Chapter 6: Multivariate Cointegration Analysis

Size: px
Start display at page:

Download "Chapter 6: Multivariate Cointegration Analysis"

Transcription

1 Chapter 6: Multivariate Cointegration Analysis 1

2 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie VI. Multivariate Cointegration Analysis - Johansen Test... 3 VI.1 The Simpelst Case: p = 1, VAR(1)... 3 VI.2 VAR(p)-Model VI.3 Model Specification VI.4 Testing the Rank of Cointegration - An Example

3 VI. Multivariate Cointegration Analysis - Johansen Test VI.1 The Simpelst Case: p = 1, VAR(1) For example, there is a three dimensional vector Y consisting of the three month interest rates for the US dollar, the Euro and the Yen. Within these three I(1) variables we can find up to two cointegrating relations due to the interest rate parity and stationary expected changes in the rate of exchange. Z 1 Z Y 1 1 Y Y 3

4 As we seen before, we have a VAR(1) model for the M I(1) variables in levels. In this simple case, we can write: Y t = µ + ΓY t-1 + ε t where: Y, µ and ε are (Mx1) vectors and Γ is a (MxM) matrix. 4

5 By subtracting the lagged vectors Y from both sides of the equation we receive the following relation: Y t - Y t-1 = µ + ΓY t-1 - Y t-1 + ε t or Y t = µ + (A 1 - I)Y t-1 + ε t Y t = µ + (Γ - I)Y t-1 + ε t In this equation we have an I(0) vector on the left hand side. On the right side there is a vector of constants as well as another I(0) vector ε. Thus, the term (Γ - I)Y t-1 must be also I(0). If the variables are not cointegrated, then the matrix Γ must be a unit matrix I. On the other hand, if there exists r cointegrated relations (Z is a (rx1) vector), this term can be written as a I(0) variable: (Γ - I)Y t-1 = λγ Y t-1 = λz t-1 where γ is the (rxm) matrix of the cointegration coefficients and λ is a (Mxr) matrix. 5

6 When multiplying with the cointegration matrix the latter results in the (MxM) matrix (Γ - I). This term is I(0) and λ can be interpreted as the matrix of the M times r error correction coefficients: Y t = µ + λz t-1 + ε t This model is a generalization of the ECM in the previous section. In the case of a VAR(1) model there appears no lagged differences in the error correction model. If the initial model constitutes a VAR(p) model then the error correction representation contains additionally (p-1) difference terms. Since the matrix (Γ - I) can be represented by the product of a (rxm) and a (Mxr) matrix, it has the rank r. This means that the number of cointegrated relations is determined by the rank of the matrix. In the marginal case r = 0, i.e Γ = I, the model reduced to a VAR model in differences (M independent random walks). If r equals M we are concerned with M stationary level data, I(0). 6

7 The approach of Johansen is based on the maximum likelihood estimation of the matrix (Γ - I) under the assumption of normal distributed error variables. Following the estimation the hypotheses r = 0, r = 1,, r = M-1 are tested using likelihood ratio (LR) tests. In the formulation of a VAR(p) model we receive the equation: y t = A 0 + Πy t-1 + Γ i y t i + Bx t + ε t As all factors in this equation except Π y t-1 are clearly stationary if the variables are cointegrated, it means that also Π y t-1 must be stationary. Furthermore, every cointegration relationship has to appear in Π. Even more, their number is given by the rank of Π. Π can be decomposed as Π = αβ, where the relevant elements of the α matrix are adjustment coefficients and the β matrix contains the cointegrating vectors. As the interest lies in α and β, the system should be reduced to one containing only them. p-1 i=1 7

8 To do that, one should regress y t on y t-1,, y t-(p-1) and then Y t-1 on the same variables. The residuals are denoted respectively R 0t and R 1t. Now the regression equation is reduced to R 0t = αβ R 1t + e t This is a multivariate regression problem: S S S S is the matrix of sums of squares and sums of products of R 0t and R 1t. Johansen (1991) shows that the asymptotic variance of β R 1t is β Σ 11 β, the asymptotic variance of R 0t is Σ 11 and the asymptotic covariance matrix of β R 1t and R 0t is β Σ 10, where Σ 00, Σ 10, and Σ 11 are the population counterparts of S 00, S 10 and S 11. The procedure is to maximize the likelihood function first with respect to α holding β constant and then maximize with respect to β. For α the result is: α = (β S 11 β) -1 β S 10 8

9 The conditional maximum of the likelihood function with respect to β is (L(β)) -2/T = S 00 -S 01 β(β S 11 β) -1 β S 10 So maximization of the likelihood function with respect to β means minimization of this determinant. By further mathematical manipulations this is equivalent to the finding of the characteristic roots of the equation: S S10S00S -λi = 0 The roots of this equation are the r canonical correlations between R 0t and R 1t. It means that those linear combinations of Y t-1 will be selected that are highly correlated to linear combinations of Y t after conditioning on the lagged variables Y t-1,, Y t-(p-1). 9

10 Denoting with λ i the characteristic value, the maximum likelihood function will be (under the assumption of normal distributed error terms): L -2 / T max = S 00 n i= 1 (1-λˆ i ) Therefore, the estimation problem is a canonical correlation analysis of the current Y t and the lagged Y. 10

