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

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Selected Resources to Learn Economics and Econometrics In Short Time

The selected resources to learn economics and econometrics in short time is presented here. These are the resources which we will have to refer most of the times when we begin learning economics or economics. The selection of resources to begin the study of economics and economists matters a lot and hence selecting the best set of books, software manuals, blogs and tutorials and related materials provides a strong foundation towards understanding of economics and economics in a recommended level. Welcome to An Economist selected resources to learn Economics and Econometrics in short time.

Economics

Micro Economics

Consumers, firms, and general equilibrium:
Arne Hallam (Iowa State), Microeconomics
Nolan Miller (Harvard), Lecture Notes on Microeconomic Theory
Robert Nau (Duke), Seminar in Choice Theory
Sten Nyberg (SSE), Advanced Microeconomics
Ariel Rubinstein (Tel Aviv), Lecture Notes in Microeconomic Theory: The Economic Agent
Max Stinchcombe (Texas), Single-Person and Multi-Person Decision Theory
Guoqiang Tian (Texas A&M), Microeconomic Theory
Nicholas Yannelis (Illinois), Lecture Notes in General Equilibrium Theory


Game theory and mechanism design:

Wayne Bialas (SUNY Buffalo), Game Theory
Bernard Caillaud (ENPC) / Benjamin Hermalin (Berkeley), Hidden Action and Incentives
Bernard Caillaud (ENPC) / Benjamin Hermalin (Berkeley), Hidden-Information Agency
Yongmin Chen (Colorado), Microeconomic Theory II
Peter Cramton (Maryland), Advanced Microeconomics
Christian Ewald (St Andrews), Games, Fixed Points and Mathematical Economics
Douglas Gale (NYU), Strategic Foundations of General Equilibrium
Benjamin Hermalin (Berkeley), Lecture Notes for Economics
Paul Klemperer (Oxford), Auctions: Theory and Practice
Levent Koçkesen (Columbia), Advanced Microeconomic Analysis I
Martin Osborne (Toronto) / Ariel Rubinstein (Tel Aviv), Bargaining and Markets
Jim Ratliff (prev. Arizona), Graduate-Level Course in Game Theory
Francesco Squintani (UCL), Notes for Non-Cooperative Game Theory
Max Stinchcombe (Texas), Dynamics and Learning
Max Stinchcombe (Texas), Notes for a Course in Game Theory


International trade:

Pol Antras (Harvard), Advanced Topics in International Trade
Donald Davis (Columbia), Notes on Competitive Trade Theory
István Kónya (Boston College), Lecture Notes in International Trade
Edward Leamer (UCLA), Sources of International Comparative Advantage
James Markusen et al. (Colorado), International Trade: Theory and Evidence
Applied and computational micro / other topics in micro:
Daron Acemoglu (MIT), Lecture Notes in Graduate Labor Economics
Ted Bergstrom (UC Santa Barbara), The Theory of Public Goods and Externalities
Christopher Carroll (JHU), Solution Methods for Microeconomic Dynamic Stochastic Optimization Problems
Alan Duncan (Nottingham), Labour Economics I & II
Bryan Ellickson (UCLA), General Equilibrium and Finance
Ariel Rubinstein (Tel Aviv), Economics and Language
Ariel Rubinstein (Tel Aviv), Modelling Bounded Rationality
David Zilberman (Berkeley), Economics & Policy of Production, Technology and Risk in Agricultural & Natural Resources

Macroeconomics

Various models:

