Review Session: Econometrics - CLEFIN (20192)
|
|
- Milo Black
- 5 years ago
- Views:
Transcription
1 Review Session: Econometrics - CLEFIN (20192) Part II: Univariate time series analysis Daniele Bianchi March 20, 2013
2 Fundamentals Stationarity A time series is a sequence of random variables x t, t = 1,..., T usually measured at equal intervals. The bulding block of time series analysis is the concept of stationarity. Two main formulations: Strict stationarity: A time series x t is said to be strictly stationary if the joitn distribution of (x t1,..., x tk ) is identical to that of (x t1 +m,..., x tk +m) for all t, with k arbitrarily positive. This is a very strong assumption which is hard to verify empirically. Weak (or covariance) stationarity: A time series x t is said to be covariance stationary if E[x t ]=μ Cov[x t, x t l ]=γ l l
3 Fundamentals Stationarity A simple example of a stationary process is the Gaussian White Noise: x t = ɛ t, ɛ t N(0, σ 2 ) E[x t ]=0, VAR(x t )=σ 2, γ l = 0, l 0
4 Fundamentals Stationarity Sample Autocorrelation Function Sample Autocorrelation Lag
5 A simple autoregressive model Formulation and distribution properties A simple AR(1) model is defined as x t = α + φx t 1 + ɛ t with ɛ t iid(0, σ 2 )
6 A simple autoregressive model Formulation and distribution properties A simple AR(1) model is defined as x t = α + φx t 1 + ɛ t with ɛ t iid(0, σ 2 ) This simple model entails a first order Markov dependence structure since E t 1 x t = α + φx t 1 Var t 1 x t = σ 2 Let us define E(x t )=μand Var(x t )=γ 0. Under stationarity the unconditional moments are given by μ = α + φμ s.t. μ = α 1 φ γ 0 = γ 0 φ 2 + σ 2 s.t. γ 0 = σ2 1 φ
7 A simple autoregressive model Properties: Stationarity The standard AR(1) model boils down to be stationarity if φ < 1, indeed therefore solving (1 φl)x t = ɛ t (1 φz) gives z = 1 which is φ 1 > 1 if φ < 1 φ given stationarity the AR(1) can be decomposed as a linear combination of white noise processes x t = α t 1 1 φ + φ i ɛ t i This can be easily proved by starting with x 2 = α + φx 1 + ɛ 2 then iterating forward. i=0
8 A simple autoregressive model Properties: Autocorrelation AR(1) with φ = AR(1) with φ =
9 A simple autoregressive model Properties: Autocorrelation The autocovariance function can be easily derived as { φγ γ l = 1 + σ 2 ifl = 0 φγ l 1 ifl > 0 By using the standard definition of correlation the autocorrelation function can be defined as ρ l = γ l γ 0 = φγ l 1 γ 0 = φρ l 1 for l 0 now since ρ 0 = 1wehaveρ l = φ l The autorrelation, namely persistency, depends on the autoregressive coefficient in the simple AR(1) model.
10 A simple autoregressive model Properties: Autocorrelation ρ l for φ=0.8 1 Sample Autocorrelation Lag Sample Autocorrelation ρ l for φ= Lag
11 The general AR(p) model The AR(1) model can be generalized for p lags as p x t = α + φ i x t i + ɛ t i=1 the unconditional moments are defined as α E(x t )= 1 p with Var(x t )=γ 0 = i=1 φ i p φ i γ i + σ 2 i=1 Cov(x t, x t i )=γ i = φ 1 γ i 1 + φ 2 γ i φ p γ j p dividing the autocovariance by γ 0 we get the ACF as p ρ j = φ i ρ j i i=1
12 A simple Moving Average model Formulation and distribution properties A simple MA(1) model is defined as x t = α + ɛ t + θɛ t 1 with ɛ t iid(0, σ 2 )
