Multivariate GARCH DCC Estimation

In this video tutorial, we will demonstrate the process of Multivariate GARCH DCC Estimation using OxMetrics 6. Multivariate Garch can be elaborate in the following written tutorial from: https://vlab.stern.nyu.edu/doc/13?topic=mdls

Definition of Multivariate GARCH

Consider n time series of returns and make the usual assumption that returns are serially uncorrelated. Then, we can define a vector of zero-mean white noises εt=rtμ, where rt is the n1 vector of returns and μ is the vector of expected returns.

Despite of being serially uncorrelated, the returns may present contemporaneous correlation. That is:

t?t1[(rtμ)(rtμ)]

may not be a diagonal matrix. Moreover, this contemporaneous variance may be time-varying, depending on past information.

The GARCH-DCC involves two steps. The first step accounts for the conditional heteroskedasticity. It consists in estimating, for each one of the n series of returns rit, its conditional volatility σit using a GARCH model (see garch documentation). Let Dt be a diagonal matrix with these conditional volatilities, i.e. Di,it=σit and, if ij, Di,jt=0. Then the standardized residuals are:

νtD-1t(rtμ)

and notice that these standardized residuals have unit conditional volatility. Now, define the matrix:

R⎯⎯⎯1Tt=1Tνtνt

This is the Bollerslev's Constant Conditional Correlation (CCC) Estimator (Bollerslev, 1990).

The second step consists in generalizing Bollerslev's CCC to capture dynamics in the correlation, hence the name Dynamic Conditional Correlation (DCC). The DCC correlations are:

Qt=R⎯⎯⎯+α(νt1νt1R⎯⎯⎯)+β(Qt1R⎯⎯⎯)

So, Qi,jt is the correlation between rit and rjt at time t, and that is what is plotted by V-Lab.

Estimation

The estimation of one GARCH model for each of the n time series of returns in the first step is standard. For details on GARCH estimation, see garch documentation.

For the second step, which is the DCC estimation per se, V-Lab estimates both parameters, α and β, simultaneously, by maximizing the log likelihood. The standardized residuals are assumed to be jointly Gaussian. To ease the computation cost of estimating a vast dimensional time-varying correlation model, V-Lab uses a technique called composite likelihood (Engle et al., 2007).

The DCC model captures a stylized facts in financial time series: correlation clustering. The correlation is more likely to be high at time t if it was also high at time t1. Another way of seeing this is noting that a shock at time t1 also impacts the correlation at time t. However, if α+β<1, the correlation itself is mean reverting, and it fluctuates around R⎯⎯⎯, the unconditional correlation.

Usual restrictions on the parameters are α,β>0. Though, it is possible to have α+β=1; the conditional correlation is then an integrated process.

Variance Targeting

Notice that if we had written the DCC model in a fashion similar to the GARCH model:

Qt=Ω+ανt1νt1+βQt1

we would have to estimate the matrix Ω also. That is, instead of estimating only two parameters, we would have to estimate 2+nn+12 parameters (it is not 2+n2 parameters due to the fact that Ω is a symmetric matrix). And then the unconditional correlation implied by the model would have been:

R⎯⎯⎯=Ω1αβ

Instead of estimating Ω, notice that we actually substituted Ω by R⎯⎯⎯(1αβ) in the DCC formula, which is a much more parsimonious way of writing the model. This is called Variance Targeting, introduced by Engle and Mezrich in 1995, and it is a very useful technique when modeling vast dimensional time-varying covariance or correlation models.

DCC(p,q)

The specific model just described can be generalized in two ways.

In the first stage, each GARCH specification used to standardize each one of the n return time series can be generalized to a GARCH(p,q) model (see garch documentation), where p and q can be chosen differently for each return time series, for instance, by Bayesian Information Criterion (BIN), also known as Schwarz Information Criterion (SIC), or by Akaike Information Criterion (AIC). The former tends to be more parsimonious than the latter. V-Lab uses p=1 and q=1 though, because this is usually the option that best fits financial time series.

In the second stage, the DCC model can be generalized to account for more lags in the conditional correlation. A DCC(p,q) model assumes that:

Qt=R⎯⎯⎯+i=1pαi(νtiνtiR⎯⎯⎯)+j=1qβj(QtjR⎯⎯⎯)

where p and q can be chosen, for instance, by information criterion. Again, V-Lab uses p=1 and q=1 though, because this is usually the option that best fits financial time series.

Bibliography

Bollerslev, T., 1990. Modeling The Coherence in Short-Run Nominal Exchange Rates: A Multivariate Generalized ARCH Model. Review of Economics and Statistics 72: 498-505.

Engle, R. F., 2002. Dynamic Conditional Correlation: A Simple Class of Multivariate GARCH Models. Journal of Business and Economic Statistics 20(3).

Engle, R. F., 2009. Anticipating Correlations: A New Paradigm for Risk Management. Princeton University Press.

Engle, R. F. and J. Mezrich, 1995. Grappling with GARCH. Risk: 112-117.

Engle, R. F., N. Shephard, and K. Sheppard, 2007. Fitting and Testing Vast Dimensional Time-Varying Covariance Models. NYU Working Paper FIN-07-046.

