AR(1)-Garch(1,1) Joint Estimation

[Farid]

Hello,

I have done separate estimations using Boxjenk for AR(1) and ARCH-GARCH for GARCH(1,1), however in Tsay (2005) states that joint distribution needs to done and AR(1) parameters have slightly changed after joint estimation in his case (Tsay, 2005, p.116).

May I get any hints for that?

Thanks in advance

[TomDoan]

Hello,

I have done separate estimations using Boxjenk for AR(1) and ARCH-GARCH for GARCH(1,1), however in Tsay (2005) states that joint distribution needs to done and AR(1) parameters have slightly changed after joint estimation in his case (Tsay, 2005, p.116).

May I get any hints for that?

Thanks in advance

The AR(1) estimates from BOXJENK are (effectively) OLS, while the AR(1) estimates embedded in the GARCH are a complicated type of GLS. The OLS estimates will be consistent, but not efficient. If there is a very significant GARCH effect in the data, the lack of efficiency of OLS will be pronounced, because there will likely be a handful of outliers which will dominate the OLS fit, but will be downweighted sharply in the GLS process.

[Farid]

Thanks for information Tom.

What you said is consistent with my results (coefficients for AR(1) either insignificant or not realistic). Although I dont' know how to get AR(1) estimates embedded in GARCH(1) in RATS. I guess that is what is called AR(1)-GARCH(1,1) joint estimation and that is what I need for my project. Could you help me in doing that?

Farid

[Farid]

Thanks for information Tom.

What you said is consistent with my results (coefficients for AR(1) either insignificant or not realistic). Although I dont' know how to get AR(1) estimates embedded in GARCH(1) in RATS. I guess that is what is called AR(1)-GARCH(1,1) joint estimation and that is what I need for my project. Could you help me in doing that?

Farid

Sorry, I missed it in mannuals:

Code: Select all

garch(p=1,q=1,regressors) / dlogdm
# const dlogdm {1}

Although my results are still insignificant for AR(1) coefficient.

Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Constant 9.8258e-004 5.6984e-004 1.72431 0.08465149
2. RETURN{1} -0.0163 0.0288 -0.56734 0.57047996
3. C 5.4171e-006 2.8316e-006 1.91307 0.05573948
4. A 0.0609 0.0153 3.97818 0.00006944
5. B 0.9278 0.0176 52.58655 0.00000000

As you see AR(1) coefficient is highly insignificant (0.57) and constant terms for both equations are also not significant (0.055 and 0.084). Do you have any idea how I can fix it?


Thanks in advance,

Farid

[TomDoan]

Why would you want to "fix" that? If the series were mean zero with no serial correlation in the mean, that's the type of behavior that you would expect.

[Farid]

Could you please look at this code and say what is wrong with it? This refers to Exercise 3.3 (Tsay, 2005, p.148), except that it was applied to my own dataset. It is AR(1)-GARCH(1,1) setup with normality assumption.

Code: Select all

OPEN DATA "C:\Documents and Settings\admin\Desktop\teze.RAT"
CALENDAR(D) 2006:1:3
DATA(FORMAT=RATS) 2006:01:03 2010:12:08 RT
***initialize the conditional variance function
set h = 0.0
***specify the parameters of the model
nonlin c0 p1 a0 a1 b1
***specify the AR(1) mean equation
frml at = rt(t) - c0 - p1*rt(t-1)
***specify the volatility equation
frml gvar = a0 + a1*h(t-1) + b1*at(t-1)**2
***specify the log-likelihood function
frml garchln = -0.5*log(h(t)=gvar(t))-0.5*at(t)**2/h(t)
***sample period used in estimation
smpl 2 511
***initial estimates
compute a0 = 0.01, a1 = 0.5, b1 = 0.1, c0 = 0.01, p1 = 0.01 
maximize(method=bhhh,recursive,iterations=150) garchln
set fv = gvar (t)
set resid = at(t)/sqrt(fv(t))
set residsq = resid(t)*resid(t)
***Checking standardized residuals
cor(qstats,number=36,span=12) resid
***Checking squared standardized residuals
cor(qstats,number=36,span=12) residsq

The error notation appears after maximization and states ""## SR10. Missing Values And/Or SMPL Options Leave No Usable Data Points". I tried it without sampling, but it didn't work again.

[TomDoan]

This is covered in the RATS (v8) User's Guide on page UG-294:

If you try to estimate a garch model using MAXIMIZE and get the error message:

Missing Values And/Or SMPL Option Leave No Usable Data Points

your estimation range probably starts too soon for your data. Check the range of your
data and adjust the MAXIMIZE instruction appropriately. If you still have problems,
you can test the settings of your formulas with (for instance),

In this case, you need to lose two data points at the start; one for the lag in the AR and one for the lagged squared residual in the ARCH term. If you look at Tsay's RATS code (which, by the way, is rather old), the AR(1)-GARCH(1,1) model starts at entry 3, while the other models that don't have the AR(1) mean start at 2.