13 A simple Moving Average model Formulation and distribution properties A simple MA(1) model is defined as x t = α + ɛ t + θɛ t 1 with ɛ t iid(0, σ 2 ) The conditional moments can be easily defined as E t 1 x t = α + θɛ t 1 Var t 1 x t = σ 2
14 A simple Moving Average model Formulation and distribution properties A simple MA(1) model is defined as x t = α + ɛ t + θɛ t 1 with ɛ t iid(0, σ 2 ) The conditional moments can be easily defined as E t 1 x t = α + θɛ t 1 Var t 1 x t = σ 2 The unconditional moments can be defined as E(x t )=α Var(x t )=γ 0 = σ 2 + θ 2 σ 2 =(1 + θ 2 )σ 2
15 A simple Moving Average model Properties: Autocorrelation MA(1) with θ = MA(12) with θ i = 0.9 for i=1,..,
16 A simple Moving Average model Properties: Autocorrelation The autocovariance function can be defined as γ 1 = E [(x t α)(x t 1 α)] = E [(ɛ t + θɛ t 1 )(ɛ t + θɛ t 1 )] = σ 2 θ being γ l = 0forl > 1, such that the autocorrelation can be defined as ρ 1 = γ 1 γ 0 = σ 2 θ (1 + θ 2 )σ 2 = θ (1 + θ 2 ) The autocorrelation depends on the moving average parameter θ
17 A simple Moving Average model Properties: Autocorrelation 1 Sample Autocorrelation ρ 1 for θ = Lag 1 Sample Autocorrelation ρ 1 for θ = Lag
18 The ARMA(1,1) model Properties We can combine the AR(1) and the MA(1) as x t = c 1 x t 1 + ɛ t + a 1 ɛ t 1 (1 c 1 L)x t =(1 + a 1 L)ɛ t x t = (1 + a 1L) (1 c 1 L) ɛ t =(1 + a 1 L)(1 + c 1 L + c 2 L )ɛ t =[1 +(a 1 + c 1 )L + c 1 (a 1 + c 1 )L 2 + c 2 1(a 1 + c 1 )L ]ɛ t
19 The ARMA(1,1) model Properties We can combine the AR(1) and the MA(1) as x t = c 1 x t 1 + ɛ t + a 1 ɛ t 1 (1 c 1 L)x t =(1 + a 1 L)ɛ t x t = (1 + a 1L) (1 c 1 L) ɛ t =(1 + a 1 L)(1 + c 1 L + c 2 L )ɛ t =[1 +(a 1 + c 1 )L + c 1 (a 1 + c 1 )L 2 + c 2 1(a 1 + c 1 )L ]ɛ t Now the unconditional moments can be derived under the assumption of weak stationarity Var(x t )= [ 1 +(a 1 + c 1 ) 2 + c 2 1(a 1 + c 1 ) ] σ 2 ɛ = [1 + (a 1 + c 1 ) 2 ] 1 c 2 σ 2 ɛ 1
20 The ARMA(1,1) model Properties We can combine the AR(1) and the MA(1) as x t = c 1 x t 1 + ɛ t + a 1 ɛ t 1 (1 c 1 L)x t =(1 + a 1 L)ɛ t x t = (1 + a 1L) (1 c 1 L) ɛ t =(1 + a 1 L)(1 + c 1 L + c 2 L )ɛ t =[1 +(a 1 + c 1 )L + c 1 (a 1 + c 1 )L 2 + c 2 1(a 1 + c 1 )L ]ɛ t Now the unconditional moments can be derived under the assumption of weak stationarity Var(x t )= [ 1 +(a 1 + c 1 ) 2 + c 2 1(a 1 + c 1 ) ] σ 2 ɛ = [1 + (a 1 + c 1 ) 2 ] 1 c 2 σ 2 ɛ 1 we can clearly see that for lim c 1 1 Var(x t)=
21 The General ARMA(p,q) model The ARMA(1,1) model can be generalized for and order p of the autoregressive structure and q for the moving average part such that x t = ρ 1 x t 1 + ρ 2 x t ρ p x t p + ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t θ q ɛ t q ( 1 ρ1 L ρ 2 L 2... ρ p L p) x t = ( 1 + θ 1 L + θ 2 L θ q L q) ɛ t such that x t ρ(l) =θ(l)ɛ t x t = θ(l) ρ(l) ɛ t = Φ(L)ɛ t
22 Estimate the ARMA model The Box-Jenkins approach Step 1: Make sure that the time series is stationary (e.g. Augmented Dickey -Fuller test). If not stationary takes first order differences. Step 2: Model selection (Information Criteria) Step 3: Model checking (Residuals tests)
23 Example: Step 1 Check for stationarity Test for stationarity of the x t variable (i.e. US Stock market returns) by using the Augmented Dickey-Fuller test x t = c + ˆδx t 1 + in order to check for stationarity k ˆφ i Δx t i 1 + ˆɛ t i=0 t-statistic Prob Test critical values: 1% level % level % level Notice the t stats is defined as t (ˆδ 1 ) /SE (ˆδ ) where the null hypothesis is H 0 : ˆδ = 1.
