This document reviews testing for causality between variables. It begins by defining Granger causality, which tests whether including one time series helps forecast another. For bivariate systems, causality can be tested by examining coefficients in a vector autoregression (VAR) model. For multivariate systems, causality is more complex and graphical models may help. The document outlines procedures for testing causality between stationary and nonstationary time series using impulse responses, vector autoregressive moving average (VARMA) models, and other techniques. It provides examples and discusses challenges like potential omitted common factors.
Granger Causality Test: A Useful Descriptive Tool for Time Series DataIJMER
Interdependency of one or more variables on the other has been in the existence over long
time when it was discovered that one variable has to move or regress toward another following the
work done by Galton (1886); Pearson & Lee (1903); Kendall & Stuart, (1961); Johnston and
DiNardo, (1997); Gujarati, (2004) etc. It was in the light of this dependency over time the researcher
uses Granger Causality as an effective tool in time series Predictive causality using Nigeria GDP and
Money Supply to know the type of causality in existence in the two time series variables under
consideration and which one can statistically predicts the other.
The research work aimed at testing for nature of causality between GDP and money supply for
Federal Republic of Nigeria for the period of thirty years using the data sourced from Central Bank
of Nigeria Statistical Bulletin. After observing the various conditions of Granger causality test such
as ensuring stationarity in the variables under consideration; adding enough number of lags in the
prescribed model before estimation as Granger causality test is sensitive to the number of lags
introduced in the model; and as well as assuming the disturbance terms in the various models are
uncorrelated, the result of the analysis indicates a bilateral relationship between Nigeria GDP and
Money Supply. It implies Nigeria GDP Granger causes money Supply and vice versa. Based on the
result of this study, both Nigeria GDP and money Supply can be successfully model using Vector
Autoregressive Model since changes in one variable has a significant effect on the other variable.
These days a lot of data being generated is in the form of time series. From climate data to users post in social media, stock prices, neurological data etc. Discovering the temporal dependence between different time series data is important task in time series analysis. It finds its application in varied fields ranging from advertising in social media, finding influencers, marketing, share markets, psychology, climate science etc. Identifying the networks of dependencies has been studied in this report.
In this report we have study how this problem has been studied in the field of econometrics. We will also study three different approaches for building causal networks between the time series and then see how this knowledge has been used in three completely different fields. At last some important issues are presented and areas in which this can be extended for further research.
Granger Causality Test: A Useful Descriptive Tool for Time Series DataIJMER
Interdependency of one or more variables on the other has been in the existence over long
time when it was discovered that one variable has to move or regress toward another following the
work done by Galton (1886); Pearson & Lee (1903); Kendall & Stuart, (1961); Johnston and
DiNardo, (1997); Gujarati, (2004) etc. It was in the light of this dependency over time the researcher
uses Granger Causality as an effective tool in time series Predictive causality using Nigeria GDP and
Money Supply to know the type of causality in existence in the two time series variables under
consideration and which one can statistically predicts the other.
The research work aimed at testing for nature of causality between GDP and money supply for
Federal Republic of Nigeria for the period of thirty years using the data sourced from Central Bank
of Nigeria Statistical Bulletin. After observing the various conditions of Granger causality test such
as ensuring stationarity in the variables under consideration; adding enough number of lags in the
prescribed model before estimation as Granger causality test is sensitive to the number of lags
introduced in the model; and as well as assuming the disturbance terms in the various models are
uncorrelated, the result of the analysis indicates a bilateral relationship between Nigeria GDP and
Money Supply. It implies Nigeria GDP Granger causes money Supply and vice versa. Based on the
result of this study, both Nigeria GDP and money Supply can be successfully model using Vector
Autoregressive Model since changes in one variable has a significant effect on the other variable.
These days a lot of data being generated is in the form of time series. From climate data to users post in social media, stock prices, neurological data etc. Discovering the temporal dependence between different time series data is important task in time series analysis. It finds its application in varied fields ranging from advertising in social media, finding influencers, marketing, share markets, psychology, climate science etc. Identifying the networks of dependencies has been studied in this report.
In this report we have study how this problem has been studied in the field of econometrics. We will also study three different approaches for building causal networks between the time series and then see how this knowledge has been used in three completely different fields. At last some important issues are presented and areas in which this can be extended for further research.
Definition of Co-integration .
Different Approaches of Co-integration.
Johansen and Juselius (J.J) Co-integration.
Error Correction Model (ECM).
Interpretation of ECM term.
Long – Run Co-integration Equation.
A Stochastic Iteration Method for A Class of Monotone Variational Inequalitie...CSCJournals
We examined a general method for obtaining a solution to a class of monotone variational inequalities in Hilbert space. Let H be a real Hilbert space, and Let T : H -> H be a continuous linear monotone operator and K be a non empty closed convex subset of H. From an initial arbitrary point x0 ∈ K. We proposed and obtained iterative method that converges in norm to a solution of the class of monotone variational inequalities. A stochastic scheme {xn} is defined as follows: x(n+1) = xn - anF* (xn), n≥0, F*(xn), n ≥ 0 is a strong stochastic approximation of Txn - b, for all b (possible zero) ∈ H and an ∈ (0,1).
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Stochastic Analysis of Van der Pol OscillatorModel Using Wiener HermiteExpans...IJERA Editor
We study a model related to Van der Poloscillatorunder an external stochastic excitation described by white
noise process. This study is limited to find the Gaussian behavior of the stochastic solution processes related to
the model. Under the application ofWiener-Hermite expansion, a deterministic system is generated to describe
the Gaussian solution parameters (Mean and Variance).The deterministic system solution is approximated by
applying the multi-stepdifferential transformedmethodand the results are compared with NDSolveMathematica
10 package. Some case studies are considered to illustrate some comparisons for the obtained results related to
the Gaussian behavior parameters.
