## goodness of fit measures for different garch models

Discussions of ARCH, GARCH, and related models

### goodness of fit measures for different garch models

Hi everyone,

I am using garch procedure to estimate different models for my research. The focus is on the mean equation specifications and the garch part is only used to deal with the arch effect in the error term.

Could anyone help me to understand the following four issues:

1. What goodness of fit measures in RATS that can be used to compare nested garch models; and also compare models based on different datasets? Ideally, these measures should be adjusted for the number of parameters in a model, e.g. similar to adj R^2 in an OLS regression.

2. Is the Wald joint test a valid test for testing some restrictions in the mean equation?

3. And how valid is the Wald join test statistics when some variables are highly correlated, e.g. with their own lags?

4. Would it be a problem if all the variables in the garch model are I(1) variables?

Anozman
anozman

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Joined: Mon Jan 30, 2012 6:16 pm

### Re: goodness of fit measures for different garch models

anozman wrote:Hi everyone,

I am using garch procedure to estimate different models for my research. The focus is on the mean equation specifications and the garch part is only used to deal with the arch effect in the error term.

Could anyone help me to understand the following four issues:

1. What goodness of fit measures in RATS that can be used to compare nested garch models; and also compare models based on different datasets? Ideally, these measures should be adjusted for the number of parameters in a model, e.g. similar to adj R^2 in an OLS regression.

Because the GARCH model doesn't focus on fitting the mean model, it's not clear how you would compare different GARCH models on the basis of how they fit the mean.

anozman wrote:2. Is the Wald joint test a valid test for testing some restrictions in the mean equation?

Yes.

anozman wrote:3. And how valid is the Wald join test statistics when some variables are highly correlated, e.g. with their own lags?

They're still valid.

anozman wrote:4. Would it be a problem if all the variables in the garch model are I(1) variables?

No, but the mean model has to reduce the residuals to being uncorrelated, which might not be easy for I(1) variables. Most GARCH models are applied to growth rates and other variables that aren't as highly serially correlated.
TomDoan

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Joined: Wed Nov 01, 2006 5:36 pm

### Re: goodness of fit measures for different garch models

Thanks Tom for your valuable feedback.

For point 3

With highly correlated variables in the mean equation, is t-test or Wald test statistic (it is essentially the same) on single parameter significance test still valid (the multicollinearity problem is not an issue here)?

For point 4

When you say “… but the mean model has to reduce the residuals to being uncorrelated, …”

are the “residuals” mentioned above referring to the raw residuals from the mean equation or the standardised residual series coming out of the GARCH model has to pass some kind of autocorrelation tests such as LB test?

Thank you very much for your help!

anozman
anozman

Posts: 23
Joined: Mon Jan 30, 2012 6:16 pm

### Re: goodness of fit measures for different garch models

anozman wrote:Thanks Tom for your valuable feedback.

For point 3

With highly correlated variables in the mean equation, is t-test or Wald test statistic (it is essentially the same) on single parameter significance test still valid (the multicollinearity problem is not an issue here)?

As long as they don't restrict unit root properties, they're asymptotically correct.

anozman wrote:For point 4

When you say “… but the mean model has to reduce the residuals to being uncorrelated, …”

are the “residuals” mentioned above referring to the raw residuals from the mean equation or the standardised residual series coming out of the GARCH model has to pass some kind of autocorrelation tests such as LB test?

The basic GARCH model has the residuals serially uncorrelated, but having dependence in the variance. The raw residuals should be serially uncorrelated, and so should the standardized residuals. Because the LB test doesn't allow for heteroscedasticity, the tests for lack of serial correlation are typically done on the standardized residuals.
TomDoan

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### Re: goodness of fit measures for different garch models

Thank you very much Tom. your explanation made me understand some issues much better now!

...they don't restrict unit root properties,..

What do you really mean? do you mean variables in the model have to be stationary or do you mean as long as the dependent variable series still has finite variance when time t -> infinite ?

anozman
anozman

Posts: 23
Joined: Mon Jan 30, 2012 6:16 pm

### Re: goodness of fit measures for different garch models

The results from Sims, Stock and Watson, Econometrica 1990 still apply when you have GARCH errors. If you have I(1) data, many hypotheses can be tested with standard asymptotics (lag length reductions for instance); some can't if they restrict the unit root behavior of the process (full exclusion tests on one of the I(1) variables, for instance).
TomDoan

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Joined: Wed Nov 01, 2006 5:36 pm

### Re: goodness of fit measures for different garch models

Thank you very much Tom. I will read the 1990 paper to gain more understanding!

I have been using t-distribution with a shape parameter estimated to control kurtosis side of the issue. for a couple of mean model specifications, the estimated shape parameters are over 2000, not within the usual 2 to 8 range.

