These slides are based on:

Slides:

Advertisements

Similar presentations

TOOLS OF THE FORECASTER

Advertisements

FINANCIAL TIME-SERIES ECONOMETRICS SUN LIJIAN Feb 23,2001.

Autocorrelation Functions and ARIMA Modelling

Nonstationary Time Series Data and Cointegration

STATIONARY AND NONSTATIONARY TIME SERIES

1 CHAPTER 10 FORECASTING THE LONG TERM: DETERMINISTIC AND STOCHASTIC TRENDS Figure 10.1 Economic Time Series with Trends González-Rivera: Forecasting for.

Time Series Building 1. Model Identification

R. Werner Solar Terrestrial Influences Institute - BAS Time Series Analysis by means of inference statistical methods.

STAT 497 APPLIED TIME SERIES ANALYSIS

Copyright © 2008 Pearson Education, Inc. Chapter 1 Linear Functions Copyright © 2008 Pearson Education, Inc.

Copyright © 2008 Pearson Education, Inc. Chapter 11 Probability and Calculus Copyright © 2008 Pearson Education, Inc.

Copyright © 2008 Pearson Education, Inc. Chapter 6 Applications of the Derivatives Copyright © 2008 Pearson Education, Inc.

Copyright © 2008 Pearson Education, Inc. Chapter 3 The Derivative Copyright © 2008 Pearson Education, Inc.

Copyright © 2008 Pearson Education, Inc. Chapter 13 The Trigonometric Functions Copyright © 2008 Pearson Education, Inc.

Modern methods The classical approach: MethodProsCons Time series regression Easy to implement Fairly easy to interpret Covariates may be added (normalization)

K. Ensor, STAT Spring 2005 The Basics: Outline What is a time series? What is a financial time series? What is the purpose of our analysis? Classification.

Slide Copyright © 2010 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems Business Statistics First Edition.

Slide 4- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Active Learning Lecture Slides For use with Classroom Response.

1 CHAPTER 14 FORECASTING VOLATILITY II Figure 14.1 Autocorrelograms of the Squared Returns González-Rivera: Forecasting for Economics and Business, Copyright.

L7: ARIMA1 Lecture 7: ARIMA Model Process The following topics will be covered: Properties of Stock Returns AR model MA model ARMA Non-Stationary Process.

Review of Probability.

Chapter 11 Simple Regression

Copyright © 2012 Pearson Education, Inc. All rights reserved. Chapter 10 Introduction to Time Series Modeling and Forecasting.

Figure 1.1 Population of the United State at 10-year Intervals ( ) González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson.

Linear Stationary Processes. ARMA models. This lecture introduces the basic linear models for stationary processes. Considering only stationary processes.

1 CHAPTER 3 STATISTICS AND TIME SERIES González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc. Figure 3.1 Examples.

FAME Time Series Econometrics Daniel V. Gordon Department of Economics University of Calgary.

Week 21 Stochastic Process - Introduction Stochastic processes are processes that proceed randomly in time. Rather than consider fixed random variables.

Analysis of sapflow measurements of Larch trees within the inner alpine dry Inn- valley PhD student: Marco Leo Advanced Statistics WS 2010/11.

K. Ensor, STAT Spring 2004 Memory characterization of a process How would the ACF behave for a process with no memory? What is a short memory series?

1 CHAPTER 7 FORECASTING WITH AUTOREGRESSIVE (AR) MODELS Figure 7.1 A Variety of Time Series Cycles González-Rivera: Forecasting for Economics and Business,

MULTIVARIATE TIME SERIES & FORECASTING 1. 2 : autocovariance function of the individual time series.

Statistics for Business and Economics 8 th Edition Chapter 7 Estimation: Single Population Copyright © 2013 Pearson Education, Inc. Publishing as Prentice.

1 EE571 PART 3 Random Processes Huseyin Bilgekul Eeng571 Probability and astochastic Processes Department of Electrical and Electronic Engineering Eastern.

1 CHAPTER 6 FORECASTING WITH MOVING AVERAGE (MA) MODELS González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

Computational Finance II: Time Series K.Ensor. What is a time series? Anything observed sequentially (by time?) Returns, volatility, interest rates, exchange.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series.

Components of Time Series Su, Chapter 2, section II.

Correlogram - ACF. Modeling for Forecast Forecast Data The Base Model Linear Trend Logistic Growth Others Models Look for a best approximation of the.

1 CHAPTER 9 FORECASTING PRACTICE II: ASSESSMENT OF FORECASTS AND COMBINATION OF FORECASTS 9.1 Optimal Forecast González-Rivera: Forecasting for Economics.

