1 MULTIVARIATE GARCH Rob Engle UCSD & NYU

2 MULTIVARIATE GARCH MULTIVARIATE GARCH MODELS ALTERNATIVE MODELS CHECKING MODEL ADEQUACY FORECASTING CORRELATIONS HEDGING CORRELATIONS APPLICATIONS AND SOFTWARE

3 TIME VARYING CORRELATIONS n All financial practitioners recognize that volatilities are time varying n Evidence is in implied volatilities and other derivative prices, and estimates over different sample periods n Similarly, correlations are time varying –Derivative prices of correlation sensitive products –Time series estimates

4 CORRELATION n Definition: n Properties: Always between (-1,1) n Measure of linear association n Conditional Correlation:

5 COVARIANCE MATRIX

6 USEFUL RELATIONS n If and only if H is positive definite, all portfolios will have correlations (-1,1)

7 MODELS n Moving Average(n) n Exponential Smoothing ( )

8 REWRITING IN MATRICES

9 Positive Definite Matrices n A matrix M is positive definite if n The sum of a positive (semi)definite and positive semidefinite matrix, is positive (semi)definite n Both Moving Average and Exponential Smoothers are positive semidefinite

10 DIAGONAL MULTIVARIATE GARCH n The reason this is “diagonal” will be clear shortly n In Matrix representation n is a Hadamard Product or element by element multiplication

11 Positive Definite Diagonal Models n If n then H is positive (semi)definite if A is positive (semi)definite n Models with A and B positive definite are useful restricted diagonal models n Scalar, squared diagonals and fuller matrices are used.

12 BEKK: Baba,Engle,Kroner,Kraft n Model with guaranteed positive definite structure n A and B can be diagonal, triangular or full. n If A and B are diagonal, then this is a “diagonal” model as written above

13 VEC n The VEC operator converts a matrix to a vector by stacking its columns n Useful Theorem: n The VEC Model

14 Special Cases n If A and B are diagonal, then we get the diagonal models n If A and B are themselves tensors, then these are BEKK models n Not all VEC models are positive definite n Because A and B are n 2 xn 2 there are many parameters!!

15 Constant Conditional Correlation Bollerslev n If conditional correlations are constant then the problem is much simpler n OR n but why should conditional correlations be constant?

16 COMPONENT BEKK n Permanent and Transitory Components n And can even add in an asymmetric term

17 FACTOR ARCH n One factor version n In Matrix notation

18 K FACTOR MODEL n or in Matrix notation

19 THE APT n In the APT correlations across assets are related to expected returns

20 Volatilities in APT n If idiosyncracies have constant variance n If idiosyncracies do not have constant variance, then they need ARCH models too n If idiosyncracies are independent of factors, and each other then univariate is sufficient

21 MAXIMUM LIKELIHOOD

22 DIAGNOSTIC CHECKING n Test standardized residuals: n Test for own autocorrelation n Test for cross asset autocorrelation n Test for cross product autocorrelation n Test for asymmetries

23 FORECASTING AND VARIANCE TARGETING n In the VEC model n Forecast recursion is: n And