Models Multivariate GARCH Models Updated: April

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1 Financial i Econometrics and Volatility Models Multivariate GARCH Models Updated: April Eric Zivot Professor and Gary Waterman Distinguished Scholar Department of Economics, University of Washington Eric Zivot. All Rights Reserved. and S&P 500 Daily Returns Sample covariance matrix Sample correlation matrix Eric Zivot. All Rights Reserved. 2 1

2 Sample Cross-lag and Autocorrelations Multivariate Series : msft.sp500.ts^2 and AC CF and ACF Lag Lag Eric Zivot. All Rights Reserved. 3 EWMA Covariance: lambda = Var Cov Var Eric Zivot. All Rights Reserved. 4 2

3 Estimating EWMA Covariance > msft.sp500.ewma2=mgarch(msft.sp500.ts~1,~ewma2,trace=f) Coefficients: C(1) C(2) ALPHA EWMA Conditional Volatilities EWMA Conditional Correlation Eric Zivot. All Rights Reserved. 5 Estimated DVEC(1,1) Model > msft.sp500.dvec = mgarch(msft.sp500.ts ~ 1, ~ dvec(1, 1), trace = F) Value Std.Error t value Pr(> t ) C(1) 2.102e e e-011 C(2) 6.812e e e-007 A(1, 1) 1.710e e e+000 A(2, 1) 2.801e e e+000 A(2, 2) 1.650e e e+000 ARCH(1; 1, 1) 6.930e e e+000 ARCH(1; 2, 1) 6.758e e e+000 ARCH(1; 2, 2) 7.724e e e+000 GARCH(1; 1, 1) 9.049e e e+000 GARCH(1; 2, 1) 9.137e e e+000 GARCH(1; 2, 2) 9.116e e e+000 AIC(11) = BIC(11) = Notice how the ARCH parameters are similar and the GARCH parameters are similar. This motivates the scalar DVEC specification Eric Zivot. All Rights Reserved. 6 3

4 Conditional Volatilities and from DEV DVEC Conditional Volatilities DVEC Conditional Correlation Eric Zivot. All Rights Reserved. 7 Estimated DVEC(1,1) Model with Covariance Targeting > msft.sp500.dvecac=mgarch(msft.sp500.ts~1,~dvecac(1,1),trace=f) Value Std.Error t value Pr(> t ) C(1) e-010 C(2) e-008 ARCH(1; 1, 1) e+000 ARCH(1; 2, 1) e+000 ARCH(1; 2, 2) e-007 GARCH(1; 1, 1) e+000 GARCH(1; 2, 1) e+000 GARCH(1; 2, 2) e-001 AIC(8) = BIC(8) = Strange result: probably converged to local minimum Eric Zivot. All Rights Reserved. 8 4

5 Conditional Volatilities and from DEV with Cov Targeting DVEC Conditional Volatilities DVEC Conditional Correlation Eric Zivot. All Rights Reserved. 9 Estimated Scalar DVEC(1,1) Model > msft.sp500.dvec.scalar = mgarch(msft.sp500.ts ~ 1, ~ dvec.scalar.scalar(1, 1), trace = F) Value Std.Error t value Pr(> t ) C(1) e-012 C(2) e-008 A(1, 1) e+000 A(2, 1) e+000 A(2, 2) e+000 ARCH(1) e+000 GARCH(1) e+000 AIC(7) = BIC(7) = Common ARCH and common GARCH parameters Eric Zivot. All Rights Reserved. 10 5

6 Conditional Volatilities and from Scalar DEV DVEC Conditional Volatilities DVEC Conditional Correlation Eric Zivot. All Rights Reserved. 11 Estimated Matrix Diagonal Model > msft.sp500.md=mgarch(msft.sp500.ts~1, ~dvec.mat.mat(1,1),trace=f) Value Std.Error t value Pr(> t ) C(1) e e-007 C(2) e e e A(1, 1) e e+000 A(2, 1) e e+000 A(2, 2) e e+000 ARCH(1; 1, 1) e e+000 ARCH(1; 2, 1) e e+000 ARCH(1; 2, 2) e e-002 GARCH(1; 1, 1) e e+000 GARCH(1; 2, 1) e e+000 GARCH(1; 2, 2) e e-001 AIC(11) = BIC(11) = Interpretation of all coefficients is not straightforward Eric Zivot. All Rights Reserved. 12 6

7 Conditional Volatilities and from Matrix Diagonal Model DVEC Conditional Volatilities DVEC Conditional Correlation Estimated correlations are smoother in this model Eric Zivot. All Rights Reserved. 13 Estimated BEKK(1,1) Model > msft.sp500.bekk=mgarch(msft.sp500.ts~1,~bekk(1,1),trace=f) Value Std.Error t value Pr(> t ) C(1) e-011 C(2) e-008 A(1, 1) e+000 A(2, 1) e+000 A(2, 2) e-004 ARCH(1; 1, 1) e+000 ARCH(1; 2, 1) e-010 ARCH(1; 1, 2) e-003 ARCH(1; 2, 2) e+000 GARCH(1; 1, 1) e+000 GARCH(1; 2, 1) e+000 GARCH(1; 1, 2) e-001 GARCH(1; 2, 2) e+000 AIC(13) = BIC(13) = Not straightforward to interpret parameters Eric Zivot. All Rights Reserved. 14 7

8 Conditional Volatilities and from BEKK Model BEKK Conditional Volatilities BEKK Conditional Correlation Eric Zivot. All Rights Reserved. 15 Estimated CCC Model > msft.sp500.ccc = mgarch(msft.sp500.ts ~ 1, ~ ccc(1, 1), trace = F) Value Std.Error t value Pr(> t ) C(1) 2.106e e e-010 C(2) 6.397e e e-006 A(1, 1) 2.898e e e+000 A(2, 2) 2.098e e e+000 ARCH(1; 1, 1) 7.087e e e+000 ARCH(1; 2, 2) 7.583e e e+000 GARCH(1; 1, 1) 8.816e e e+000 GARCH(1; 2, 2) 9.079e e e+000 Estimated Conditional Constant Correlation Matrix: AIC(8) = BIC(8) = Eric Zivot. All Rights Reserved. 16 8

9 Conditional Volatilities and from CCC Model CCC Conditional Volatilities CCC Conditional Correlation Conditional correlation restricted to sample correlation Eric Zivot. All Rights Reserved. 17 Volatility and Correlation Predictions from DVEC(1,1) Predicted volatility for Predicted correlation b/w and Predicted volatility for Eric Zivot. All Rights Reserved. 18 9

10 Comparison of Predicted Volatilities Predicted volatility for DVEC MD BEKK CCC Predicted volatility for DVEC MD BEKK CCC Eric Zivot. All Rights Reserved. 19 Comparison of Correlation Predictions Predicted correlation b/w and DVEC MD BEKK CCC Eric Zivot. All Rights Reserved