11 The trace statistic is Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie λ n = -T ln(1-λˆ ) trace i i= r+ 1 where λˆ r+ 1,, λˆ n are the smallest characteristic roots. If the statistic is bigger than the critical value, the null hypothesis of at most r cointegrating vectors is rejected. The maximum eigenvalue statistic is λ max = -Tln(1-λˆ r + 1) If the statistic is bigger than the critical value, the null hypothesis of exactly r cointegrated vectors is rejected. The critical values for both test are derived from the trace and maximum eigenvalue of the stochastic matrix and depend on whether we include a trend (either linear or quadratic) or a constant in the VAR model. Since we have not to deal with stationary variables, but with I(1) variables, the test values are not χ 2 and follow a different distribution that is tabulated by Johansen and Juselius. 11

12 VI.2 VAR(p)-Model Consider a VAR of order p with M I(1) variables in levels: y t = A 0 + A 1 y t-1 + A 2 y t A p y t-p + Bx t + ε t y t = A 0 + (A 1 - I)y t-1 + A 2 y t-2 + A 3 y t A p y t-p + Bx t + ε t y t = A 0 + (A 1 - I)y t-1 - (A 1 - I)y t-2 + (A 1 - I)y t-2 + A 2 y t-2 + A 3 y t A p y t-p + Bx t + ε t y t = A 0 + (A 1 - I) y t-1 + (A 2 + A 1 - I)y t-2 + A 3 y t A p y t-p + Bx t + ε t y t = A 0 + (A 1 - I) y t-1 + (A 2 + A 1 - I)y t-2 + (A 2 + A 1 - I)y t-3 + (A 2 + A 1 - I)y t-3 + A 3 )y t A p y t-p + Bx t + ε t y t = A 0 + (A 1 - I) y t-1 + (A 2 + A 1 - I) y t-2 + (A 3 + A 2 + A 1 - I)y t A p y t-p + Bx t + ε t with: Γ i = (A i + A i A 1 ), I = unit vector where: y t-p is I(1) and Γ p y t-p is I(0) y t = A 0 + Γ 1 y t-1 + Γ 2 y t Γ p-1 y t-p-1 + Γ p y t-p + Bx t + ε t 12

13 Γ p calculates stationary linear combinations of the non-stationary y and the rows of Γ p are the cointegrating vectors for the elements of y. z p := Γ p y t-p is I(0) or y t = A 0 + Πy t-1 + Γ y + Bx + ε t where y t is a k-vector of non-stationary I(1) variables, x t is a d-vector of deterministic variables, and ε t is a vector of innovations. We may rewrite the VAR as, p-1 i=1 i t i t with: Π = p A i - I and p Γ = - i i=1 j=i+ 1 Aj 13

14 VI.3 Model Specification Eviews considers the following five cases considered by Johansen (1995): 1. The level data y t have no deterministic trends and the cointegrating equations do not have intercepts: H(r): Πy t-1 + Bx t = αβ y t-1 2. The level data y t have no deterministic trends and the cointegrating equations have intercepts: H(r): Πy t-1 + Bx t = α(β y t-1 + ρ 0 ) 3. The level data y t have linear trends but the cointegrating equations have only intercepts: H(r): Πy t-1 + Bx t = α(β y t-1 + ρ 0 ) + α γ 0 14

15 4. The level data y t and the cointegrating equations have linear trends: H(r): Πy t-1 + Bx t = α(β y t-1 + ρ 0 + ρ 1 t) + α γ 0 5. The level data y t have quadratic trends and the cointegrating equations have linear trends: H(r): Πy t-1 + Bx t = α(β y t-1 + ρ 0 + ρ 1 t) + α (γ 0 + γ 1 t) The terms associated with α are the deterministic terms outside the cointegrating relations. When a deterministic term appears both inside and outside the cointegrating relation, the decomposition is not uniquely identified. Johansen (1995) identifies the part that belongs inside the error correction term by orthogonally projecting the exogenous terms on to the α space so that α is the null space of α such that α α = 0. EViews uses a different identification method so that the error correction term has a sample mean of zero. More specifically, we identify the part inside the error correction term by regressing the cointegration relations β y t on a constant (and linear trend). 15

16 VI.4 Testing the Rank of Cointegration - An Example a) The Choice of the optimal Lag Length Lag LogL LR FPE AIC SC HQ NA 6.74e * indicates lag order selected by the criterion e LR: sequential modified LR test statistic (each test at 5% level) e * FPE: Final prediction error e AIC: Akaike information criterion e SC: Schwarz information criterion e * HQ: Hannan-Quinn information criterion * 1.25e-15* * e e e e

17 b) Trace statistics Unrestricted Cointegration Rank Test (Trace) Hypothesize d Trace 0.05 No. of CE(s) Eigenvalue Statistic Critical Value Prob.** None * At most 1 * At most E Trace test indicates 2 cointegrating eqn(s) at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values 17

18 The portion of the output tells you whether there is cointegration and the number of cointegrated vectors. Here one cannot reject the null of two cointegrating vectors using the trace test. We saw in class the differences between the trace and maximal e igenvalue tests. The latter can be evaluated from the column of eigenvalues provided. The trace statistic reports in the first block tests the null hypothesis of r cointegrated relations against the alternative of k cointegrating relations, where k is the number of endogenous variables. We can see from the second column that the first two eigenvalues are much higher compared to the last eigenvalue, which lies near zero. This suggests that there exist two cointegrated relations. The null hypothesis r = 0 and r 1 can clearly be rejected. The calculated test value of 48,75 lies outside the interval between 0 and 29,79. Also the second test value of 15,91 is higher than 15,49. 18

19 c) Maximum eigenvalues statistics Unrestricted Cointegration Rank Test (Maximum Eigenvalue) Hypothesized Max-Eigen 0.05 No. of CE(s) Eigenvalue Statistic Critical Value Prob.** None * At most 1 * At most E Max-eigenvalue test indicates 2 cointegrating eqn(s) at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values 19

Chapter 4: Vector Autoregressive Models

Chapter 4: Vector Autoregressive Models Chapter 4: Vector Autoregressive Models 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie IV.1 Vector Autoregressive Models (VAR)...