Stochastic Analysis of Van der Pol OscillatorModel Using Wiener HermiteExpans...IJERA Editor
We study a model related to Van der Poloscillatorunder an external stochastic excitation described by white
noise process. This study is limited to find the Gaussian behavior of the stochastic solution processes related to
the model. Under the application ofWiener-Hermite expansion, a deterministic system is generated to describe
the Gaussian solution parameters (Mean and Variance).The deterministic system solution is approximated by
applying the multi-stepdifferential transformedmethodand the results are compared with NDSolveMathematica
10 package. Some case studies are considered to illustrate some comparisons for the obtained results related to
the Gaussian behavior parameters.
Time Delay and Mean Square Stochastic Differential Equations in Impetuous Sta...ijtsrd
This paper specially exhibits about the time delay and mean square stochastic differential equations in impetuous stabilization is analyzed. By projecting a delay differential inequality and using the stochastic analysis technique, a few present stage for mean square exponential stabilization are survived. It is express that an unstable stochastic delay system can be achieved some stability by impetuous. This example is also argued to derived the efficiency of the obtained results. G. Pramila | S. Ramadevi"Time Delay and Mean Square Stochastic Differential Equations in Impetuous Stabilization" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-3 , April 2018, URL: http://www.ijtsrd.com/papers/ijtsrd11062.pdf http://www.ijtsrd.com/humanities-and-the-arts/other/11062/time-delay-and-mean-square-stochastic-differential-equations-in-impetuous-stabilization/g-pramila
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 5
Module 5
Chapter 10: Correlation and Regression
Chapter 11: Goodness of Fit and Contingency Tables
Chapter 12: Analysis of Variance
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 11: Goodness-of-Fit and Contingency Tables
11.2: Contingency Tables
RESIDUALS AND INFLUENCE IN NONLINEAR REGRESSION FOR REPEATED MEASUREMENT DATAorajjournal
All observations don’t have equal significance in regression analysis. Diagnostics of observations is an important aspect of model building. In this paper, we use diagnostics method to detect residuals and influential points in nonlinear regression for repeated measurement data. Cook distance and Gauss newton method have been proposed to identify the outliers in nonlinear regression analysis and parameter estimation. Most of these techniques based on graphical representations of residuals, hat matrix and case deletion measures. The results
show us detection of single and multiple outliers cases in repeated measurement data. We use these techniques
to explore performance of residuals and influence in nonlinear regression model.
Definition of Co-integration .
Different Approaches of Co-integration.
Johansen and Juselius (J.J) Co-integration.
Error Correction Model (ECM).
Interpretation of ECM term.
Long – Run Co-integration Equation.
A Stochastic Iteration Method for A Class of Monotone Variational Inequalitie...CSCJournals
We examined a general method for obtaining a solution to a class of monotone variational inequalities in Hilbert space. Let H be a real Hilbert space, and Let T : H -> H be a continuous linear monotone operator and K be a non empty closed convex subset of H. From an initial arbitrary point x0 ∈ K. We proposed and obtained iterative method that converges in norm to a solution of the class of monotone variational inequalities. A stochastic scheme {xn} is defined as follows: x(n+1) = xn - anF* (xn), n≥0, F*(xn), n ≥ 0 is a strong stochastic approximation of Txn - b, for all b (possible zero) ∈ H and an ∈ (0,1).
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Stochastic Analysis of Van der Pol OscillatorModel Using Wiener HermiteExpans...IJERA Editor
We study a model related to Van der Poloscillatorunder an external stochastic excitation described by white
noise process. This study is limited to find the Gaussian behavior of the stochastic solution processes related to
the model. Under the application ofWiener-Hermite expansion, a deterministic system is generated to describe
the Gaussian solution parameters (Mean and Variance).The deterministic system solution is approximated by
applying the multi-stepdifferential transformedmethodand the results are compared with NDSolveMathematica
10 package. Some case studies are considered to illustrate some comparisons for the obtained results related to
the Gaussian behavior parameters.
Stochastic Analysis of Van der Pol OscillatorModel Using Wiener HermiteExpans...IJERA Editor
We study a model related to Van der Poloscillatorunder an external stochastic excitation described by white
noise process. This study is limited to find the Gaussian behavior of the stochastic solution processes related to
the model. Under the application ofWiener-Hermite expansion, a deterministic system is generated to describe
the Gaussian solution parameters (Mean and Variance).The deterministic system solution is approximated by
applying the multi-stepdifferential transformedmethodand the results are compared with NDSolveMathematica
10 package. Some case studies are considered to illustrate some comparisons for the obtained results related to
the Gaussian behavior parameters.
Time Delay and Mean Square Stochastic Differential Equations in Impetuous Sta...ijtsrd
This paper specially exhibits about the time delay and mean square stochastic differential equations in impetuous stabilization is analyzed. By projecting a delay differential inequality and using the stochastic analysis technique, a few present stage for mean square exponential stabilization are survived. It is express that an unstable stochastic delay system can be achieved some stability by impetuous. This example is also argued to derived the efficiency of the obtained results. G. Pramila | S. Ramadevi"Time Delay and Mean Square Stochastic Differential Equations in Impetuous Stabilization" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-3 , April 2018, URL: http://www.ijtsrd.com/papers/ijtsrd11062.pdf http://www.ijtsrd.com/humanities-and-the-arts/other/11062/time-delay-and-mean-square-stochastic-differential-equations-in-impetuous-stabilization/g-pramila
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 5
Module 5
Chapter 10: Correlation and Regression
Chapter 11: Goodness of Fit and Contingency Tables
Chapter 12: Analysis of Variance
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 11: Goodness-of-Fit and Contingency Tables
11.2: Contingency Tables
RESIDUALS AND INFLUENCE IN NONLINEAR REGRESSION FOR REPEATED MEASUREMENT DATAorajjournal
All observations don’t have equal significance in regression analysis. Diagnostics of observations is an important aspect of model building. In this paper, we use diagnostics method to detect residuals and influential points in nonlinear regression for repeated measurement data. Cook distance and Gauss newton method have been proposed to identify the outliers in nonlinear regression analysis and parameter estimation. Most of these techniques based on graphical representations of residuals, hat matrix and case deletion measures. The results
show us detection of single and multiple outliers cases in repeated measurement data. We use these techniques
to explore performance of residuals and influence in nonlinear regression model.
