Robert Engle

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DYNAMIC CONDITIONAL CORRELATIONS

• Robert Engle
• UCSD and NYU and Robert F. Engle, Econometric
  Services

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WHAT WE KNOW

• VOLATILITIES AND CORRELATIONS VARY OVER TIME,
  SOMETIMES ABRUPTLY
• RISK MANAGEMENT, ASSET ALLOCATION, DERIVATIVE
  PRICING AND HEDGING STRATEGIES ALL DEPEND UPON UP
  TO DATE CORRELATIONS AND VOLATILITIES

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AVAILABLE METHODS

• MOVING AVERAGES
• Length of moving average determines smoothness
  and responsiveness
• EXPONENTIAL SMOOTHING
• Just one parameter to calibrate for memory decay
  for all vols and correlations
• MULTIVARIATE GARCH
• Number of parameters becomes intractable for many
  assets

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DYNAMIC CONDITIONAL CORRELATIONA NEW SOLUTION

• THE STRATEGY
• ESTIMATE UNIVARIATE VOLATILITY MODELS FOR ALL
  ASSETS
• CONSTRUCT STANDARDIZED RESIDUALS (returns divided
  by conditional standard deviations)
• ESTIMATE CORRELATIONS BETWEEN STANDARDIZED
  RESIDUALS WITH A SMALL NUMBER OF PARAMETERS

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MOTIVATION

• Assume structure for conditional correlations
• Simplest assumption- constancy
• Alternatives
• Integrated Processes
• Mean Reverting Processes

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DEFINITION CONDITIONAL CORRELATIONS
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BOLLERSLEV(1990) CONSTANT CONDITIONAL
CORRELATION
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DISCUSSION

• Likelihood is simple when estimating jointly
• Even simpler when done in two steps
• Can be used for unlimited number of assets
• Guaranteed positive definite covariances
• BUT IS THE ASSUMPTION PLAUSIBLE?

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CORRELATIONS BETWEEN PORTFOLIOS
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HOWEVER

• EVEN IF ASSETS HAVE CONSTANT CONDITIONAL
  CORRELATIONS, LINEAR COMBINATIONS OF ASSETS WILL
  NOT

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DYNAMIC CONDITIONAL CORRELATIONS

• STRATEGYestimate the time varying correlation
  between standardized residuals
• MODELS
• Moving Average calculate simple correlations
  with a rolling window
• Exponential Smoothing select a decay parameter
  ? and smooth the cross products to get
  covariances, variances and correlations
• Mean Reverting ARMA

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Multivariate Formulation

• Let r be a vector of returns and D a diagonal
  matrix with standard deviations on the diagonal
• R is a time varying correlation matrix

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Log Likelihood
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Conditional Likelihood

• Conditional on fixed values of D , the
  likelihood is maximized with the last two terms.
• In the bivariate case this is simply

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Two Step Maximum Likelihood

• First, estimate each return as GARCH possibly
  with other variables or returns as inputs, and
  construct the standardized residuals
• Second, maximize the conditional likelihood with
  respect to any unknown parameters in rho

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Specifications for Rho

• Exponential Smoother
• i.e.

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Mean Reverting Rho

• Just as in GARCH
• and

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Alternatives to MLE

• Instead of maximizing the likelihood over the
  correlation parameters
• For exponential smoother, estimate IMA
• For ARMA, estimate

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Monte Carlo Experiment

• Six experiments - Rho is
• Constant .9
• Sine from 0 to .9 - 4 year cycle
• Step from .9 to .4
• Ramp from 0 to 1
• Fast sine - one hundred day cycle
• Sine with t-4 shocks
• One series is highly persistent, one is not

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DIMENSIONS

• SAMPLE SIZE 1000
• REPLICATIONS 200

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METHODS

• SCALAR BEKK (variance targeting)
• DIAGONAL BEKK (variance targeting)
• DCC - LOG LIKELIHOOD WITH MEAN REVERSION
• DCC - LOG LIKELIHOOD FOR INTEGRATED CORRELATIONS
• DCC - INTEGRATED MOVING AVERAGE ESTIMATION

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MORE METHODS

• EXPONENTIAL SMOOTHER .06
• MOVING AVERAGE 100
• ORTHOGONAL GARCH (first series is first factor,
  second is orthogonalized by regression and GARCH
  estimated for each)

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CRITERIA

• MEAN ABSOLUTE ERROR IN CORRELATION ESTIMATE
• AUTOCORRELATION FOR SQUARED JOINT STANDARDIZED
  RESIDUALS - SERIES 2, SERIES 1
• DYNAMIC QUANTILE TEST FOR VALUE AT RISK

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JOINT STANDARDIZED RESIDUALS

• In a multivariate context the joint standardized
  residuals are given by
• There are many matrix square roots - the Cholesky
  root is chosen

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TESTING FOR AUTOCORRELATION

• REGRESS SQUARED JOINT STANDARDIZED RESIDUAL ON
• ITS OWN LAGS - 5
• 5 LAGS OF THE OTHER
• 5 LAGS OF CROSS PRODUCTS
• AN INTERCEPT
• TEST THAT ALL COEFFICIENTS ARE EQUAL TO ZERO
  EXCEPT INTERCEPT

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RESULTS-MeanAbsoluteError
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FRACTION OF DIAGNOSTIC FAILURES(2)
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FRACTION OF DIAGNOSTIC FAILURES (1)
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DQT for VALUE AT RISK
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CONCLUSIONS

• VARIOUS METHODS FOR ESTIMATING DCC HAVE BEEN
  PROPOSED and TESTED
• IN THESE EXPERIMENTS, THE LIKELIHOOD BASED
  METHODS ARE SUPERIOR
• THE MEAN REVERTING METHODS ARE SLIGHTLY BETTER
  THAN THE INTEGRATED METHODS

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EMPIRICAL EXAMPLES

• DOW JONES AND NASDAQ
• STOCKS AND BONDS
• CURRENCIES

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CONCLUSIONS

• VARIOUS METHODS FOR ESTIMATING DCC HAVE BEEN
  PROPOSED and TESTED
• IN THESE EXPERIMENTS, THE LIKELIHOOD BASED
  METHODS ARE SUPERIOR
• THE MEAN REVERTING METHODS ARE SLIGHTLY BETTER
  THAN THE INTEGRATED METHODS