24 Example: Step 2 Model selection The ACF and the PACF might be misleading the lag structure of the ARMA(p,q) can be investigate by using the information criteria AIC: The Akaike s Information Criteria is defined as AIC = 2log(L)+2(p q) SBC: The Schwarz Bayesian Information Criteria is defined as SBC = 2log(L)+log(T)(p q) where L is the value of the maximized likelihood and T the number of observations Model AIC SBC ARMA(1,1) ARMA(1,2) ARMA(1,3) ARMA(2,1) ARMA(3,1)
25 Example: Step 3 Model estimates and checking Variable Coefficient Std. Error t-statistic Prob. C AR(1) MA(1) Adjusted R-squared Log likelihood Akaike info criterion Schwarz criterion Hannan-Quinn criter
26 Example: Step 3 Model estimates and checking Distribution of the residuals
27 Example: Step 3 Model estimates and checking 0.8 Residuals Autocorrelation Function 0.6 Sample Autocorrelation Lag
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
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 informationProblem Set 2: Box-Jenkins methodology
Problem Set : Box-Jenkins methodology 1) For an AR1) process we have: γ0) = σ ε 1 φ σ ε γ0) = 1 φ Hence, For a MA1) process, p lim R = φ γ0) = 1 + θ )σ ε σ ε 1 = γ0) 1 + θ Therefore, p lim R = 1 1 1 +
More informationLecture 2: ARMA(p,q) models (part 2)
Lecture 2: ARMA(p,q) models (part 2) Florian Pelgrin University of Lausanne, École des HEC Department of mathematics (IMEA-Nice) Sept. 2011 - Jan. 2012 Florian Pelgrin (HEC) Univariate time series Sept.
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 informationBrief Sketch of Solutions: Tutorial 3. 3) unit root tests
Brief Sketch of Solutions: Tutorial 3 3) unit root tests.5.4.4.3.3.2.2.1.1.. -.1 -.1 -.2 -.2 -.3 -.3 -.4 -.4 21 22 23 24 25 26 -.5 21 22 23 24 25 26.8.2.4. -.4 - -.8 - - -.12 21 22 23 24 25 26 -.2 21 22
More informationLecture 1: Fundamental concepts in Time Series Analysis (part 2)
Lecture 1: Fundamental concepts in Time Series Analysis (part 2) Florian Pelgrin University of Lausanne, École des HEC Department of mathematics (IMEA-Nice) Sept. 2011 - Jan. 2012 Florian Pelgrin (HEC)
More 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 informationEcon 427, Spring Problem Set 3 suggested answers (with minor corrections) Ch 6. Problems and Complements:
Econ 427, Spring 2010 Problem Set 3 suggested answers (with minor corrections) Ch 6. Problems and Complements: 1. (page 132) In each case, the idea is to write these out in general form (without the lag
More informationRomanian 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
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 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 information3. 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
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 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 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 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 informationEconometría 2: Análisis de series de Tiempo
Econometría 2: Análisis de series de Tiempo Karoll GOMEZ kgomezp@unal.edu.co http://karollgomez.wordpress.com Segundo semestre 2016 II. Basic definitions A time series is a set of observations X t, each
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 informationClass 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
More informationEconometrics II Heij et al. Chapter 7.1
Chapter 7.1 p. 1/2 Econometrics II Heij et al. Chapter 7.1 Linear Time Series Models for Stationary data Marius Ooms Tinbergen Institute Amsterdam Chapter 7.1 p. 2/2 Program Introduction Modelling philosophy
More 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 informationECON/FIN 250: Forecasting in Finance and Economics: Section 6: Standard Univariate Models
ECON/FIN 250: Forecasting in Finance and Economics: Section 6: Standard Univariate Models Patrick Herb Brandeis University Spring 2016 Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN
More informationFinal Exam Financial Data Analysis at the University of Freiburg (Winter Semester 2008/2009) Friday, November 14, 2008,
Professor Dr. Roman Liesenfeld Final Exam Financial Data Analysis at the University of Freiburg (Winter Semester 2008/2009) Friday, November 14, 2008, 10.00 11.30am 1 Part 1 (38 Points) Consider the following
More information10. Time series regression and forecasting
10. Time series regression and forecasting Key feature of this section: Analysis of data on a single entity observed at multiple points in time (time series data) Typical research questions: What is the
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 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 informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY & Contents PREFACE xiii 1 1.1. 1.2. Difference Equations First-Order Difference Equations 1 /?th-order Difference
More 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 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 informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY PREFACE xiii 1 Difference Equations 1.1. First-Order Difference Equations 1 1.2. pth-order Difference Equations 7
More 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 informationModule 3. Descriptive Time Series Statistics and Introduction to Time Series Models
Module 3 Descriptive Time Series Statistics and Introduction to Time Series Models Class notes for Statistics 451: Applied Time Series Iowa State University Copyright 2015 W Q Meeker November 11, 2015
More 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 informationNon-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
More informationTime Series 2. Robert Almgren. Sept. 21, 2009