Comment expliquer le développement de l’économétrie des séries temporelles ? Pourquoi ne s’être pas contenté des méthodes économétriques classiques, en utilisant simplement la régression simple et multiple par les « Moindres Carrés Ordinaires », et leurs divers prolongements (MCG, etc.) ? On peut répondre à ces questions selon deux optiques. Tout d’abord, le développement de la macro-dynamique théorique, des modèles de théorie financière moderne a débouché sur un certain nombre de problèmes empiriques qui nécessitent la mise au point d’outils appropriés et nouveaux. Ensuite, dans un certain nombre de cas, la méthode d’estimation des MCO ne s’applique pas, tout simplement et si on l’applique nous induirons en erreur tout un processus d’explications possible. Le traitement économétrique des modèles nécessite une analyse préalable des données. Notre attention se portera successivement sur les profils théoriques des séries temporelles, sur la détection de la non-stationnarité, sur le travail concret d’estimation et enfin sur les problèmes d’analyse des résultats.
Knowledge of cause-effect relationships is central to the field of climate science, supporting mechanistic understanding, observational sampling strategies, experimental design, model development and model prediction. While the major causal connections in our planet's climate system are already known, there is still potential for new discoveries in some areas. The purpose of this talk is to make this community familiar with a variety of available tools to discover potential cause-effect relationships from observed or simulation data. Some of these tools are already in use in climate science, others are just emerging in recent years. None of them are miracle solutions, but many can provide important pieces of information to climate scientists. An important way to use such methods is to generate cause-effect hypotheses that climate experts can then study further. In this talk we will (1) introduce key concepts important for causal analysis; (2) discuss some methods based on the concepts of Granger causality and Pearl causality; (3) point out some strengths and limitations of these approaches; and (4) illustrate such methods using a few real-world examples from climate science.
Bayesian inference for mixed-effects models driven by SDEs and other stochast...Umberto Picchini
An important, and well studied, class of stochastic models is given by stochastic differential equations (SDEs). In this talk, we consider Bayesian inference based on measurements from several individuals, to provide inference at the "population level" using mixed-effects modelling. We consider the case where dynamics are expressed via SDEs or other stochastic (Markovian) models. Stochastic differential equation mixed-effects models (SDEMEMs) are flexible hierarchical models that account for (i) the intrinsic random variability in the latent states dynamics, as well as (ii) the variability between individuals, and also (iii) account for measurement error. This flexibility gives rise to methodological and computational difficulties.
Fully Bayesian inference for nonlinear SDEMEMs is complicated by the typical intractability of the observed data likelihood which motivates the use of sampling-based approaches such as Markov chain Monte Carlo. A Gibbs sampler is proposed to target the marginal posterior of all parameters of interest. The algorithm is made computationally efficient through careful use of blocking strategies, particle filters (sequential Monte Carlo) and correlated pseudo-marginal approaches. The resulting methodology is is flexible, general and is able to deal with a large class of nonlinear SDEMEMs [1]. In a more recent work [2], we also explored ways to make inference even more scalable to an increasing number of individuals, while also dealing with state-space models driven by other stochastic dynamic models than SDEs, eg Markov jump processes and nonlinear solvers typically used in systems biology.
[1] S. Wiqvist, A. Golightly, AT McLean, U. Picchini (2020). Efficient inference for stochastic differential mixed-effects models using correlated particle pseudo-marginal algorithms, CSDA, https://doi.org/10.1016/j.csda.2020.107151
[2] S. Persson, N. Welkenhuysen, S. Shashkova, S. Wiqvist, P. Reith, G. W. Schmidt, U. Picchini, M. Cvijovic (2021). PEPSDI: Scalable and flexible inference framework for stochastic dynamic single-cell models, bioRxiv doi:10.1101/2021.07.01.450748.
In this paper we focus on mixed model analysis for regression model to take account of over dispersion in random effects. Moreover, we present the Data Exploration, Box plot, QQ plot, Analysis of variance, linear models, linear mixed –effects model for testing the over dispersion parameter in the mixed model. A mixed model is similar in many ways to a linear model. It estimates the effects of one or more explanatory variables on a response variable. In this article, the mixed model analysis was analyzed with the R-Language. The output of a mixed model will give you a list of explanatory values, estimates and confidence intervals of their effect sizes, P-values for each effect, and at least one measure of how well the model fits. The application of the model was tested using open-source dataset such as using numerical illustration and real datasets
Optimal Estimating Sequence for a Hilbert Space Valued ParameterIOSR Journals
Some optimality criteria used in estimation of parameters in finite dimensional space has been extended to a separable Hilbert space. Different optimality criteria and their equivalence are established for estimating sequence rather than estimator. An illustrious example is provided with the estimation of the mean of a Gaussian process
call for papers, research paper publishing, where to publish research paper, journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJEI, call for papers 2012,journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, research and review articles, engineering journal, International Journal of Engineering Inventions, hard copy of journal, hard copy of certificates, journal of engineering, online Submission, where to publish research paper, journal publishing, international journal, publishing a paper, hard copy journal, engineering journal
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...ijistjournal
This paper derives new results for the global chaos synchronization of identical hyperchaotic Qi systems (2008), identical hyperchaotic Jha systems (2007) and non-identical hyperchaotic Qi and Jha systems. Active nonlinear control is the method adopted to achieve the complete synchronization of the identical and different hyperchaotic Qi and Jha systems. Our stability results derived in this paper are established using Lyapunov stability theory. Numerical simulations are shown to validate and illustrate the effectiveness of the synchronization results derived in this paper.