More information

Chapter 5: Bivariate Cointegration Analysis

Chapter 5: Bivariate Cointegration Analysis Chapter 5: Bivariate Cointegration Analysis 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie V. Bivariate Cointegration Analysis...

More information

Testing The Quantity Theory of Money in Greece: A Note

Testing The Quantity Theory of Money in Greece: A Note ERC Working Paper in Economic 03/10 November 2003 Testing The Quantity Theory of Money in Greece: A Note Erdal Özmen Department of Economics Middle East Technical University Ankara 06531, Turkey ozmen@metu.edu.tr

More information

Vector Time Series Model Representations and Analysis with XploRe

Vector Time Series Model Representations and Analysis with XploRe 0-1 Vector Time Series Model Representations and Analysis with plore Julius Mungo CASE - Center for Applied Statistics and Economics Humboldt-Universität zu Berlin mungo@wiwi.hu-berlin.de plore MulTi Motivation

More information

COINTEGRATION AND CAUSAL RELATIONSHIP AMONG CRUDE PRICE, DOMESTIC GOLD PRICE AND FINANCIAL VARIABLES- AN EVIDENCE OF BSE AND NSE *

COINTEGRATION AND CAUSAL RELATIONSHIP AMONG CRUDE PRICE, DOMESTIC GOLD PRICE AND FINANCIAL VARIABLES- AN EVIDENCE OF BSE AND NSE * Journal of Contemporary Issues in Business Research ISSN 2305-8277 (Online), 2013, Vol. 2, No. 1, 1-10. Copyright of the Academic Journals JCIBR All rights reserved. COINTEGRATION AND CAUSAL RELATIONSHIP

More information

A Trading Strategy Based on the Lead-Lag Relationship of Spot and Futures Prices of the S&P 500

A Trading Strategy Based on the Lead-Lag Relationship of Spot and Futures Prices of the S&P 500 A Trading Strategy Based on the Lead-Lag Relationship of Spot and Futures Prices of the S&P 500 FE8827 Quantitative Trading Strategies 2010/11 Mini-Term 5 Nanyang Technological University Submitted By:

More information

STATISTICA Formula Guide: Logistic Regression. Table of Contents

STATISTICA Formula Guide: Logistic Regression. Table of Contents : Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 Sigma-Restricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary

More information

Trends and Breaks in Cointegrated VAR Models

Trends and Breaks in Cointegrated VAR Models Trends and Breaks in Cointegrated VAR Models Håvard Hungnes Thesis for the Dr. Polit. degree Department of Economics, University of Oslo Defended March 17, 2006 Research Fellow in the Research Department

More information

TIME SERIES ANALYSIS OF CHINA S EXTERNAL DEBT COMPONENTS, FOREIGN EXCHANGE RESERVES AND ECONOMIC GROWTH RATES. Hüseyin Çetin

TIME SERIES ANALYSIS OF CHINA S EXTERNAL DEBT COMPONENTS, FOREIGN EXCHANGE RESERVES AND ECONOMIC GROWTH RATES. Hüseyin Çetin TIME SERIES ANALYSIS OF CHINA S EXTERNAL DEBT COMPONENTS, FOREIGN EXCHANGE RESERVES AND ECONOMIC GROWTH RATES Hüseyin Çetin Phd Business Administration Candidate Okan University Social Science Institute,

More information

THE EFFECTS OF BANKING CREDIT ON THE HOUSE PRICE

THE EFFECTS OF BANKING CREDIT ON THE HOUSE PRICE THE EFFECTS OF BANKING CREDIT ON THE HOUSE PRICE * Adibeh Savari 1, Yaser Borvayeh 2 1 MA Student, Department of Economics, Science and Research Branch, Islamic Azad University, Khuzestan, Iran 2 MA Student,

More information

Adoptability of Korean Growth Model to Developing Economies: The Case Study of Pakistan

Adoptability of Korean Growth Model to Developing Economies: The Case Study of Pakistan Middle-East Journal of Scientific Research 13(Special Issue of Economics): 43-49, 2013 ISSN 1990-9233 IDOSI Publications, 2013 DOI: 10.5829/idosi.mejsr.2013.13.e.13010 Adoptability of Korean Growth Model

More information

Time Series Analysis III

Time Series Analysis III Lecture 12: Time Series Analysis III MIT 18.S096 Dr. Kempthorne Fall 2013 MIT 18.S096 Time Series Analysis III 1 Outline Time Series Analysis III 1 Time Series Analysis III MIT 18.S096 Time Series Analysis

More information

Gastvortrag: Solvency II, Asset Liability Management, and the European Bond Market Theory and Empirical Evidence

Gastvortrag: Solvency II, Asset Liability Management, and the European Bond Market Theory and Empirical Evidence Risikomanagement von Finanzinstituten Gastvortrag: Solvency II, Asset Liability Management, and the European Bond Market Theory and Empirical Evidence Meik Friedrich Hannover, 16.06.2009 Gliederung Solvency

More information

Forecasting the US Dollar / Euro Exchange rate Using ARMA Models

Forecasting the US Dollar / Euro Exchange rate Using ARMA Models Forecasting the US Dollar / Euro Exchange rate Using ARMA Models LIUWEI (9906360) - 1 - ABSTRACT...3 1. INTRODUCTION...4 2. DATA ANALYSIS...5 2.1 Stationary estimation...5 2.2 Dickey-Fuller Test...6 3.

