1. Tutorial 10 SEG7550

2. Outline ARCH and GARCH model part three
Assignment 2 suggested solution
Project title received list

3. ARCH and GARCH model part three
Order determination
In assignment 2 Question 3, you are asked to obtain the AR model and GARCH model from Cheung Kong stock price data and HSBC stock price data
Motivation: when dealing with real data (not generated randomly), order must be determined before estimating parameters

4. Order determination Autocorrelation function.
Consider a weakly stationary return series . When linear dependence between and its past is of interest, the concept of correlation is generalized into autocorrelation. The correlation coefficient between and is called lag-l autocorrelation of and is commonly denoted by which under the weak stationary assumption is a function of l only.

5. Order determination
Definition of ACF:
Where is used.

6. Order determination for AR model
Partial autocorrelation function
The PACF of a time series is a function of its ACF and is a useful tool to determination the order of AR model. A simple yet efficient way to introduce PACF is to considering the following AR models in consecutive orders:

7. Order determination for AR model
Where , and are respectively, the constant term, the coefficient term of , and error term of AR(j) model.
The models can be estimated by least squares method and as a matter of fact, they are arranged in a sequential order that enables us to apply the idea of partial linear F test in multiply linear regression analysis.
The estimated parameter of first equation is called the lag-1 sample PACF of The estimate of the second equation is called lag-2 sample PACF of , and so on.

8. Order determination for AR model
From the definition, shows the added contribution of to over the AR(1) model. The lag-3 PACF shows the added contribution of to over the AR(2) model. And so on
Therefore for an AR(p) model, the lag-p PACF should not be zero. But should be zero for Under some regularity conditions, it can be shown that sample PACF of an AR(p) process has the following properties:

9. Order determination for AR model
PACF in matlab
Use function parcorr.

10. Order determination for AR model
Other than PACF, there are several information criteria to determine the order of AR model. For example, the well-known Akaike information criterion is defined as:
Where the likelihood function is evaluated at the maximum likelihood estimates and T is the sample size. For Gaussian AR model, AIC reduces to:

11. Order determination for AR model
In practice, we can calculate AIC(l) for l=1,2,…,k, where k is a prespecified value. And select the order with minimum AIC value.

12. Order determination for ARCH model
Recall that ARCH model can be rewritten into an AR model by letting :
Thus we can use the same method in AR model to determine the order.

13. Order determination for ARCH model
Specify order of GARCH model is not easy. In most application only lower order is used, say GARCH(1,1), GARCH(1,2) and GARCH(2,1).
Refer to “Information criteria for GARCH model selection” by Chris Brooks and Simon Burke.
or other papers from internet.

14. Suggested solution for assignment2

15. Student ID list of project title I received
, , , , , , , , , , , , , , , , , , , ,