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...ijistjournal
This paper derives new results for the global chaos synchronization of identical hyperchaotic Qi systems (2008), identical hyperchaotic Jha systems (2007) and non-identical hyperchaotic Qi and Jha systems. Active nonlinear control is the method adopted to achieve the complete synchronization of the identical and different hyperchaotic Qi and Jha systems. Our stability results derived in this paper are established using Lyapunov stability theory. Numerical simulations are shown to validate and illustrate the effectiveness of the synchronization results derived in this paper.
STATISTICAL TESTS TO COMPARE k SURVIVAL ANALYSIS FUNCTIONS INVOLVING RECURREN...Carlos M Martínez M
The objective of this paper is to propose statistical tests to compare k survival curves involving recurrent events. Recurrent events occur in many important scientific areas: psychology, bioengineering, medicine, physics, astronomy,
biology, economics and so on. Such events are very common in the real world: viral diseases, seizure, carcinogenic tumors, fevers, machinery and equipment failures, births, murders, rain, industrial accidents, car accidents and so on. The idea is to generalize the weighted statistics used to compare survival curves in classical models. The estimation of the survival functions is based on a non-parametric model proposed by Peña et al., using counting processes. Rlanguage programs using known routines like survival and survrec were designed to make the calculations. The database Byar experiment is used and the time (months) of recurrence of tumors in 116 sick patients with superficia bladder cancer is measured. These patients were randomly allocated to the following treatments: placebo (47 patients), pyridoxine (31 patients) and thiotepa (38 patients). The aim is to compare the survival curves of the three groups and to determine if there are significant differences between treatments.
Discretization of a Mathematical Model for Tumor-Immune System Interaction wi...mathsjournal
The present study deals with the analysis of a Lotka-Volterra model describing competition between tumor and immune cells. The model consists of differential equations with piecewise constant arguments and based on metamodel constructed by Stepanova. Using the method of reduction to discrete equations, it is obtained a system of difference equations from the system of differential equations. In order to get local and global stability conditions of the positive equilibrium point of the system, we use Schur-Cohn criterion and Lyapunov function that is constructed. Moreover, it is shown that periodic solutions occur as a consequence of Neimark-Sacker bifurcation.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Introduction to AI for Nonprofits with Tapp Network
isi
1. Notes on Testing Causality
Jin-Lung Lin
Institute of Economics, Academia Sinica
Department of Economics, National Chengchi University
May 27, 2008
2. Abstract
This note reviews the definition, distribution theory and modeling strategy of test-
ing causality. Starting with the definition of Granger Causality, we discuss various
issues on testing causality within stationary and nonstationary systems. In addi-
tion, we cover the graphical modeling and spectral domain approaches which are
relatively unfamiliar to economists. We compile a list of Do and Don’t Do on causal-
ity testing and review several empirical examples.
3. 1 Introduction
Testing causality among variables is one of the most important and, yet, one of the
most difficult issues in economics. The difficulty arises from the non-experimental
nature of social science. For natural science, researchers can perform experiments
where all other possible causes are kept fixed except for the sole factor under in-
vestigation. By repeating the process for each possible cause, one can identify the
causal structures among factors or variables. There are no such luck for social sci-
ence, and economics is no exception. All different variables affect the same variable
simultaneously and repeated experiments under control are infeasible (experimen-
tal economics is no solution, at least, not yet).
Two most difficult challenges are :
1. Correlation does not imply causality. Distinguishing between these two is by
no means an easy task.
2. There always exist the possibility of ignored common factors. The causal
relationship among variables might disappear when the previously ignored
common causes are considered.
While there are no satisfactory answer to these two questions and there might
never be one, philosophers and social scientists have attempted to use graphical
models to address the second issue. As for the first issue, time series analysts look
for rescue from the unique unidirectional property of time arrow: cause precedes
effect. Based upon this concept, Clive W.J. Granger has proposed a working defi-
nition of causality, using the foreseeability as a yardstick which is called Granger
causality. This note examines and reviews the key issues in testing causality in eco-
nomics.
In additional to this introduction, Section 2 discusses the definition of Granger
causality. Testing causality for stationary processes are reviewed in Section 3 and
Section 4 focuses on nonstationary processes. We turn to graphical models in sec-
tion 5. A to-do and not-to-do list is put together in Section 6.
2 Defining Granger causality
2.1 Two assumptions
1. The future cannot cause the past. The past causes the present or future. (How
about expectation?)
1
4. 2. A cause contains unique information about an effect not available elsewhere.
2.2 Definition
Xt is said not to Granger-cause Yt if for all h > 0
F(Yt+h Ωt) = F(yt+h Ωt − Xt)
where F denotes the conditional distribution and Ωt − Xt is all the information in
the universe except series Xt. In plain words, Xt is said to not Granger-cause Yt if
X cannot help predict future Y.
Remarks:
• The whole distribution F is generally difficult to handle empirically and we
turn to conditional expectation and variance.
• It is defined for all h > 0 and not only for h = 1. Causality at different h does
not imply each other. They are neither sufficient nor necessary.
• Ωt contains all the information in the universe up to time t that excludes the
potential ignored common factors problem. The question is: how to mea-
sure Ωt in practice? The unobserved common factors are always a potential
problem for any finite information set.
• Instantaneous causality Ωt+h − xt+h and feedback is difficult to interpret un-
less on has additional structural information.
A refined definition become as below:
Xt does not Granger cause Yt+h with respect to information Jt, if
E(Yt+h Jt, Xt) = E(Yt+h Jt)
Remark: Note that causality here is defined as relative to. In other words, no
effort is made to find the complete causal path and possible common factors.
2.3 Equivalent definition
For a l-dimension stationary process, Zt, there exists a canonical MA representa-
tion
Zt = µ + Φ(B)ut
= µ +
∞
∑
i=1
Φiut−i, Φ0 = Il
2
5. A necessary and sufficient condition for variable k not Granger-cause variable j is
that Φjk,i = 0, for i = 1, 2, ⋯. If the process is invertible, then
Zt = C + A(B)Zt−1 + ut
= C +
∞
∑
i=1
Ai Zt−i + ut
If there are only two variables, or two-group of variables, j and k, then a necessary
and sufficient condition for variable k not to Granger-cause variable j is that Ajk,i =
0, for i = 1, 2, ⋯. The condition is good for all forecast horizon, h.