I was wondering whether I have done something wrong here or it is simplly the case that a 2000 number is the best and valid estimate and is needed to control the fat tail feature in the series?

thank you very much again for your help!

anozman
anozman

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Joined: Mon Jan 30, 2012 6:16 pm

### Re: goodness of fit measures for different garch models

Hi Tom,

I have read the Sims, Stock and Watson 1990 paper. I have one more question I hope you could help me to understand.

When I used OLS for intial analysis, the raw residauls coming out of most of the equations are I(0), impling these I(1) variables are cointegrated although the residuals are nonnormal and have some autocorrelation and arch effect. Does this result (I(0) residuals) have an implication on the validity of those t- and Wald-tests when I set up the models properly within a garch framework and estimate them by MLE?

anozman
anozman

Posts: 23
Joined: Mon Jan 30, 2012 6:16 pm

### Re: goodness of fit measures for different garch models

anozman wrote:Hi Tom,

I have read the Sims, Stock and Watson 1990 paper. I have one more question I hope you could help me to understand.

When I used OLS for intial analysis, the raw residauls coming out of most of the equations are I(0), impling these I(1) variables are cointegrated

Actually, it's not implying that the I(1) variables are cointegrated. You have to organize the calculations in a very specific way in order to test for cointegration.

anozman wrote:although the residuals are nonnormal and have some autocorrelation and arch effect. Does this result (I(0) residuals) have an implication on the validity of those t- and Wald-tests when I set up the models properly within a garch framework and estimate them by MLE?

anozman

If they weren't I(0) you would have real problems. You want your mean model to reduce the residuals to white noise, not just stationarity. If the residuals in the GARCH model are conditionally Normal, the tests will generally be OK. If they might have fatter tails that Normal, you can add the ROBUSTERRORS option to get corrected QMLE estimates for the covariance matrix.
TomDoan

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Joined: Wed Nov 01, 2006 5:36 pm

### Re: goodness of fit measures for different garch models

Thank Tom.

Sorry I used the word "cointegrated" when I mean "integrated".

I have read some papers discussing Wald test results when joint restrictions were applied to I(1)s in the integrated processes. The general finding is that the Wald tests tend to over reject the null. I was wondering if this is the bias due to I(1)s, then in practice, if one uses 1% significance level instead of 5%, this could potentially reduce the bias to some degree although the fundemantal cause of the problem is not fixed.

Could you please tell me whether my thinking is valid?

One paper (ZHIJIE XIAO and PETER C. B. PHILLIPS 2004) has suggested that a higher order approximation of the Wald test can improve the testing accuracy, I was wondering whether RATS has similar codes so that one can modify.

In terms of the garch model, I have one or two mean equation specifications that give me odd t-distribution shape parameters about 2000. Could you please tell me whether this kind of magnitude is still valid as it is used in order to adjust the distribution to fit the data?

Thanks Tom, really appreciate your help!

anozman
anozman

Posts: 23
Joined: Mon Jan 30, 2012 6:16 pm

### Re: goodness of fit measures for different garch models

I'm not sure I've ever seen a shape parameter of 2000 on a t. Infinite degrees of freedom=Normal, so there's nothing wrong with that, but it just is very hard for an estimate to get out that far since there is almost no difference between the log likelihood for 100 d.f. and 2000 d.f. Perhaps your GARCH effect isn't all that strong.
TomDoan

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Joined: Wed Nov 01, 2006 5:36 pm

### Re: goodness of fit measures for different garch models

Yes Tom. The garch effect is not strong. in fact, a single shift paramter in the variance equation is sufficient to make the standardised residuals behave well. i am just not sure whether the violation of the skewness will have an impact on my results as t and GED can deal with fat tails, but not the skewness. unfortunately, almost all financial series related models suffer from this little problem. i was wondering how other people adjust their models to deal with this problem using RATS. can LOGSKEWTDENSITY be used in this case?

thanks Tom
anozman
anozman

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Joined: Mon Jan 30, 2012 6:16 pm

### Re: goodness of fit measures for different garch models

Hi Tom,

It seems Bruce Hansen (1994)'s skew-t-distribution can be used to deal with skewed t errors. Could you please tell me whethter the normal wald tests we discussed in this topic are still valid under the skew-t-distribution?

anozman
anozman

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Joined: Mon Jan 30, 2012 6:16 pm

### Re: goodness of fit measures for different garch models

anozman wrote:Hi Tom,

It seems Bruce Hansen (1994)'s skew-t-distribution can be used to deal with skewed t errors. Could you please tell me whethter the normal wald tests we discussed in this topic are still valid under the skew-t-distribution?

anozman

If you use the robusterrors option, yes.
TomDoan

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Joined: Wed Nov 01, 2006 5:36 pm

### Re: goodness of fit measures for different garch models

anozman
anozman

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Joined: Mon Jan 30, 2012 6:16 pm

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