More information

The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series.

The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series. Cointegration The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series. Economic theory, however, often implies equilibrium

More information

Working Papers. Cointegration Based Trading Strategy For Soft Commodities Market. Piotr Arendarski Łukasz Postek. No. 2/2012 (68)

Working Papers. Cointegration Based Trading Strategy For Soft Commodities Market. Piotr Arendarski Łukasz Postek. No. 2/2012 (68) Working Papers No. 2/2012 (68) Piotr Arendarski Łukasz Postek Cointegration Based Trading Strategy For Soft Commodities Market Warsaw 2012 Cointegration Based Trading Strategy For Soft Commodities Market

More information

Normalization and Mixed Degrees of Integration in Cointegrated Time Series Systems

Normalization and Mixed Degrees of Integration in Cointegrated Time Series Systems Normalization and Mixed Degrees of Integration in Cointegrated Time Series Systems Robert J. Rossana Department of Economics, 04 F/AB, Wayne State University, Detroit MI 480 E-Mail: r.j.rossana@wayne.edu

More information

Introduction to Matrix Algebra

Introduction to Matrix Algebra Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary

More information

Performing Unit Root Tests in EViews. Unit Root Testing

Performing Unit Root Tests in EViews. Unit Root Testing Página 1 de 12 Unit Root Testing The theory behind ARMA estimation is based on stationary time series. A series is said to be (weakly or covariance) stationary if the mean and autocovariances of the series

More information

Introduction to General and Generalized Linear Models

Introduction to General and Generalized Linear Models Introduction to General and Generalized Linear Models General Linear Models - part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby

More information

Review Jeopardy. Blue vs. Orange. Review Jeopardy

Review Jeopardy. Blue vs. Orange. Review Jeopardy Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?

More information

IS THERE A LONG-RUN RELATIONSHIP

IS THERE A LONG-RUN RELATIONSHIP 7. IS THERE A LONG-RUN RELATIONSHIP BETWEEN TAXATION AND GROWTH: THE CASE OF TURKEY Salih Turan KATIRCIOGLU Abstract This paper empirically investigates long-run equilibrium relationship between economic

More information

On the long run relationship between gold and silver prices A note

On the long run relationship between gold and silver prices A note Global Finance Journal 12 (2001) 299 303 On the long run relationship between gold and silver prices A note C. Ciner* Northeastern University College of Business Administration, Boston, MA 02115-5000,

More information

Department of Economics

Department of Economics Department of Economics Working Paper Do Stock Market Risk Premium Respond to Consumer Confidence? By Abdur Chowdhury Working Paper 2011 06 College of Business Administration Do Stock Market Risk Premium

More information

Financial Integration of Stock Markets in the Gulf: A Multivariate Cointegration Analysis

Financial Integration of Stock Markets in the Gulf: A Multivariate Cointegration Analysis INTERNATIONAL JOURNAL OF BUSINESS, 8(3), 2003 ISSN:1083-4346 Financial Integration of Stock Markets in the Gulf: A Multivariate Cointegration Analysis Aqil Mohd. Hadi Hassan Department of Economics, College

More information

The Relationship Between Crude Oil and Natural Gas Prices

The Relationship Between Crude Oil and Natural Gas Prices The Relationship Between Crude Oil and Natural Gas Prices by Jose A. Villar Natural Gas Division Energy Information Administration and Frederick L. Joutz Department of Economics The George Washington University

More information

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written

More information

Univariate and Multivariate Methods PEARSON. Addison Wesley

Univariate and Multivariate Methods PEARSON. Addison Wesley Time Series Analysis Univariate and Multivariate Methods SECOND EDITION William W. S. Wei Department of Statistics The Fox School of Business and Management Temple University PEARSON Addison Wesley Boston

More information

TEMPORAL CAUSAL RELATIONSHIP BETWEEN STOCK MARKET CAPITALIZATION, TRADE OPENNESS AND REAL GDP: EVIDENCE FROM THAILAND

TEMPORAL CAUSAL RELATIONSHIP BETWEEN STOCK MARKET CAPITALIZATION, TRADE OPENNESS AND REAL GDP: EVIDENCE FROM THAILAND I J A B E R, Vol. 13, No. 4, (2015): 1525-1534 TEMPORAL CAUSAL RELATIONSHIP BETWEEN STOCK MARKET CAPITALIZATION, TRADE OPENNESS AND REAL GDP: EVIDENCE FROM THAILAND Komain Jiranyakul * Abstract: This study

More information

SYSTEMS OF REGRESSION EQUATIONS

SYSTEMS OF REGRESSION EQUATIONS SYSTEMS OF REGRESSION EQUATIONS 1. MULTIPLE EQUATIONS y nt = x nt n + u nt, n = 1,...,N, t = 1,...,T, x nt is 1 k, and n is k 1. This is a version of the standard regression model where the observations