Note that for a VAR(1) process with dimension equal or greater than 3, Ajk,i =
0, for i = 1, 2, ⋯ is sufficient for non-causality at h = 1 but insufficient for h > 1.
Variable k might affect variable j in two or more period in the future via the effect
through other variables. For example,
⎡
⎢
⎢
⎢
⎢
⎢
⎣
y1t
y2t
y3t
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
.5 0 0
.1 .1 .3
0 .2 .3
⎤
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎣
y1t−1
y2t−1
y3t−1
⎤
⎥
⎥
⎥
⎥
⎥
⎦
+
⎡
⎢
⎢
⎢
⎢
⎢
⎣
u1t
u2t
u3t
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Then,
y0 =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
u10
u20
u30
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
; y1 = A1 y0 =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
.5
.1
0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
; y2 = A2
1 y0 =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
.25
.06
.02
⎤
⎥
⎥
⎥
⎥
⎥
⎦
To summarize,
1. For bivariate or two groups of variables, IR analysis is equivalent to applying
Granger-causality test to VAR model;
2. For testing the impact of one variable on the other within a high dimensional
(≥ 2) system, IR analysis can not be substituted by the Granger-causality test.
For example, for an VAR(1) process with dimension greater than 2, it does not
suffice to check the upper right-hand corner element of the coefficient matrix
in order to determine if the last variable is noncausal for the first variable.
Test has to be based upon IR.
See Lutkepohl(1993) and Dufor and Renault (1998) for detailed discussion.
3
6. 3 Testing causality for stationary series
3.1 Impulse response and causal ordering
It is well known that residuals from a VAR model are generally correlated and ap-
plying the Cholesky decomposition is equivalent to assuming recursive causal or-
dering from the top variable to the bottom variable. Changing the order of the
variables could greatly change the results of the impulse response analysis.
3.2 Causal analysis for bivariate VAR
For a bivariate system, yt, xt defined by
[
yt
xt
] = [
A11(B) A12(B)
A21(B) A22(B)
][
yt−1
xt−1
] + [
uyt
uxt
]
= [
Φ11(B) Φ12(B)
Φ21(B) Φ22(B)
][
uyt−1
uxt−1
] + [
uyt
uxt
]
xt does not Granger-cause yt if Φ12(B) = 0 or Φ12,i = 0, for i = 1, 2, ⋯. This
condition is equivalent to A12,i = 0, for i = 1, 2, ⋯,p. In other words, this corre-
sponds to the restrictions that all cross-lags coefficients are all zeros which can be
tested by Wald statistics.
We now turn to determining the causal direction for bivariate VAR system. For
ease of illustration, we shall focus upon bivariate AR(1) process so that Aij(B) =
Aij, i, j = 1, 2 as defined above. The results can be easily generalized to AR(p) case.
Four possible causal directions between x and y are:
1. Feedback, H0, x ↔ y
H0 = (
A11 A12
A21 A22
)
2. Independent, H1 x ⊥ y
H1 = (
A11 0
0 A22
)
3. x causes y but y does not cause x, H2, y /→ x
H2 = (
A11 A12
0 A22
)
4
7. 4. y causes x but x does not cause y, H3, x /→ y
H3 = (
A11 0
A21 A22
)
Caines, Keng and Sethi(1981) proposed a two-stage testing procedure for deter-
mining causal directions. In first stage, test H1 (null) against H0, H2 (null) against
H0, and H3 (null) against H0. If necessary, test H1 (null) against H2, and H1 (null)
against H3. See Liang, Chou and Lin(1995) for an application.
3.3 Causal analysis for multivariate VAR
Possible causal structure grows exponentially as number of variables increase. Pair-
wise causal structure might change when different conditioning variables are added.
Caines, Keng and Sethi (1981) provided a reasonable procedure.
1. For a pair (X, Y), construct bivariate VAR with order chosen to minimize
multivariate final prediction error (MFPE);
2. Apply the stagewise procedure to determine the causal structure of X, Y;
3. If a process X, has n multiple causal variables, y1, . . . , yn, rank these variables
according to the decreasing order of their specific gravity which is the inverse
of MFPE(X, yi);
4. For each caused variable process, X, first construct the optimal univariate AR
model using FPE to determine the lag order. The, add the causal variable,
one at a time according to their causal rank and use FPE to determine the
optimal orders at each step. Finally, we get the optimal ordered univariate
multivariate AR model of X against its causal variables;
5. Pool all the optimal univariate AR models above and apply the Full Infor-
mation Maximum Likelihood (FIML) method to estimate the system. Fi-
nally perform the diagnostic checking with the whole system as maintained
model.
3.4 Causal analysis for Vector ARMA model (h = 1)
Let X be n × 1 vector generated by
Φ(B)Xt = Θ(B)at
5
8. Xi does not cause Xj if and only if
det(Φi(z), Θ(j)(z)) = 0
where Φi(B) is the ith column of the matrix Φ(z) and Θ(j)(z) is the matrix Θ(z)
without its jth column.
