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# some concepts (2)

#### ps_ricky_son

##### Member
Hi all

Grateful if you could help the below concepts again.

1. In the variance-variance matrix, there is a concept of positive semi-definitive which the Factor model qualifies, what is the concept about? quite confused with the transpose of vector/ vector mentioned in the book

2. the GARCH model is said to perform a number of excellent/very good/good jobs regarding the explanation on volatility, what in fact account for the differences among those descriptions? Can I say excellenet job means greater explanatory power?

3. the AR model's auto correlation is said to be oscillate, why is it so?

4. in the calculation of long run variance, sometimes we need to multiply the answer with the square of the frequency in a year to derive the annualized volatility. So, what is the unit of the long run variance measurement then? (i.e. is the original answer before multiply measured in days or according to what the question specifies? )

Many thanks again.

#### ShaktiRathore

##### Well-Known Member
Subscriber
Hi
2) Garch models incorporates mean reversion which better reflects reality/practice because in reality volatility does shows mean reversion,that is why it performs better job in explaining volatility. Garch model is the most preffered for this reason as compared to other models.
4) long run variance will be same as volatility squared unit,even if you multiply by frequency sqaure the unit shall remain the same of the long run variance because frequency is just a number.
Thanks

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#### ps_ricky_son

##### Member
thx again ShaktiRathore

regarding Q4, I may not be clear. What I would like to know is how the mean reversion is compounded? say daily, weekly or yearly? Thanks

#### ami44

##### Well-Known Member
Subscriber
1. In the variance-variance matrix, there is a concept of positive semi-definitive which the Factor model qualifies, what is the concept about? quite confused with the transpose of vector/ vector mentioned in the book

I'm not sure what the connection to the Factor model is (what Factor model?), but a variance-covariance matrix is always positive semi-definite by construction. Sometimes if you try to estimate the matrix, dependent of your method and model, you run in danger that your estimate is not positive semi-definite. This can happen for example, if you estimate variances and correlations with different methods. You should do something to fix that then.

The mathematical condition for a n x n matrix C to be positive semi definite is:
wᵀ · C · w ≥ 0
where w is a n x 1 column vector and wᵀ it's transposed.

Why are variance-covariance matrices positive semi-definite?
If X1, X2, ... Xn are the random variables, that C is the variance-covariance matrix of, than wᵀ · C · w is the variance of the following linear combination
w1 * X1 + w2 * X2 + ... + wn * Xn
with wi being the components of the vector w.
As a variance wᵀ · C · w must be equal or greater 0.

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#### ps_ricky_son

##### Member
Thx all of the above.

Let me try to rephrase.

-> use the same value of parameters when calculating variance and co variance
-> consistency of variance and co variance
-> w(transpose T)w => 0
-> variance-covariance matrix are positive-semidefinite.

Kindly correct me if any. Thanks a lot.

#### ami44

##### Well-Known Member
Subscriber
Thx all of the above.
-> w(transpose T)w => 0

I'm not so sure what this means. Maybe I'm not seeing it, but where is the matrix in your formula?
The formula is wᵀ · C · w ≥ 0
with C being the variance-covariance matrix.

#### ps_ricky_son

##### Member
hi ami44~~ sorry my typo again, should be wᵀ · C · w ≥ 0, thx