Time Series 2 Robert Almgren Sept. 21, 2009 This week we will talk about linear time series models: AR, MA, ARMA, ARIMA, etc. First we will talk about theory and after we will talk about fitting the models
More information13. 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
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 informationECONOMETRIA II. CURSO 2009/2010 LAB # 3
ECONOMETRIA II. CURSO 2009/2010 LAB # 3 BOX-JENKINS METHODOLOGY The Box Jenkins approach combines the moving average and the autorregresive models. Although both models were already known, the contribution
More 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 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 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 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 informationCovariance Stationary Time Series. Example: Independent White Noise (IWN(0,σ 2 )) Y t = ε t, ε t iid N(0,σ 2 )
Covariance Stationary Time Series Stochastic Process: sequence of rv s ordered by time {Y t } {...,Y 1,Y 0,Y 1,...} Defn: {Y t } is covariance stationary if E[Y t ]μ for all t cov(y t,y t j )E[(Y t μ)(y
More 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 informationFrequency Forecasting using Time Series ARIMA model
Frequency Forecasting using Time Series ARIMA model Manish Kumar Tikariha DGM(O) NSPCL Bhilai Abstract In view of stringent regulatory stance and recent tariff guidelines, Deviation Settlement mechanism
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 informationLecture 1: Stationary Time Series Analysis
Syllabus Stationarity ARMA AR MA Model Selection Estimation Lecture 1: Stationary Time Series Analysis 222061-1617: Time Series Econometrics Spring 2018 Jacek Suda Syllabus Stationarity ARMA AR MA Model
More informationECON 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
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 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 informationArma-Arch Modeling Of The Returns Of First Bank Of Nigeria
Arma-Arch Modeling Of The Returns Of First Bank Of Nigeria Emmanuel Alphonsus Akpan Imoh Udo Moffat Department of Mathematics and Statistics University of Uyo, Nigeria Ntiedo Bassey Ekpo Department of
More informationStationary Stochastic Time Series Models
Stationary Stochastic Time Series Models When modeling time series it is useful to regard an observed time series, (x 1,x,..., x n ), as the realisation of a stochastic process. In general a stochastic
More informationLecture 3: Autoregressive Moving Average (ARMA) Models and their Practical Applications
Lecture 3: Autoregressive Moving Average (ARMA) Models and their Practical Applications Prof. Massimo Guidolin 20192 Financial Econometrics Winter/Spring 2018 Overview Moving average processes Autoregressive
More 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 informationProblem set 1 - Solutions
EMPIRICAL FINANCE AND FINANCIAL ECONOMETRICS - MODULE (8448) Problem set 1 - Solutions Exercise 1 -Solutions 1. The correct answer is (a). In fact, the process generating daily prices is usually assumed
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 informationOil price volatility in the Philippines using generalized autoregressive conditional heteroscedasticity
Oil price volatility in the Philippines using generalized autoregressive conditional heteroscedasticity Carl Ceasar F. Talungon University of Southern Mindanao, Cotabato Province, Philippines Email: carlceasar04@gmail.com
More informationNote: The primary reference for these notes is Enders (2004). An alternative and more technical treatment can be found in Hamilton (1994).
Chapter 4 Analysis of a Single Time Series Note: The primary reference for these notes is Enders (4). An alternative and more technical treatment can be found in Hamilton (994). Most data used in financial
More informationMCMC analysis of classical time series algorithms.
MCMC analysis of classical time series algorithms. mbalawata@yahoo.com Lappeenranta University of Technology Lappeenranta, 19.03.2009 Outline Introduction 1 Introduction 2 3 Series generation Box-Jenkins
More informationFinancial 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
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 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 informationIntroduction to Stochastic processes
Università di Pavia Introduction to Stochastic processes Eduardo Rossi Stochastic Process Stochastic Process: A stochastic process is an ordered sequence of random variables defined on a probability space
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 informationEconometrics of financial markets, -solutions to seminar 1. Problem 1
Econometrics of financial markets, -solutions to seminar 1. Problem 1 a) Estimate with OLS. For any regression y i α + βx i + u i for OLS to be unbiased we need cov (u i,x j )0 i, j. For the autoregressive
More informationStochastic Processes: I. consider bowl of worms model for oscilloscope experiment:
Stochastic Processes: I consider bowl of worms model for oscilloscope experiment: SAPAscope 2.0 / 0 1 RESET SAPA2e 22, 23 II 1 stochastic process is: Stochastic Processes: II informally: bowl + drawing
More informationLecture 1: Stationary Time Series Analysis
Syllabus Stationarity ARMA AR MA Model Selection Estimation Forecasting Lecture 1: Stationary Time Series Analysis 222061-1617: Time Series Econometrics Spring 2018 Jacek Suda Syllabus Stationarity ARMA
More informationMAT 3379 (Winter 2016) FINAL EXAM (SOLUTIONS)
MAT 3379 (Winter 2016) FINAL EXAM (SOLUTIONS) 15 April 2016 (180 minutes) Professor: R. Kulik Student Number: Name: This is closed book exam. You are allowed to use one double-sided A4 sheet of notes.