For bivariate (two-group) case,
(
Φ11(B) Φ12(B)
Φ21(B) Φ22(B)
)(
Xit
X2t
) = (
Θ11(B) Θ12(B)
Θ21(B) Θ22(B)
)(
a1t
a2t
)
Then, Xi does not cause Xj if and only if
Φ21(z) − Θ21(z)Θ−1
(z)11Φ11(z) = 0
If n1 = n2 = 1, Then, Xi does not cause Xj if and only if
Θ11(z)Φ12(z) − Θ21(z)Φ11(z) = 0
General testing procedures is:
1. Build a multivariate ARMA model for Xt,
2. Derive the noncausality conditions in term of AR and MA parameters, say
Rj(βl ) = 0, j = 1, . . . , K
3. Choose a test criterion, Wald, LM or LR test.
Let
T( ˆβl ) = (
∂Rj(B)
∂βl
βl − ˆβl
)k×k
Let V(βl ) be the asymptotic covariance matrix of
√
N( ˆβl = βl ). Then the Wald
and LR test statistics are:
ξW = NR( ˆβl )′
[T( ˆβl )′
V( ˆβl )T( ˆβl )]−1
R( ˆβl ),
ξLR = 2(L( ˆβ, X) − L( ˆβ∗, X))
where ˆβ∗ is the MLE of β under the constraint of noncausality.
6
9. To illustrate, let Xt be a invertible 2-dimensional ARMA(1,1) model.
(
1 − ϕ11B −ϕ12B
−ϕ21B 1 − ϕ22B
)(
X1t
X2t
) = (
1 − θ11B θ12B
θ21B θ22B
)(
a1t
a2t
)
X1 does not cause X2 if and only if
Θ11(z)Φ21(z) − Θ21(z)Φ11(z) = 0
(ϕ21 − θ21)z + (θ11θ21 − ϕ21θ11)z2
= 0
ϕ21 − θ21 = 0, ϕ11θ21 − ϕ21θ11 = 0
For the vector, βl = (ϕ11, ϕ21, θ11, θ21)′, the matrix
T(βl ) =
⎛
⎜
⎜
⎜
⎝
0 θ21
1 −θ11
0 −ϕ21
−1 ϕ11
⎞
⎟
⎟
⎟
⎠
might not be nonsingular under the null of H0 X1 does not cause X2.
Remarks:
• The conditions are weaker than ϕ21 = θ21 = 0
• ϕ21 − θ21 = 0 is a necessary condition for H0, ϕ21 = θ21 = 0 is sufficient condi-
tion and ϕ21 − θ21 = 0, &ϕ11 = θ11 are sufficient for H0.
Let H0 X1 does not cause X2. Consider the following hypotheses:
H1
0 ϕ21 − θ21 = 0;
H2
0 ϕ21 = θ21 = 0
H3
0 ϕ21 ≠ 0, ϕ21 − θ21 = 0, and ϕ11 − θ11 = 0
˜H3
0 ϕ11 − θ11 = 0
Then, H3
0 = ˜H3
0 ∩ H1
0, H2
0 ⊆ H0 ⊆ H1
0, H3
0 ⊆ H0 ⊆ H1
0.
Testing procedures:
1. Test H1
0 at level α1. If H1
0 is rejected, then H0 is rejected. Stop.
2. If H1
0 is not rejected, test H2
0 at level α2. If H2
0 is not rejected, H0 cannot be
rejected. Stop
3. If H2
0 is rejected, test ˜H3
0 ϕ11 − θ11 = 0 at level α2. If ˜H3
0 is rejected, then H0 is
also rejected. If ˜H3
0 is not reject ed, then H0 is also not rejected.
7
10. 4 Causal analysis for nonstationary processes
The asymptotic normal or χ2 distribution in previous section is build upon the
assumption that the underlying processes Xt is stationary. The existence of unit
root and cointegration might make the traditional asymptotic inference invalid.
Here, I shall briefly review unit root and cointegration and their relevance with
testing causality. In essence, cointegration, causality test, VAR model and IR are
closely related and should be considered jointly.
4.1 Unit root:
What is unit root?
The time series yt as defined in Ap(B)yt = C(B)єt has an unit root if Ap(1) =
0, C(1) ≠ 0.
Why do we care about unit root?
• For yt, the existence of unit roots implies that a shock in єt has permanent
impacts on yt.
• If yt has a unit root, then the traditional asymptotic normality results usually
no longer apply. We need different asymptotic theorems.
4.2 Cointegration:
What is cointegration?
When linear combination of two I(1) process become an I(0) process, then these
two series are cointegrated.
Why do we care about cointegration?
• Cointegration implies existence of long-run equilibrium;
• Cointegration implies common stochastic trend;
• With cointegration, we can separate short- and long- run relationship among
variables;
• Cointegration can be used to improve long-run forecast accuracy;
8
11. • Cointegration implies restrictions on the parameters and proper accounting
of these restrictions could improve estimation efficiency.
Let Yt be k-dimensional VAR(p) series with r cointegration vector β(p × r).
Ap(B)Yt = Ut
∆Yt = ΠYt−1 +
p−1
∑
i=1
Γi∆Yt−i + ΦDt + Ut
Yt = C
t
∑
i=1
(Ut + ΦDi) + C∗
(B)(Ut + ΦDt) + Pβ⊥ Y0
Ap(1) = −Π = αβ′
C = β⊥(α′
⊥Γβ⊥)−1
α′
⊥
• Cointegration introduces one additional causal channel (error correction
term) for one variable to affect the other variables. Ignoring this additional
channel will lead to invalid causal analysis.
• For cointegrated system, impulse response estimates from VAR model in
level without explicitly considering cointegration will lead to incorrect con-
fidence interval and inconsistent estimates of responses for long horizons.
Recommended procedures for testing cointegration:
1. Determine order of VAR(p). Suggest choosing the minimal p such that the
residuals behave like vector white noise;
2. Determine type of deterministic terms: no intercept, intercept with con-
straint, intercept without constraint, time trend with constraint, time trend
without constraint. Typically, model with intercept without constraint is pre-
ferred;
3. Use trace or λmax tests to determine number of unit root;
4. Perform diagnostic checking of residuals;
5. Test for exclusion of variables in cointegration vector;
6. Test for weak erogeneity to determine if partial system is appropriate;
9
12. 7. Test for stability;
8. Test for economic hypotheses that are converted to homogeneous restric-
tions on cointegration vectors and/or loading factors.
