Econometrics of Panel Data Jakub Mućk Meeting # 9 Jakub Mućk Econometrics of Panel Data Meeting # 9 1 / 22 Outline 1 Time series analysis Stationarity Unit Root Tests for Nonstationarity 2 Panel Unit Root
Stat 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
Unit 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
Ch. 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.
Decision 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
Covers Chapter 10-12, some of 16, some of 18 in Wooldridge. Regression Analysis with Time Series Data
Covers Chapter 10-12, some of 16, some of 18 in Wooldridge Regression Analysis with Time Series Data Obviously time series data different from cross section in terms of source of variation in x and y temporal
11. 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,
Heteroskedasticity 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.