Unlike univariate cases, multivariate volatility forecasting is regarded as complex, mainly because: The dimension of the variance-covariance matrix required: k(K+1)/2 The dependencies of the returns in the portfolio and their correlations Lack of positive semi-definiteness (ensure portfolio variance is non negative) Therefore most models have practical issues when implementing Focus on models such as O-GARCH and CCC that addresses these issues

1. Multivariate Volatility
forecasting
Abukar Ali

2. Table of contents
• Brief Introduction on Multivariate volatility
• Forecasting EWMA,
• Orthogonal Garch,
• Constant Conditional Correlations (CCC)
• Review and analysis of key points
• Estimation comparison

3. Introduction
Unlike univariate cases, multivariate volatility forecasting is regarded as complex, mainly
because:
• The dimension of the variance-covariance matrix required: k(K+1)/2
• The dependencies of the returns in the portfolio and their correlations
• Lack of positive semi-definiteness (ensure portfolio variance is non negative)
• Therefore most models have practical issues when implementing
• Focus on models such as O-GARCH and CCC that addresses these issues

4. EWMA – exponential weighted moving average
Recall from the univariate case:
𝜎𝑡
2 = 𝜆𝜎𝑡−1
2
+ (1 − 𝜆) 𝑦𝑡−1
2
where 𝜆 is given constant = 0.94
In the multivariate case:
𝛴𝑡 = 𝜆𝛴𝑡−1 + (1 − 𝜆) 𝑦𝑡−1
′
𝑦𝑡−1
Where 𝛴𝑡−1 represents the covariance matrix at t-1 period
*alternative to simpler Moving average model.
*uses more weights on recent information
*issue is that it may suffer from the so called “curse of dimensionality”
First lets review simple models such as the EWMA in multivariate scenario

5. EWMA – exponential weighted moving average
Next we fit EWMA to a portfolio of equity returns (Eurostoxx50 and msft):
𝛴𝑡 = 𝜆𝛴𝑡−1 + (1 − 𝜆) 𝑦𝑡−1
′
𝑦𝑡−1
Where 𝛴𝑡−1 represents the covariance matrix at t-1 period
*From the variance-covariance matrix, we do have the marginal variances as well
as well as time-dependent portfolio correlation.
*next slide, we can see the time-dependent correlations instead of constant correlation.
*we need better model to eliminate the covariance such that the returns are independent
or uncorrelated.
First lets review simple models such as the EWMA in multivariate scenario

7. O-GARCH – Orthogonal Garch
*In multivariate volatility modelling we need to estimate the entire conditional covariance
matrix in one go – this is usually a problem
*In practise: the univariate variances are separated, from the covariances, such that we can
model the individual volatility the same as the univariate case.
*Apply Orthogonal approach: transform the observed returns matrix into uncorrelated
portfolios.
Method:
- PCA: transform correlated returns into uncorrelated portfolios via principal
component analysis
- GARCH: use Garch model to forecast the volatility of each uncorrelated returns
separately
- PCA guarantees the positive definiteness of the covariance matrix

8. Multivariate Garch(1,1):
Very Quick review:
Recall from the univariate case:
In multivariate case:
- Maybe we may not need all the off-diagonal covariances – too much noise.
- Calculating the covariances individually is not valid option either
- There is no guarantee then that the matrix is positive definite

9. Constant conditional Correlations (CCC)
- The first step: compute the sample (n,n) correlation matrix 𝑅
- Compute the diagonal matrix of time dependent variances: D = diag(𝜎1
2
, … … … 𝜎 𝑛
2
)
- The covariance forecast is given by: =
𝜎𝑡1
2
0
0 𝜎𝑡2
2
1 𝜌
𝜌 1
𝜎𝑡1
2
0
0 𝜎𝑡2
2
𝛴𝑡 =𝐷𝑡 𝑅𝐷𝑡
- Where :
- Since the matrix D has only diagonal elements, we can estimate each volatility separately:
* simple model
* guarantees the positive definiteness of cov-matrix
* due to the diagonality of the matrix, we can estimate the volatility separately
CCC model advantages & disadvantages :
* Correlation is constant over time

10. Dynamic conditional Correlations (CCC)
- The DCC allows time varying conditional correlations, Engle (2002), Engle and Sheppard (2001)
- Define correlation matrix:
- Where 𝑄𝑡
𝑅 𝑡 = 𝑄𝑡′ 𝑄𝑡
𝑄𝑡 is given by:
𝑖𝑠 𝑡ℎ𝑒 𝑢𝑛𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝑜𝑓 𝑢 and 𝛿1 𝑎𝑛𝑑 𝛿2 𝑎𝑟𝑒 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 𝑡𝑜 𝑏𝑒 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑
**depending on parameter restriction the conditional correlation matrix is guaranteed to be positive definite
**drawback: two additional parameters are required to drive the dynamics of all correlations.
***Next, we fit our portfolio to all the above models but we use period
of 2007 - 2010

11. EWMA
DCC
Orth.Garch
**correlations under DCC, stable across time
**correlations under OO improves on EWMA but still seems volatile
** EMWA shows cyclical or cluster in volatility. high vol period followed by low vol period.