4.3 Unit root, Cointegration and causality
For a VAR system, Xt with possible unit root and cointegration, the usual causal-
ity test for the level variables could be misleading. Let Xt = (X1t, X2t, X3t)′ with
n1, n2, n3 dimension respectively. The VAR level model is:
Xt = J(B)Xt−1 + ut
=
k
∑
i=1
Ji Xt−i + ut
The null hypothesis of X3 does not cause X1 can be formulated as:
H0 J1,13 = J2,13 = ⋯ = Jk,13 = 0
Let FLS be the Wald statistics for testing H0.
1. If Xt has unit root and is not cointegrated, FLS converges to a limiting distri-
bution which is the sum of χ2 and unit root distribution. The test is similar
and critical values can be constructed. Yet, it is more efficient and easier to
difference Xt and test causality for the differenced VAR.
2. If there is sufficient cointegration for X3 then FLS → χ2
n1n2k. , More specifi-
cally, let A = (A1, A2, A3) be the cointegration vector. The usual asymptotic
distribution results hold if rank(A3) = n3, ie. all X3 appear in the cointegra-
tion vector.
3. If there is not sufficient cointegration, ie. not all X3 appears in the cointe-
gration vector, then the limiting distribution contain unit root and nuisance
parameters.
For the error correction model,
∆Xt = J∗
(B)∆Xt−1 + ΓA′
Xt−1 + ut
where Γ, A are respectively the loading matrix and cointegration vector. Partition
Γ, A conforming to X1, X2, X3. Then, if rank(A3) = n3 or rank(Γ1) = n1, FML →
10
13. χ2
n1n3k. In other words, testing with ECM the usual asymptotic distribution hold
when there are sufficient cointegrations or sufficient loading vector.
Remark: The Johansen test seems to assume sufficient cointegration or suffi-
cient loading vector.
Toda and Yamamoto (1995) proposed a test of causality without pretesting coin-
tegration. For an VAR(p) process and each series is at most I(k), then estimate the
augmented VAR(p+k) process even the last k coefficient matrix is zero.
Xt = A1Xt−1 + ⋯ + ApXt−k + ⋯ + Ap+k Xt−p−k + Ut
and perform the usual Wald test Akj,i = 0, i = 1, ⋯, p. The test statistics is
asymptotical χ2 with degree of freedom m being the number of constraints. The
result holds no matter whether Xt is I(0) or I(1) and whether there exist cointegra-
tion.
As there is no free lunch under the sun, the Toda-Yamamoto test suffer the
following weakness.
• Inefficient as compared with ECM where cointegration is explicitly consid-
ered.
• Cannot distinguish between short run and long run causality.
• Cannot test for hypothesis on long run equilibrium, say PPP which is for-
mulated on cointegration vector.
One more remark: Cointegration between two variables implies existence of
long-run causality for at least one direction. Testing cointegration and causality
should be considered jointly.
5 Causal analysis using graphical models
A directed graph assigns a contemporaneous causal flow among a set of variables
based on correlations and partial correlations. The edge relationship of each pair
of variables characterizes the causal relationship between them. No edge indicates
(conditional) independence between two variables, whereas an undirected edge
(X − Y) signifies a correlation with no particular causal interpretation. A directed
edge (Y → X) means Y causes X but X does not cause Y conditional upon other
variables. A bidirected edge (X ↔ Y) indicates bidirectional causality between
these two variables. In other words, there is contemporaneous feedback between
X and Y.
11
14. To illustrate the main idea, let X, Y, Z be three variables under investigation.
Y ← X → Z represents the fact that X is the common cause of Y and Z. Uncondi-
tional correlation between Y and Z is nonzero but conditional correlation between
Y and Z given X is zero. On the other hand, Y → X ← Z says that both Y and Z
cause X. Thus, unconditional correlation between Y and Z is zero but conditional
correlation between Y and Z given X is nonzero. Similarly, Y → X → Z states
the fact that Y causes X and X causes Z. Again, being conditional upon X, Y is
uncorrelated with Z. The direction of the arrow is then transformed into the zero
constraints of A(i, j), i ≠ j. Let ut = (Xt, Yt, Zt)′ and then the corresponding A
matrix for the three cases discussed above denoted as A1, A2 and A3 are:
A1 =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 0 0
a21 1 0
a31 0 1
⎤
⎥
⎥
⎥
⎥
⎥
⎦
; A2 =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 a12 a13
0 1 0
0 0 1
⎤
⎥
⎥
⎥
⎥
⎥
⎦
; A3 =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 a12 0
0 1 0
a31 0 1
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Several search algorithms are available and the PC algorithm seems to be the
most popular one (see Pearl (2000), and Spirtes, Glymour and Scheines (1993) for
the details). In this paper, we adopt the PC algorithm and outline the main algo-
rithm as shown below. First, we start with a graph in which each variable is con-
nected by an edge with every other variable. We then compute the unconditional
correlation between each pair of variables and remove the edge for the insignificant
pairs. We then compute the 1-th order conditional correlation between each pair
of variables and eliminate the edge between the insignificant ones. We repeat the
procedure to compute the i-th order conditional correlation until i = N-2, where
N is the number of variables under investigation. Fisher’s z statistic is used in the
significance test:
z(i, j K) = 1/2(n − K − 3)
(1/2)
ln(
1 + r[i, j K]
1 − r[i, j K]
)
where r([i, j K]) denotes conditional correlation between variables, which i and j
being conditional upon the K variables, and K the number of series for K.
Under some regularity conditions, z approximates the standard normal distri-
bution. Next, for each pair of variables (Y, Z) that are unconnected by a direct
edge but are connected through an undirected edge through a third variable X, we
assign Y → X ← Z if and only if the conditional correlations of Y and Z condi-
tional upon all possible variable combinations with the presence of the X variable
are nonzero. We then repeat the above process until all possible cases are exhausted.
If X → Z, Z −Y and X and Y are not directly connected, we assign Z → Y. If there
12
15. is a directed path between X and Y (say X → Z → Y) and there is an undirected
edge between X and Y, we then assign X → Y.