Willem Buiter (Cambridge), Lectures on Really Useful Ad Hoc Macroeconomics
John Driscoll (Fed), Lecture Notes in Macroeconomics
Brian Krauth (Simon Fraser), Macroeconomic Theory
Roland Meeks (Oxford), Economic Growth
Gregor Smith (Queen's), Macroeconomics Lecture Notes
Paul Söderlind (St Gallen), Macro II
Stephen Williamson (WUSTL), Notes on Macroeconomic Theory
Recursive (dynamic programming) treatments and dynamic methods:
Chris Edmond (NYU), Advanced Macroeconomic Techniques
Jeremy Greenwood (Rochester), Lecture Notes on Dynamic Competitive Analysis
Nezih Guner (Penn State), Advanced Macroeconomic Theory
Lars-Peter Hansen (Chicago) / Thomas Sargent (NYU), Recursive Models of Dynamic Linear Economies
Lars-Peter Hansen (Chicago) / Thomas Sargent (NYU), Robustness
John Hassler (Stockholm U), Math II (Dynamic Systems)
Christopher House (Michigan), Macroeconomics II
David Kendrick (Texas), Stochastic Control for Economic Models
Miles Kimball (Michigan), Advanced Mathematical Methods for Macroeconomics .doc
Ian King (Auckland), A Simple Introduction to Dynamic Programming in Macroeconomic Models
Paul Klein (Western Ontario), Solving the Growth Model by Linearizing the Euler Equations
Dirk Krüger (Frankfurt), Macroeconomic Theory
Dirk Krüger (Frankfurt), Quantitative Macroeconomics: An Introduction
Per Krusell (Princeton), Lecture Notes for Macroeconomics I
Lars Ljungqvist (SSE) / Thomas Sargent (NYU), Recursive Macroeconomic Theory .ps
Rody Manuelli (Wisconsin), Notes on Discrete Time Economic Models: The Growth Model
Rody Manuelli (Wisconsin), Topics in Macroeconomics: An Introduction to Stochastic Calculus
Maurice Obstfeld (Berkeley), Dynamic Optimization in Continuous-Time Economic Models I & II
Nicola Pavoni (University College), Notes on Dynamic Methods in Macroeconomics
Shouyong Shi (Toronto), Macro Theory I
John Stachurski (Melbourne), Stochastic Economic Dynamics
Nancy Stokey (Chicago), Brownian Models in Economics
Stijn Van Nieuwenburg (NYU) / Pierre-Olivier Weill (NYU), Exercises in Recursive Macroeconomic Theory
Randall Wright (Penn), Macroeconomics
Asset pricing, financial economics and financial mathematics:
Christian Ewald (St Andrews), Discrete-Time Finance
Christian Ewald (St Andrews), Mathematical Finance: Introduction to Continuous-Time Financial Market Models
Robert Kohn (NYU), Continuous-Time Finance
Antonio Mele (LSE), Lecture Notes in Financial Economics
Steven Shreve (Carnegie Mellon), Stochastic Calculus and Finance
Tyler Shumway (Michigan), Introduction to Finance
Tyler Shumway (Michigan), Introduction to Continuous-Time Asset Pricing
Paul Söderlind (St. Gallen), Financial Theory I & II
A.W. van der Vaart (Vrije U), Financial Stochastics
A.W. van der Vaart (Vrije U), Martingales, Diffusions, and Financial Mathematics

Other macro and computational methods:

Miles Kimball (Michigan), Q-Theory and Real Business Cycle Analytics
Miles Kimball (Michigan), Real Business Cycle Theory: A Semiparametric Approach
Dirk Krüger (Frankfurt), Dynamic Fiscal Policy
Dirk Krüger (Frankfurt), Consumption and Saving: Theory and Evidence
Eric Leeper (Indiana), Bundesbank Mini-Course on Monetary Economics
Stephanie Schmitt-Grohe (Duke) / Martin Uribe (Duke), International Macroeconomics
Shouyong Shi (Toronto), Topics in Monetary Theory
Paul Söderlind (St Gallen), Macroeconomic and Financial Forecasting
Paul Söderlind (St Gallen), Empirical Macroeconomics
Paul Söderlind (St Gallen), Monetary Policy
Harald Uhlig (Humboldt U Berlin), A Toolkit for Analyzing Nonlinear Stochastic Models Easily
Martin Uribe (Duke), Lectures in Open Economy Macroeconomics


Econometrics

Probability and mathematical statistics:

Richard Bass (Connecticut), The Basics of Financial Mathematics
Graham Brightwell (LSE), Probability for Finance and Economics
Robert Gray (Stanford), Probability, Random Processes, and Ergodic Properties
Charles Grinstead (Swarthmore) / Laurie Snell (Dartmouth), Introduction to Probability
Rachel Fewster (Auckland), Statistical Theory
Arne Hallam (Iowa State), Econometrics I
Guido Imbens (UCLA), Probability and Statistics
Oliver Knill (Harvard), Probability
Daniel McFadden (Berkeley), Statistical Tools
D.S.G. Pollock (Queen Mary College), Lectures in Mathematical Statistics
S.R.S. Varadhan (NYU), Probability Theory
S.R.S. Varadhan (NYU), Stochastic Processes
Ivan Wilde (King's College London),  Measure, Integration & Probability
Robert Wolpert (Duke), Probability and Measure Theory

Econometrics (general):

Herman Bierens (Penn State), Econometrics Lecture Notes
Erik Biorn (Oslo), Econometrics - Advanced
Michael Creel (Barcelona), Graduate Economics Lecture Notes
Bruce Hansen (Wisconsin), Econometrics
Daniel McFadden (Berkeley), Econometrics
Ariel Pakes (Harvard), Advanced Applied Econometrics
Ariel Pakes (Harvard) / Oliver Linton (LSE), Nonlinear Methods for Econometrics
D.S.G. Pollock (Queen Mary College), A Course of Econometrics
D.S.G. Pollock (Queen Mary College), Introductory Econometrics
D.S.G. Pollock (Queen Mary College), Topics in Econometric Theory
Paul Söderlind (St. Gallen), Lecture Notes for Econometrics

Macroeconometrics (time series) / financial econometrics:

John Cochrane (Chicago), Time Series for Macroeconomics and Finance
D.S.G. Pollock (Queen Mary College), The Methods of Time Series Analysis
Paul Söderlind (St. Gallen), Lecture Notes in Financial Econometrics
A.W. van der Vaart (Vrije U), Time Series
Microeconometrics and other econometrics:
Alan Duncan (Nottingham), Cross-Sectional and Panel Data Econometrics
Stepan Jurajda (Charles U), Econometrics of Panel Data and Limited Dependent Variable Models
James LeSage (Toledo), Spatial Econometrics
Charles Manski (Northwestern) / Daniel McFadden (Berkeley), Structural Analysis of Discrete Data and Econometric Applications
Kenneth Train (Berkeley), Discrete Choice Methods with Simulation
Melvyn Weeks (Cambridge), Advanced Econometrics: Microeconometrics


Mathematics

Mathematics for economists:

Julio Dávila (Penn), Mathematics for Economic Theory
Arne Hallam (Iowa State), Quantitative Methods in Economic Analysis
John Hillas / Dmitriy Kvasov (Auckland), Foundations of Economic Analysis
Michael Manove (Boston U), Mathematics for Micro
Markus Möbius (Harvard), Mathematics for Economists
Efe Ok (NYU), Real Analysis & Probability Theory with Economic Applications
Martin Osborne (Toronto), Mathematical Methods for Economic Theory
Guoqiang Tian (Texas A&M), Mathematical Economics
Viatcheslav Vinogradov (Charles U), A Cook-Book of Mathematics


Optimization:

Steve Alpern (LSE), Optimization Theory
Stephen Boyd (Stanford) / Lieven Vandenberghe (UCLA), Convex Optimization
Michael Burger (UCLA), Infinite-Dimensional Optimization and Optimal Design
Jan van den Heuvel (LSE) / Graham Brightwell (LSE), Optimization Theory
Robert Vanderbei (Princeton), Linear Programming: Foundations and Extensions
Pravin Varaiya (Berkeley), Lecture Notes on Optimization
Klaus Wälde (U Würzburg), Applied Intertemporal Optimization
Richard Woodward (Texas A&M), Dynamic Optimization


