# GARCH (1,1)

• GARCH stands for Generalized Autoregressive Conditional Heteroscedasticity
• Heteroscedasticity means variance is changing with time.
• Conditional means variance is changing conditional on latest volatility.
• Autoregressive refers to positive correlation between volatility today and volatility yesterday.
• (1,1) means that only the latest values of the variables.
• GARCH model recognizes that variance tends to show mean â€“ reversion i.e. it gets pulled to a long-term Volatility rate over time.

• Generally Î³*VL  is replaced by Ï‰
• Since the sum of all the weights is equal to 1 we get the following equation as well:

Example

Suppose a GARCH model is estimated using MLE from daily data as follows:

Suppose that on a particular day â€˜tâ€™; actual return was -1% and the volatility (std. deviation) estimate for that was 1.8%. Calculate the volatility estimate for next day (t+1) and long-term average volatility (to which the model shows reversion over-time)

Solution
• In the GARCH model, 12% is the weight given to latest squared return (reactive factor). 85% is the weight given to latest variance estimate (persistence factor). Therefore,
• 1-0.12-0.85 = 3% is weight given to long-term average Volatility.
• Therefore, 3%*VL = 0.000005 i.e. VL  = 0.017%
• Also, variance estimate for t+1 = .000005 + 0.12*(-1%)^2 + 0.85*(1.8%)^2 = 0.0292%
• Volatility (Std. Dev.) estimate for t+1 = sqrt (0.0292%) = 1.71%
• For a stable GARCH model, alpha + Beta <=1. If alpha + Beta>1, then weight given to long-term volatility is negative and the model becomes â€˜mean-fleeingâ€™

Example

Which of the following GARCH models will take the shortest time to revert to its mean?

1. ht = 0.05 + 0.03r2t-1 + 0.96ht-1
2. ht = 0.03 + 0.02r2t-1 + 0.95ht-1
3. ht = 0.02 + 0.01r2t-1 + 0.97ht-1
4. ht = 0.01 + 0.01r2t-1 + 0.98ht-1

Solution

B.
A. Incorrect. The model that will take the shortest time to revert to its mean is the model with the lowest persistence defined by Î±1 + Î². In this case the persistence factor is the second largest:
Î±1 + Î² = 0.03 + 0.96 = 0.99.

B. Correct. The model that will take the shortest time to revert to its mean is the model with the lowest persistence defined by Î±1 + Î². In this case the persistence factor is the second lowest:
Î±1 + Î² = 0.02 + 0.95 = 0.97.

C. Incorrect. The model that will take the shortest time to revert to its mean is the model with the lowest persistence defined by Î±1 + Î². In this case the persistence factor is the largest:
Î±1 + Î² = 0.01 + 0.97 = 0.98.

D. Incorrect. The model that will take the shortest time to revert to its mean is the model with the lowest persistence defined by Î±1 + Î². In this case the persistence factor is the lowest:
Î±1 + Î² = 0.01 + 0.98 = 0.99.

Example
• Suppose the long-run variance rate is 0.0002 so that the long-run volatility per day is 1.4%

• Suppose that the current estimate of the volatility is 1.6% per day and the most recent percentage change in the market variable is 1%. What is the new variance rate?
Solution

The new volatility is 1.53% per day