More information

Time Series Analysis

Time Series Analysis Time Series Analysis Forecasting with ARIMA models Andrés M. Alonso Carolina García-Martos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and García-Martos (UC3M-UPM)

More information

Least Squares Estimation

Least Squares Estimation Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN-13: 978-0-470-86080-9 ISBN-10: 0-470-86080-4 Editors Brian S Everitt & David

More information

Explaining Cointegration Analysis: Part II

Explaining Cointegration Analysis: Part II Explaining Cointegration Analysis: Part II David F. Hendry and Katarina Juselius Nuffield College, Oxford, OX1 1NF. Department of Economics, University of Copenhagen, Denmark Abstract We describe the concept

More information

An Econometric Measurement of the Impact of Marketing Communication on Sales in the Indian Cement Industry

An Econometric Measurement of the Impact of Marketing Communication on Sales in the Indian Cement Industry An Econometric Measurement of the Impact of Marketing Communication on Sales in the Indian Cement Industry Somroop Siddhanta 1, Neelotpaul Banerjee 2 1 Department of Management Studies, NSHM College of

More information

Factors affecting online sales

Factors affecting online sales Factors affecting online sales Table of contents Summary... 1 Research questions... 1 The dataset... 2 Descriptive statistics: The exploratory stage... 3 Confidence intervals... 4 Hypothesis tests... 4

More information

Advanced Forecasting Techniques and Models: ARIMA

Advanced Forecasting Techniques and Models: ARIMA Advanced Forecasting Techniques and Models: ARIMA Short Examples Series using Risk Simulator For more information please visit: www.realoptionsvaluation.com or contact us at: admin@realoptionsvaluation.com

More information

" Y. Notation and Equations for Regression Lecture 11/4. Notation:

 Y. Notation and Equations for Regression Lecture 11/4. Notation: Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

More information

IMPACT OF AGRICULTURE, MANUFACTURING AND SERVICE INDUSTRY ON THE GDP GROWTH OF PAKISTAN

IMPACT OF AGRICULTURE, MANUFACTURING AND SERVICE INDUSTRY ON THE GDP GROWTH OF PAKISTAN IMPACT OF AGRICULTURE, MANUFACTURING AND SERVICE INDUSTRY ON THE GDP GROWTH OF PAKISTAN ABDUL RAZZAQ NAZISH (Corresponding Author) ABDULLAH IQBAL Dr. MUHAMMAD RAMZAN SUPERIOR UNIVERSITY, Raiwind Road,

More information

Indices of Model Fit STRUCTURAL EQUATION MODELING 2013

Indices of Model Fit STRUCTURAL EQUATION MODELING 2013 Indices of Model Fit STRUCTURAL EQUATION MODELING 2013 Indices of Model Fit A recommended minimal set of fit indices that should be reported and interpreted when reporting the results of SEM analyses:

More information

Is the Basis of the Stock Index Futures Markets Nonlinear?

Is the Basis of the Stock Index Futures Markets Nonlinear? University of Wollongong Research Online Applied Statistics Education and Research Collaboration (ASEARC) - Conference Papers Faculty of Engineering and Information Sciences 2011 Is the Basis of the Stock

More information

Air passenger departures forecast models A technical note

Air passenger departures forecast models A technical note Ministry of Transport Air passenger departures forecast models A technical note By Haobo Wang Financial, Economic and Statistical Analysis Page 1 of 15 1. Introduction Sine 1999, the Ministry of Business,

More information

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they

More information

Non-Stationary Time Series andunitroottests

Non-Stationary Time Series andunitroottests Econometrics 2 Fall 2005 Non-Stationary Time Series andunitroottests Heino Bohn Nielsen 1of25 Introduction Many economic time series are trending. Important to distinguish between two important cases:

More information

Chapter 5: The Cointegrated VAR model

Chapter 5: The Cointegrated VAR model Chapter 5: The Cointegrated VAR model Katarina Juselius July 1, 2012 Katarina Juselius () Chapter 5: The Cointegrated VAR model July 1, 2012 1 / 41 An intuitive interpretation of the Pi matrix Consider

More information

A Primer on Forecasting Business Performance

A Primer on Forecasting Business Performance A Primer on Forecasting Business Performance There are two common approaches to forecasting: qualitative and quantitative. Qualitative forecasting methods are important when historical data is not available.

More information

Jim Gatheral Scholarship Report. Training in Cointegrated VAR Modeling at the. University of Copenhagen, Denmark

Jim Gatheral Scholarship Report. Training in Cointegrated VAR Modeling at the. University of Copenhagen, Denmark Jim Gatheral Scholarship Report Training in Cointegrated VAR Modeling at the University of Copenhagen, Denmark Xuxin Mao Department of Economics, the University of Glasgow x.mao.1@research.gla.ac.uk December

More information

NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( ) Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

More information

Statistical Machine Learning

Statistical Machine Learning Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes

More information

REASSESSMENT OF SUSTAINABILITY OF CURRENT ACCOUNT DEFICIT IN INDIA

REASSESSMENT OF SUSTAINABILITY OF CURRENT ACCOUNT DEFICIT IN INDIA South-Eastern Europe Journal of Economics 1 (2012) 67-79 REASSESSMENT OF SUSTAINABILITY OF CURRENT ACCOUNT DEFICIT IN INDIA AVIRAL KUMAR TIWARI * ICFAI University, Tripura Abstract In this study, we examined

