Robert Engle UCSD and NYU and Robert F. Engle, Econometric Services DYNAMIC CONDITIONAL CORRELATIONS

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

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

DYNAMIC CONDITIONAL CORRELATION A NEW SOLUTION n 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

MOTIVATION n Assume structure for conditional correlations n Simplest assumption- constancy n Alternatives –Integrated Processes –Mean Reverting Processes

DEFINITION: CONDITIONAL CORRELATIONS

BOLLERSLEV(1990): CONSTANT CONDITIONAL CORRELATION

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

CORRELATIONS BETWEEN PORTFOLIOS

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

DYNAMIC CONDITIONAL CORRELATIONS n STRATEGY:estimate the time varying correlation between standardized residuals n 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

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

Log Likelihood

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

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

Specifications for Rho n Exponential Smoother n i.e.

Mean Reverting Rho n Just as in GARCH n and

Alternatives to MLE n Instead of maximizing the likelihood over the correlation parameters: n For exponential smoother, estimate IMA n For ARMA, estimate

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

DIMENSIONS n SAMPLE SIZE 1000 n REPLICATIONS 200

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

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

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

JOINT STANDARDIZED RESIDUALS n In a multivariate context the joint standardized residuals are given by n There are many matrix square roots - the Cholesky root is chosen:

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

RESULTS-MeanAbsoluteError

FRACTION OF DIAGNOSTIC FAILURES(2)

FRACTION OF DIAGNOSTIC FAILURES (1)

DQT for VALUE AT RISK

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

EMPIRICAL EXAMPLES n DOW JONES AND NASDAQ n STOCKS AND BONDS n CURRENCIES

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