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

3. 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

4. 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

5. 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

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

7. DEFINITION: CONDITIONAL CORRELATIONS

8. BOLLERSLEV(1990): CONSTANT CONDITIONAL CORRELATION

9. 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?

10. CORRELATIONS BETWEEN PORTFOLIOS

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

13. 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

14. 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

15. Log Likelihood

16. 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

17. 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

18. Specifications for Rho n Exponential Smoother n i.e.

19. Mean Reverting Rho n Just as in GARCH n and

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

22. Monte Carlo Experiment n 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 n One series is highly persistent, one is not

23. DIMENSIONS n SAMPLE SIZE 1000 n REPLICATIONS 200

25. 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

26. 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)

27. 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

28. 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:

29. 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

31. RESULTS-MeanAbsoluteError

32. FRACTION OF DIAGNOSTIC FAILURES(2)

33. FRACTION OF DIAGNOSTIC FAILURES (1)

34. DQT for VALUE AT RISK

35. 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

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

47. 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