More information

FULLY MODIFIED OLS FOR HETEROGENEOUS COINTEGRATED PANELS

FULLY MODIFIED OLS FOR HETEROGENEOUS COINTEGRATED PANELS FULLY MODIFIED OLS FOR HEEROGENEOUS COINEGRAED PANELS Peter Pedroni ABSRAC his chapter uses fully modified OLS principles to develop new methods for estimating and testing hypotheses for cointegrating

More information

Time Series Analysis

Time Series Analysis Time Series Analysis Identifying possible ARIMA models Andrés M. Alonso Carolina García-Martos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and García-Martos

More information

Relationship between Commodity Prices and Exchange Rate in Light of Global Financial Crisis: Evidence from Australia

Relationship between Commodity Prices and Exchange Rate in Light of Global Financial Crisis: Evidence from Australia Relationship between Commodity Prices and Exchange Rate in Light of Global Financial Crisis: Evidence from Australia Omar K. M. R. Bashar and Sarkar Humayun Kabir Abstract This study seeks to identify

More information

EXPORT INSTABILITY, INVESTMENT AND ECONOMIC GROWTH IN ASIAN COUNTRIES: A TIME SERIES ANALYSIS

EXPORT INSTABILITY, INVESTMENT AND ECONOMIC GROWTH IN ASIAN COUNTRIES: A TIME SERIES ANALYSIS ECONOMIC GROWTH CENTER YALE UNIVERSITY P.O. Box 208269 27 Hillhouse Avenue New Haven, Connecticut 06520-8269 CENTER DISCUSSION PAPER NO. 799 EXPORT INSTABILITY, INVESTMENT AND ECONOMIC GROWTH IN ASIAN

More information

Business Cycles and Natural Gas Prices

Business Cycles and Natural Gas Prices Department of Economics Discussion Paper 2004-19 Business Cycles and Natural Gas Prices Apostolos Serletis Department of Economics University of Calgary Canada and Asghar Shahmoradi Department of Economics

More information

2. Linear regression with multiple regressors

2. Linear regression with multiple regressors 2. Linear regression with multiple regressors Aim of this section: Introduction of the multiple regression model OLS estimation in multiple regression Measures-of-fit in multiple regression Assumptions

More information

Applied Statistics. J. Blanchet and J. Wadsworth. Institute of Mathematics, Analysis, and Applications EPF Lausanne

Applied Statistics. J. Blanchet and J. Wadsworth. Institute of Mathematics, Analysis, and Applications EPF Lausanne Applied Statistics J. Blanchet and J. Wadsworth Institute of Mathematics, Analysis, and Applications EPF Lausanne An MSc Course for Applied Mathematicians, Fall 2012 Outline 1 Model Comparison 2 Model

More information

Week TSX Index 1 8480 2 8470 3 8475 4 8510 5 8500 6 8480

Week TSX Index 1 8480 2 8470 3 8475 4 8510 5 8500 6 8480 1) The S & P/TSX Composite Index is based on common stock prices of a group of Canadian stocks. The weekly close level of the TSX for 6 weeks are shown: Week TSX Index 1 8480 2 8470 3 8475 4 8510 5 8500

More information

ANALYSIS OF EUROPEAN, AMERICAN AND JAPANESE GOVERNMENT BOND YIELDS

ANALYSIS OF EUROPEAN, AMERICAN AND JAPANESE GOVERNMENT BOND YIELDS Applied Time Series Analysis ANALYSIS OF EUROPEAN, AMERICAN AND JAPANESE GOVERNMENT BOND YIELDS Stationarity, cointegration, Granger causality Aleksandra Falkowska and Piotr Lewicki TABLE OF CONTENTS 1.

More information

Factorization Theorems

Factorization Theorems Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization

More information

Testing for Granger causality between stock prices and economic growth

Testing for Granger causality between stock prices and economic growth MPRA Munich Personal RePEc Archive Testing for Granger causality between stock prices and economic growth Pasquale Foresti 2006 Online at http://mpra.ub.uni-muenchen.de/2962/ MPRA Paper No. 2962, posted

More information

5. Multiple regression

5. Multiple regression 5. Multiple regression QBUS6840 Predictive Analytics https://www.otexts.org/fpp/5 QBUS6840 Predictive Analytics 5. Multiple regression 2/39 Outline Introduction to multiple linear regression Some useful

More information

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a

More information

Statistics Graduate Courses

Statistics Graduate Courses Statistics Graduate Courses STAT 7002--Topics in Statistics-Biological/Physical/Mathematics (cr.arr.).organized study of selected topics. Subjects and earnable credit may vary from semester to semester.

More information

Regression Analysis: A Complete Example

Regression Analysis: A Complete Example Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty

More information

The Impact of Macroeconomic Fundamentals on Stock Prices Revisited: Evidence from Indian Data

The Impact of Macroeconomic Fundamentals on Stock Prices Revisited: Evidence from Indian Data Eurasian Journal of Business and Economics 2012, 5 (10), 25-44. The Impact of Macroeconomic Fundamentals on Stock Prices Revisited: Evidence from Indian Data Pramod Kumar NAIK *, Puja PADHI ** Abstract

More information

DEPARTMENT OF ECONOMICS CREDITOR PROTECTION AND BANKING SYSTEM DEVELOPMENT IN INDIA

DEPARTMENT OF ECONOMICS CREDITOR PROTECTION AND BANKING SYSTEM DEVELOPMENT IN INDIA DEPARTMENT OF ECONOMICS CREDITOR PROTECTION AND BANKING SYSTEM DEVELOPMENT IN INDIA Simon Deakin, University of Cambridge, UK Panicos Demetriades, University of Leicester, UK Gregory James, University