Chapter 6 Random Processes

Analysis of Financial Data Spring 2012 Lecture 4: Time Series Models - 1 Priyantha Wijayatunga Department of Statistics, Umeå University

Stochastic Process - Introduction

Multiple Random Variables and Joint Distributions

Introduction to Time Series Analysis

Time Series Analysis and Its Applications

Chapter 6: Autoregressive Integrated Moving Average (ARIMA) Models

FORECASTING PRACTICE I

FORECASTING WITH A SYSTEM OF EQUATIONS: VECTOR AUTOREGRESSION

REVIEW OF THE LINEAR REGRESSION MODEL

STATIONARY AND NONSTATIONARY TIME SERIES

A LINK TO ECONOMIC MODELS

REVIEW OF THE LINEAR REGRESSION MODEL

Machine Learning Week 4.

Introduction to Econometrics

FORECASTING THE LONG TERM AND

Financial Forecasting M.Sc. in Finance – 2018/19 – 1st Semester

Introduction to Time Series

These slides are based on:

INTRODUCTION AND CONTEXT

Introduction to Time Series Analysis

Introduction to Time Series Analysis

Chapter 6 Random Processes

Chapter 4 Mathematical Expectation.

Chapter 7 The Normal Distribution and Its Applications

Time Series introduction in R - Iñaki Puigdollers

Basic descriptions of physical data

Ch.1 Basic Descriptions and Properties

FORECASTING VOLATILITY I

BOX JENKINS (ARIMA) METHODOLOGY

Presentation transcript:

These slides are based on: CHAPTER 3 STATISTICS AND TIME SERIES A time series is a collection of records indexed by time. Figure 3.1 Examples of Time Series Dow Jones Index These slides are based on: González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc. Slides adapted for this course. We thank Gloria González-Rivera and assume full responsibility for all errors due to our changes which are mainly in red

12% Y 3.1 Stochastic Process and Time Series Random variables can be characterized in two ways: probability density/mass function (full characterization) moments (partial characterization) 10 12% Y pdf Figure 3.2 Probability Density Function González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

Figure 3.3 Graphical Representation of a Stochastic Process A stochastic process is a collection of random variables indexed by time. 1 2 t T …………….. Figure 3.3 Graphical Representation of a Stochastic Process González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

3.1.1 Time Series A time series is a sample realization of a stochastic process. 1 2 t T …………….. . ………. Figure 3.4 Graphical Representation of a Stochastic Process and a Time Series (Thick Line) González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

Figure 3.5 Nonstationary and Stationary Stochastic Process 3.2.1 Stationarity Figure 3.5 Nonstationary and Stationary Stochastic Process 1 2 t T …………….. Nonstationary Stationary González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

A stochastic process is said to be first order strongly stationary if all random variables have the same probability mass/density function. A stochastic process is said to be first order weakly stationary if all random variables have the same mean. 1 T 2 t …………….. Figure 3.6 Strongly Stationary Stochastic Process A stochastic process is said to be second order weakly stationary (or covariance stationary) if all random variables have the same mean and the same variance and covariances do not depend on time, only on lag. González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

3.2.2 Useful transformations of Nonstationary Processes To transform a nonstationary series into a first-moment-stationary series, we can apply first differences : Lag operator To stabilize the variance we can use the Box-Cox transformation: (before taking differences) González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

3.2.2 Useful transformations of Nonstationary Processes Table 3.1 Dow Jones Index and Returns Date González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

Figure 3.7 Dow Jones Index and Its Transformation to Returns González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

3.3 A New Tool of Analysis: The Autocorrelation Functions Given two random variables Y and X, the correlation coefficient is a measure of the linear association. Autocorrelation coefficient: For second order stationary processes : The autocorrelation function (ACF) is the function González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

3.3 A New Tool of Analysis: The Autocorrelation Functions Figure 3.8 Annual Hours Worked per Person Employed in Germany Autocorrelation function k 1 2 3 4 5 6 7 8 9 10 .22 .29 -.10 .16 -.01 .19 -.06 -.04 .09 .20 González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

Table 3.2 Percentage Change in Working Hours in Germany: Calculation of the Autocorrelation Coefficients 1978 -1.0604 1979 -0.6699 1980 -1.1018 1981 -1.2413 1982 -0.6497 1983 -0.7536 1984 -0.6826 1985 -1.3733 1986 -1.1438 1987 -1.3533 1988 -0.3196 1989 -1.4574 1990 -1.4536 1991 -2.0234 1992 -0.5904 1993 -1.4550 1994 0.0264 1995 -0.7087 1996 -0.9752 1997 -0.5249 1998 -0.2026 1999 -0.7057 2000 -1.2126 2001 -0.8375 2002 -0.7745 2003 -0.4141 2004 0.2950 2005 -0.2879 2006 -0.0492 Mean: -0.8026 Variance: 0.2905 (k= 1,3) 0.0651 -0.0282 0.2240 -0.0970 González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

Figure 3.9 Percentage Change in Working Hours in Germany: Autocorrelations of Order 1 and 3 González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

Autocorrelation between and controlling for the effect of The partial autocorrelation function (PACF) is the function: González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

Figure 3.10 Annual Working Hours per Employee in the United States González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

3.3.2 Statistical Tests for Autocorrelation Coefficients Figure 3.11 Time Series: Annual Working Hours per Employee in the United States. Autocorrelation Function González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.