Video On Multivariate GARCH DCC Estimation

Applied Statistics for Business and Management

Statistics for Business and Management professionals and academic researchers is of critical importance. Although there are specialized courses in Statistics for the students of Business and Management but none of the courses offer specialization in either, Stata, Eviews or SPSS and other computer packages. The current course introduces the application of Stata, Eviews and SPSS and other computer application while emphasis is laid on the development of theoretical background of the learners as well as with strong application of the techniques in Statistics for Business. The contents of the course offered to students include (but are not limited) to:
Basic Data Management and Presentation
Hypothesis Testing and Research Problems
Causal Relationships, Regression Analysis and Applications
Qualitative Data Analysis for Binary and Multi-chotomous variables

Also the students of the course would be provided additional support via other mediums like phone, SMS, email beside the traditional and favorite WizIQ!
This course will benefit the students of MBA, BBA and Doctorate students in Business and Management courses have to study and develop the skills of Quantitative Data Analysis. This course will master these students in their Statistical courses.
Course Highlights
Apply Statistical Techniques to real world data
Apply Data Analysis to relevant research problems
Wide range of contents which are usually required in broader contexts

The course package:
15 LIVE online classes
Duration: Two to Four Weeks (Negotiable)
All Days: 0500 PM to 1000 PM GMT (1000 PM to 0200 AM Pakistan Times)
The course outline:
Deterministic Data and Random Data
Multivariate Tables, Scatter Plots and D Plots
Measures of Association for Continuous Variables
Measures of Association for Ordinal Variables
Measures of Association for Nominal Variables
Estimating Data Parameters
Parametric Tests of Hypotheses
Non-Parametric Tests of Hypotheses
Statistical Classification
Multiple Regression
General Linear Regression Model
General Linear Regression in Matrix Terms
Multiple Correlation
Inferences on Regression Parameters
ANOVA and Extra Sums of Squares
Polynomial Regression and Other Models
Building and Evaluating the Regression Model
Principal Components
Dimensional Reduction
Principal Components of Correlation Matrices
Factor Analysis
Survival Analysis
The Exponential Model
The Weibull Model
The Cox Regression Model

Recommended books for this course:
Probability and Statistics, Fourth Edition by Morris H. DeGroot & Mark J. Schervish
SPSS for Introductory Statistics Second Edition by Morgan et al
A First Course in Business Statistics, Eight Edition by JAMES T. McCLAVE, P. GEORGE BENSON & TERRYSlNClCH
Note: Students must purchase these books themselves. These are not available as part of the course. Discount coupons may be asked from the teacher who would be happy to share if any available right now.

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Financial Econometrics Using Matlab: Two Weeks Course

Matlab has been one of the important Modeling, Simulations and Data Analysis software widely used by Financial Econometricians. Two Weeks long course has been developed to introduce basic training in Financial Econometrics using Matlab and making the registered students enabled to independently apply their skills to Data Analysis and Modeling in the Areas of Econometrics and Finance. The course is equally important to the beginners and advanced learners in Finance, Economics and Financial Econometrics who want to develop a sense of modeling and analysis. The main objective of the course is to introduce the theory and application of Financial Econometrics and how to use the powerful Mathematical Modeling Software Tool, Matlab for analysis and simulation.
The course is planned in three sections:
·         Registration starts today: Today
·         Registration ends:November 10, 2016
·         Course Begins:November 10, 2016

Main contents are:
Exempted Topics for Advanced Learners in Statistics and Econometrics
·         Solution of Stochastic Differential Equations
·         General Approach to the Valuation of Contingent Claims
·         Pricing Options using Monte Carlo Simulations
·         Term Structure of Interest Rates and Interest Rate Derivatives
·         Credit Risk and the Valuation of Corporate Securities
·         Valuation of Portfolios of Financial Guarantees
·         Risk Management and Value at Risk (VaR)
·         Value at Risk (VaR) and Principal Components Analysis (PCA)

All these topics will be discussed in context of Matlab

The course fee structure:
Individuals: 350GBP
Groups: 500GBP

MatLab M-files, EBooks, Manuals, Example datasets will be provided upon enrolment for the course. EBooks and weekly lecture slides will be provided before commencement of each lecture.

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Introduction to Matlab for Finance

Matlab has been one of the important Modeling, Simulations and Data Analysis software widely used by Financial Econometricians. The one week short course has been developed to introduce basic training in using Matlab for Finance enabling the registered students to independently apply their skills to Data Analysis and Modeling in the area of Finance. The course is equally important to the beginners and advanced learners in Finance, Economics and Financial Econometrics who want to develop a sense of modeling and analysis. The main objective of the course is to introduce the theory and application of Financial Econometrics and how to use the powerful Mathematical Modeling Software Tool, Matlab for analysis and simulation.

The course is planned in three sections:
·         Registration starts today: Today
·         Registration ends:November 14, 2016
·         Course Begins:November 15, 2016

Contents of the Course:
Selected Topics from the following will be discussed. Student has the choice to select one or many topics.
·         Solution of Stochastic Differential Equations
·         General Approach to the Valuation of Contingent Claims
·         Pricing Options using Monte Carlo Simulations
·         Term Structure of Interest Rates and Interest Rate Derivatives
·         Credit Risk and the Valuation of Corporate Securities
·         Valuation of Portfolios of Financial Guarantees
·         Risk Management and Value at Risk (VaR)
·         Value at Risk (VaR) and Principal Components Analysis (PCA)

All these topics will be discussed in context of Matlab

The course fee structure:
Individuals: 300GBP payable on registration for the course
Groups: 750GBP, payable after the invoice has been sent to the group leader

MatLab M-files, EBooks, Manuals, Example datasets and Lecture Recordings into DVD format will be provided upon enrolment for the course. EBooks and weekly lecture slides will be provided before commencement of each lecture. DVD recordings will be posted on the portal after the course completes and editing has been completed for improving quality

The course materials will be provided to those who register on https://elearning.aneconomist.com. Registration should be confirmed within 24 Hours by paying the course fee.  Only 10 Places are Available for live, online and person-to-personal online lecturing to maintain proper balance in discussion and participation.