More information

Orthogonal Diagonalization of Symmetric Matrices

Orthogonal Diagonalization of Symmetric Matrices MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding

More information

Introduction to Principal Components and FactorAnalysis

Introduction to Principal Components and FactorAnalysis Introduction to Principal Components and FactorAnalysis Multivariate Analysis often starts out with data involving a substantial number of correlated variables. Principal Component Analysis (PCA) is a

More information

Mehtap Ergüven Abstract of Ph.D. Dissertation for the degree of PhD of Engineering in Informatics

Mehtap Ergüven Abstract of Ph.D. Dissertation for the degree of PhD of Engineering in Informatics INTERNATIONAL BLACK SEA UNIVERSITY COMPUTER TECHNOLOGIES AND ENGINEERING FACULTY ELABORATION OF AN ALGORITHM OF DETECTING TESTS DIMENSIONALITY Mehtap Ergüven Abstract of Ph.D. Dissertation for the degree

More information

Chapter 5. Analysis of Multiple Time Series. 5.1 Vector Autoregressions

Chapter 5. Analysis of Multiple Time Series. 5.1 Vector Autoregressions Chapter 5 Analysis of Multiple Time Series Note: The primary references for these notes are chapters 5 and 6 in Enders (2004). An alternative, but more technical treatment can be found in chapters 10-11

More information

Does Insurance Promote Economic Growth? Evidence from the UK

Does Insurance Promote Economic Growth? Evidence from the UK Does Insurance Promote Economic Growth? Evidence from the UK Maurice Kugler and Reza Ofoghi * July 2005 Abstract The first conference of UNCTAD in 1964 acknowledged the development of national insurance

More information

Introduction to Regression and Data Analysis

Introduction to Regression and Data Analysis Statlab Workshop Introduction to Regression and Data Analysis with Dan Campbell and Sherlock Campbell October 28, 2008 I. The basics A. Types of variables Your variables may take several forms, and it

More information

Simple Linear Regression Inference

Simple Linear Regression Inference Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

More information

Implied volatility transmissions between Thai and selected advanced stock markets

Implied volatility transmissions between Thai and selected advanced stock markets MPRA Munich Personal RePEc Archive Implied volatility transmissions between Thai and selected advanced stock markets Supachok Thakolsri and Yuthana Sethapramote and Komain Jiranyakul Public Enterprise

More information

THE EFFECT OF MONETARY GROWTH VARIABILITY ON THE INDONESIAN CAPITAL MARKET

THE EFFECT OF MONETARY GROWTH VARIABILITY ON THE INDONESIAN CAPITAL MARKET 116 THE EFFECT OF MONETARY GROWTH VARIABILITY ON THE INDONESIAN CAPITAL MARKET D. Agus Harjito, Bany Ariffin Amin Nordin, Ahmad Raflis Che Omar Abstract Over the years studies to ascertain the relationship

More information

Cointegration And Causality Analysis of Government Expenditure And Economic Growth In Nigeria

Cointegration And Causality Analysis of Government Expenditure And Economic Growth In Nigeria Cointegration And Causality Analysis of Government Expenditure And Economic Growth In Nigeria Chiawa, M. M, Torruam, J. T, Abur, C. C Abstract:- The study investigates government expenditure and economic

More information

CHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression

CHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the

More information

Foreign Investors and Noise Trade in Istanbul Stock Exchange

Foreign Investors and Noise Trade in Istanbul Stock Exchange Abstract Foreign Investors and Noise Trade in Istanbul Stock Exchange Prof. Dr. Güven Sevil School of Physical Education and Sports Anadolu University Turkey. Prof. Dr. Mustafa Özer Faculty of Economics

More information

SAS Software to Fit the Generalized Linear Model

SAS Software to Fit the Generalized Linear Model SAS Software to Fit the Generalized Linear Model Gordon Johnston, SAS Institute Inc., Cary, NC Abstract In recent years, the class of generalized linear models has gained popularity as a statistical modeling

More information

Overview of Factor Analysis

Overview of Factor Analysis Overview of Factor Analysis Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 35487-0348 Phone: (205) 348-4431 Fax: (205) 348-8648 August 1,

More information

PITFALLS IN TIME SERIES ANALYSIS. Cliff Hurvich Stern School, NYU

PITFALLS IN TIME SERIES ANALYSIS. Cliff Hurvich Stern School, NYU PITFALLS IN TIME SERIES ANALYSIS Cliff Hurvich Stern School, NYU The t -Test If x 1,..., x n are independent and identically distributed with mean 0, and n is not too small, then t = x 0 s n has a standard

More information

Business cycles and natural gas prices

Business cycles and natural gas prices Business cycles and natural gas prices Apostolos Serletis and Asghar Shahmoradi Abstract This paper investigates the basic stylised facts of natural gas price movements using data for the period that natural

More information

Cointegration and error correction

Cointegration and error correction EVIEWS tutorial: Cointegration and error correction Professor Roy Batchelor City University Business School, London & ESCP, Paris EVIEWS Tutorial 1 EVIEWS On the City University system, EVIEWS 3.1 is in

More information

Time Series Analysis

Time Series Analysis Time Series Analysis hm@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby 1 Outline of the lecture Identification of univariate time series models, cont.:

More information

Are the US current account deficits really sustainable? National University of Ireland, Galway

Are the US current account deficits really sustainable? National University of Ireland, Galway Provided by the author(s) and NUI Galway in accordance with publisher policies. Please cite the published version when available. Title Are the US current account deficits really sustainable? Author(s)

More information

Empirical Properties of the Indonesian Rupiah: Testing for Structural Breaks, Unit Roots, and White Noise

Empirical Properties of the Indonesian Rupiah: Testing for Structural Breaks, Unit Roots, and White Noise Volume 24, Number 2, December 1999 Empirical Properties of the Indonesian Rupiah: Testing for Structural Breaks, Unit Roots, and White Noise Reza Yamora Siregar * 1 This paper shows that the real exchange

More information

Part 2: Analysis of Relationship Between Two Variables

Part 2: Analysis of Relationship Between Two Variables Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable

More information

Stock Market Liberalizations: The South Asian Experience

Stock Market Liberalizations: The South Asian Experience Stock Market Liberalizations: The South Asian Experience Fazal Husain and Abdul Qayyum Pakistan Institute of Development Economics P. O. Box 1091, Islamabad PAKISTAN March 2005 I. Introduction Since 1980s

More information

Internet Appendix to Stock Market Liquidity and the Business Cycle

Internet Appendix to Stock Market Liquidity and the Business Cycle Internet Appendix to Stock Market Liquidity and the Business Cycle Randi Næs, Johannes A. Skjeltorp and Bernt Arne Ødegaard This Internet appendix contains additional material to the paper Stock Market

More information

Keywords: Baltic stock markets, unit root, Engle-Granger approach, Johansen cointegration test, causality, impulse response, variance decomposition.

Keywords: Baltic stock markets, unit root, Engle-Granger approach, Johansen cointegration test, causality, impulse response, variance decomposition. Department of Economics Master thesis January 28 Dynamic linkages between Baltic and International stock markets Author: Julija Moroza Supervisor: Hossein Asgharian Abstract 1 The fact is that high integration

More information

Factor analysis. Angela Montanari

Factor analysis. Angela Montanari Factor analysis Angela Montanari 1 Introduction Factor analysis is a statistical model that allows to explain the correlations between a large number of observed correlated variables through a small number

More information

TRACKING ERRORS AND SOVEREIGN DEBT CRISIS

TRACKING ERRORS AND SOVEREIGN DEBT CRISIS EUROPEAN BOND ETFs TRACKING ERRORS AND SOVEREIGN DEBT CRISIS Mikica Drenovak, Branko Urošević, and Ranko Jelic National Bank of Serbia National Bank of Serbia First Annual Conference of Young Serbian Economists

More information

The relationship between stock market parameters and interbank lending market: an empirical evidence

The relationship between stock market parameters and interbank lending market: an empirical evidence Magomet Yandiev Associate Professor, Department of Economics, Lomonosov Moscow State University mag2097@mail.ru Alexander Pakhalov, PG student, Department of Economics, Lomonosov Moscow State University

More information

1 Short Introduction to Time Series

1 Short Introduction to Time Series ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The

More information

AUTOMATION OF ENERGY DEMAND FORECASTING. Sanzad Siddique, B.S.

AUTOMATION OF ENERGY DEMAND FORECASTING. Sanzad Siddique, B.S. AUTOMATION OF ENERGY DEMAND FORECASTING by Sanzad Siddique, B.S. A Thesis submitted to the Faculty of the Graduate School, Marquette University, in Partial Fulfillment of the Requirements for the Degree

More information

LOGISTIC REGRESSION. Nitin R Patel. where the dependent variable, y, is binary (for convenience we often code these values as

LOGISTIC REGRESSION. Nitin R Patel. where the dependent variable, y, is binary (for convenience we often code these values as LOGISTIC REGRESSION Nitin R Patel Logistic regression extends the ideas of multiple linear regression to the situation where the dependent variable, y, is binary (for convenience we often code these values

More information

Department of Economics

Department of Economics Department of Economics On Testing for Diagonality of Large Dimensional Covariance Matrices George Kapetanios Working Paper No. 526 October 2004 ISSN 1473-0278 On Testing for Diagonality of Large Dimensional

More information

Trading Basket Construction. Mean Reversion Trading. Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com

Trading Basket Construction. Mean Reversion Trading. Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com Trading Basket Construction Mean Reversion Trading Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com Speaker Profile Dr. Haksun Li CEO, Numerical Method Inc. (Ex-)Adjunct Professors, Industry

More information

Elucidating the Relationship among Volatility Index, US Dollar Index and Oil Price

Elucidating the Relationship among Volatility Index, US Dollar Index and Oil Price 23-24 July 25, Sheraton LaGuardia East Hotel, New York, USA, ISBN: 978--92269-79-5 Elucidating the Relationship among Volatility Index, US Dollar Index and Oil Price John Wei-Shan Hu* and Hsin-Yi Chang**

More information

Booth School of Business, University of Chicago Business 41202, Spring Quarter 2015, Mr. Ruey S. Tsay. Solutions to Midterm

Booth School of Business, University of Chicago Business 41202, Spring Quarter 2015, Mr. Ruey S. Tsay. Solutions to Midterm Booth School of Business, University of Chicago Business 41202, Spring Quarter 2015, Mr. Ruey S. Tsay Solutions to Midterm Problem A: (30 pts) Answer briefly the following questions. Each question has

More information