Linear algebra / calculus / differential equations

:
Alan Bain (prev. Cambridge), Stochastic Calculus
Michael Berry (Tennessee) et al., Templates for the Solution of Linear Systems
George Cain (Georgia Tech) / James Herod (Georgia Tech), Multivariable Calculus
William Chen (Macquarie), First-Year Calculus
William Chen (Macquarie), Linear Algebra
Ian Craw (Aberdeen), Advanced Calculus and Analysis
Lawrence Evans (Berkeley), An Introduction to Stochastic Differential Equations
John Friedlander / Peter Rosenthal (Toronto), Calculus Lecture Notes
Jonathan Goodman (NYU), Stochastic Calculus
Jim Hefferon (St. Michael's College), Linear Algebra
Robert Kohn (NYU), Partial Differential Equations for Finance
Thomas Kurtz (Wisconsin), Lectures in Stochastic Analysis
Lee Lady (Hawaii), Topics in Calculus
Keith Matthews (Queensland), Elementary Linear Algebra
Kaare Petersen / Michael Petersen (Technical U Denmark), The Matrix Cookbook
Dinakar Ramakrishnan (Caltech), Calculus, Number Theory & Vector Calculus
Klaus Schmitt (Utah), Nonlinear Analysis and Differential Equations
Ruslan Shapirov (Bashkir State U, Russia), Course of Linear Algebra and Multidimensional Geometry
Dan Sloughter (Furman), Difference to Differential Equations
Dan Sloughter (Furman), The Calculus of Functions of Several Variables
Gerald Teschl (Vienna), Ordinary Differential Equations and Dynamical Systems
Sergei Treil (Brown), Linear Algebra Done Wrong


Analysis / measure theory / topology:

Robert Anderson (Berkeley), Measure Theory
Douglas Arnold (Penn State), Complex Analysis
Douglas Arnold (Penn State), Functional Analysis
George Cain (Georgia Tech), Complex Analysis
William Chen (Macquarie), Fundamentals of Analysis
William Chen (Macquarie), Introduction to Complex Analysis
William Chen (Macquarie), Introduction to Lebesgue Integration
William Chen (Macquarie), Linear Functional Analysis
William Chen (Macquarie), Multivariable and Vector Analysis
Paul Garrett (Minnesota), Functional Analysis
Lee Larson (Louisville), Real Analysis Lecture Notes
Vitali Liskevich (Bristol), Measure Theory and Functional Analysis
Aisling McCluskey (York, Ca.) / Brian McMaster (York, Ca.), Topology Course Notes
Sidney Morris (Ballarat), Topology without Tears
Sylvia Serfaty (NYU), Functional Analysis Notes
Bert Wachsmuth (Seton Hall), Interactive Real Analysis
Elias Zakon (Windsor), Mathematical Analysis I


Mathematical game theory and logic, other math:

Stefan Bilaniuk (Trent), A Problem Course in Mathematical Logic
William Chen (Macquarie), Discrete Mathematics
William Chen (Macquarie), Congruences, Polynomials, and Group Theory
George Collins, II (Case Western), Fundamental Numerical Methods and Data Analysis
Germund Dahlquist (prev. RIT Sweden) / Ake Bjork (Linkoping), Numerical Mathematics in Scientific Computation
Thomas Ferguson (UCLA), Game Theory
Steven Pav (UCSD), Numerical Methods Course Notes
Stephen Simpson (Penn State), Mathematical Logic
Steven Sugden (Bond U, Australia), Discrete Mathematics
Michal Walicki (Bergen), Introduction to Logic


Reference:

Alexandre Stevanov's Listing of Math Lecture Notes
Eric Weissenstein's Mathworld
Wikipedia

Software

Matlab:

Ian Cavers (UBC), An Introductory Guide to Matlab
Paul Fackler (North Carolina State), Matlab Primer
Edward Neuman (Southern Illinois University), Matlab Tutorials
Christian Roessler (Melbourne), Matlab Basics
Kermit Sigmon (Florida), Matlab Primer
Kermit Sigmon (Florida), Matlab Tutorial
Matlab Summary and Tutorial at Florida


Gauss:

Marc Nerlove (Maryland), Notes on GAUSS
Felix Ritchie (Trig Consulting), Guide to Programming in GAUSS
Mark Watson (Princeton), GAUSS Basics
GAUSS 5.0 User Guide at Aptech


Stata:

Robert Yaffee (NYU), Getting Started with STATA for MS Windows: A Brief Introduction
STATA Tutorial at Princeton


Latex:

Peter Flynn (Silmaril Consultants), A Beginner's Introduction to Typesetting with LaTex

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[youtube https://www.youtube.com/watch?v=ndpEhC6mysM]