Pearl (2000) and Spirtes, Glymour, and Scheines (1993) provide a detailed ac-
count of this approach. Demiralp and Hoover (2003) present simulation results to
show how the efficacy of the PC algorithm varies with signal strength. In general,
they find the directed graph method to be a useful tool in structural causal analysis.
6 Causality on the spectral domain
Causality on the time domain is qualitative but the strength of causality at each
frequency can be measured on spectral domain. To my mind, this is an ideal model
for analyzing permanent consumption theory. Let (xt, yt) be generated by
[
xt
yt
] = [
Λ11(B) Λ12(B)
Λ21(B) Λ22(B)
][
ext
eyt
]
Rewrite the above as
[
xt
yt
] = [
Γ11(B) Γ12(B)
Γ21(B) Γ22(B)
][
˜ext
˜eyt
]
where
[
Γ11(B) Γ12(B)
Γ21(B) Γ22(B)
] = [
Λ11(B) Λ12(B)
Λ21(B) Λ22(B)
][
1 0
ρ 1
]
and
[
˜ext
˜eyt
] = [
1 0
−ρ 1
][
ext
eyt
]
fx(w) =
1
2π
{ Γ11(z) 2
+ Γ12(z) 2
(1 − ρ2
)}
where z = e−iw.
Hosoya’s measure of one-way causality is defined as:
My→x(w) = log[
fx(w)
1/2π Γ11(z) 2
]
= log[1 +
Λ12(z) 2(1 − ρ2)
Λ11(z) + ρΛ12(z)2
]
13
16. 6.1 Error correction model
Let xt, yt be I(1) and ut = yt − Axt be an I(0). The the error correction model is:
∆xt = λ1ut−1 + a1(B)∆xt−1 + b1(B)∆yt−1 + ext
∆yt = λ2ut−1 + a2(B)∆xt−1 + b2(B)∆yt−1 + eyt
[
D(B)xt
D(B)yt
] = [
(1 − B)(1 − b2B)λ2B λ1B + b1B(1 − B)
(1 − B)a2B − λ2AB λ1AB + (1 − a1B)(1 − B)
][
ext
eyt
]
where D(B) arises from matrix inversion. Then,
My→x(w) = log[1 +
λ1 + b1(1 − z) 2(1 − ρ2)
{(1 − z)(¯z − b2) − λ2} + {λ1 + b1(1 − z)} ρ 2
]
where ¯z = eiw.
7 Softwares
Again, the usual disclaim applies. They are subjective. Your choices might be as
good as mine. See Lin(2004) for a detailed account.
1. Impulse responses: Reduced form and structural form
• VAR.SRC/RATS by Norman Morin
• SVAR.SRC/RATS by Antonio Lanzarotti and Mario Seghelini
• VAR/View/Impulse/Eviews
• FinMetrics/Splus
2. Cointegration:
• CATS/RATS
• COINT2/GAUSS
• VAR/Eviews
• urca/R
• FinMetrics/Splus
14
17. 3. Impulse response under cointegration constraint:
CATS,CATSIRFS/RATS
4. Stability analysis:
• CATS/RATS
• Eviews
• FinMetrics/Splus
8 Do and Don’t Do list
8.1 Don’t Do
1. Don’t do single equation causality testing and draw inference on the causal
direction,
2. Don’t test causality between each possible pair of variables and then draw
conclusions on the causal directions among variables,
3. Do not employ the two-step causality testing procedure though it is not an
uncommon practice.
People often test for cointegration first and then treat the error-correction
term as an independent regressor and then apply the usual causality testing.
This procedure is flawed for the following reasons. First, EC term is esti-
mated and using it as an regressor in the next step will give rise to generated
regressor problem. That is, the usual standard deviation in the second step
is not right. Second, there could be more than one cointegration vectors and
linear combination of them are also cointegrated vectors.
8.2 Do
1. Examine the graphs first. Look for pattern, mismatch of seasonality, abnor-
mality, outliers, etc.
2. Always perform diagnostic checking of residuals:
Time series modelling does not obtain help from economic theory and de-
pends heavily upon statistical aspects of correct model specification. White-
ness of residuals are the key assumption.
15
18. 3. Often graph the residuals and check for abnormality and outliers.
4. Be aware of seasonality for data not seasonally adjusted.
5. Apply the Wald test within the Johansen framework where one can test for
hypothesis on long- and short- run causality.
6. When you employ several time series methods or analyze several similar
models, be careful about the consistency among them.
7. Always watch for balance between explained and explanatory variables in
regression analysis. For example, if the dependent variable has a time-trend
but explanatory variables are limited between 0 and 1, then the regression
coefficient can never be a fixed constant. Be careful about mixing I(0) and
I(1) variables in one equation.
8. For VAR, number of parameters grows rapidly with number of variables and
lags. Removing the insignificant parameters to achieve estimation efficiency
is strongly recommended. The resulting IR will be more accurate.
9 Empirical Examples
1. Evaluating the effectiveness of interest rate policy in Taiwan: an impulse re-
sponses analysis
Lin(2003a).
2. Modelling information flow among four stock markets in China
Lin and Wu (2006).
3. Causality between export expansion and manufacturing growth (if time per-
mits)
Liang, Chou and Lin (1995).
Reference Books:
1. Banerjee, Anindya and David F. Hendry eds. (1997), The Econometrics of
Economic Policy, Oxford: Blackwell Publishers
2. Hamilton, James D. Time Series Analysis, New Jersey: Princeton University
Press, 1994
16
19. 3. Clive Granger Forecasting Economic Time Series, 2nd edition Academic
Press 1986.
4. Johansen, S. (1995) Likelihood-based inference in cointegrated vector au-
toregressive models, Oxford: Oxford University Press
5. Lutkepohl, Helmut Introduction to multiple time series analysis, 2nd ed. Springer-
Verlag, 1991.
6. Pena, D, G. Tiao, and R. Tsay, eds. A course in time series analysis, New York:
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