More informationTime Series Analysis -- 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 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 informationThe 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 Elena Roxana Stan Faculty of Internal and International Economy of Tourism RomanianAmerican University,
More informationMidterm Suggested Solutions
CUHK Dept. of Economics Spring 2011 ECON 4120 Sung Y. Park Midterm Suggested Solutions Q1 (a) In time series, autocorrelation measures the correlation between y t and its lag y t τ. It is defined as. ρ(τ)
More 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 informationNon-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
More information5 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
More informationEstimating AR/MA models
September 17, 2009 Goals The likelihood estimation of AR/MA models AR(1) MA(1) Inference Model specification for a given dataset Why MLE? Traditional linear statistics is one methodology of estimating
More informationTime Series Forecasting: A Tool for Out - Sample Model Selection and Evaluation
AMERICAN JOURNAL OF SCIENTIFIC AND INDUSTRIAL RESEARCH 214, Science Huβ, http://www.scihub.org/ajsir ISSN: 2153-649X, doi:1.5251/ajsir.214.5.6.185.194 Time Series Forecasting: A Tool for Out - Sample Model
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 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 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 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 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 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 informationTIME SERIES AND FORECASTING. Luca Gambetti UAB, Barcelona GSE Master in Macroeconomic Policy and Financial Markets
TIME SERIES AND FORECASTING Luca Gambetti UAB, Barcelona GSE 2014-2015 Master in Macroeconomic Policy and Financial Markets 1 Contacts Prof.: Luca Gambetti Office: B3-1130 Edifici B Office hours: email:
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 informationAustrian Inflation Rate
Austrian Inflation Rate Course of Econometric Forecasting Nadir Shahzad Virkun Tomas Sedliacik Goal and Data Selection Our goal is to find a relatively accurate procedure in order to forecast the Austrian
More informationNon-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
More informationEnd-Semester Examination MA 373 : Statistical Analysis on Financial Data
End-Semester Examination MA 373 : Statistical Analysis on Financial Data Instructor: Dr. Arabin Kumar Dey, Department of Mathematics, IIT Guwahati Note: Use the results in Section- III: Data Analysis using
More informationTime Series Solutions HT 2009
Time Series Solutions HT 2009 1. Let {X t } be the ARMA(1, 1) process, X t φx t 1 = ɛ t + θɛ t 1, {ɛ t } WN(0, σ 2 ), where φ < 1 and θ < 1. Show that the autocorrelation function of {X t } is given by
More 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 informationAuthor: Yesuf M. Awel 1c. Affiliation: 1 PhD, Economist-Consultant; P.O Box , Addis Ababa, Ethiopia. c.
ISSN: 2415-0304 (Print) ISSN: 2522-2465 (Online) Indexing/Abstracting Forecasting GDP Growth: Application of Autoregressive Integrated Moving Average Model Author: Yesuf M. Awel 1c Affiliation: 1 PhD,
More informationLesson 9: Autoregressive-Moving Average (ARMA) models
Lesson 9: Autoregressive-Moving Average (ARMA) models Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@ec.univaq.it Introduction We have seen
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 informationModel selection using penalty function criteria
Model selection using penalty function criteria Laimonis Kavalieris University of Otago Dunedin, New Zealand Econometrics, Time Series Analysis, and Systems Theory Wien, June 18 20 Outline Classes of models.
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 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 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 informationGaussian Copula Regression Application
International Mathematical Forum, Vol. 11, 2016, no. 22, 1053-1065 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2016.68118 Gaussian Copula Regression Application Samia A. Adham Department
More informationMAT3379 (Winter 2016)
MAT3379 (Winter 2016) Assignment 4 - SOLUTIONS The following questions will be marked: 1a), 2, 4, 6, 7a Total number of points for Assignment 4: 20 Q1. (Theoretical Question, 2 points). Yule-Walker estimation
More information