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

2. 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

3. 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

4. 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

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

6. DEFINITION: CONDITIONAL CORRELATIONS

7. BOLLERSLEV(1990): CONSTANT CONDITIONAL CORRELATION

8. 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?

9. CORRELATIONS BETWEEN PORTFOLIOS

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

12. DYNAMIC CONDITIONAL CORRELATIONS • STRATEGY:estimate 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

13. 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

14. Log Likelihood

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

16. 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

17. Specifications for Rho • Exponential Smoother • i.e.

18. Mean Reverting Rho • Just as in GARCH • and

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

21. 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

22. DIMENSIONS • SAMPLE SIZE 1000 • REPLICATIONS 200

24. 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

25. 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)

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

27. 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:

28. 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

30. RESULTS-MeanAbsoluteError

31. FRACTION OF DIAGNOSTIC FAILURES(2)

32. FRACTION OF DIAGNOSTIC FAILURES (1)

33. DQT for VALUE AT RISK

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

35. EMPIRICAL EXAMPLES • DOW JONES AND NASDAQ • STOCKS AND BONDS • CURRENCIES

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