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  • I know that if the order of Arch(m) is over 3, we should use GARCH and GARCH(1,1) was proved to be the best. But was GARCH(1,1) proved to be available for any country's stock market?

  • My result show that GARCH(1,1) is not statistically significant (although i bsed on the result of ACF/PACF of Squared error). However, the Garch(2,1) (3,1) (4,1) (5,1) (6,1) (7,1) (8,1) are statistically significant.

  • Consequently, i conflict that the method based on the ACF/PACF of the Squared return or Squared error to define the Order of GARCH are not available. How can we estimate the order of GARCH(m.s)?

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    $\begingroup$ Why not using the information critera to choose the order? $\endgroup$ – Egodym May 25 '16 at 21:29
  • $\begingroup$ -Continously, I have many materials about GARCH model (Applied Time series econometrics,page198 ; Econometrics by example- Damodar Gujarati p.238; Introductory econometrics for finance - Chris Brooks p.379) to figure out the Order of Garch(m,s). ​-All indicate that if the order of ARCH is over 3, use GARCH. And as the order of ARCH increases to infinity, ARCH(m) is equivalent to GARCH(1,1). $\endgroup$ – Maciel May 26 '16 at 2:50
  • $\begingroup$ -Also, GARCH(1,1) is proved to be useful to model the return of financial asset and rarely used in any higher order model.​ - But my result show that the coefficent of mean equation (Logreturn)is not significant with the P of 0.148. It show the rejection of GARCH(1,1). But another GARCH(2,1) and (3,1) is significant. Please give me suggestion ! Thank you! $\endgroup$ – Maciel May 26 '16 at 2:51
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    $\begingroup$ I would just calculate AIC and BIC, and choose the best model based on these criteria. $\endgroup$ – Egodym May 26 '16 at 20:57
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    $\begingroup$ Look at the significance of the parameters of your GARCH (2,1) model: it may be that the first GARCH parameter is not significant; in that case, you can set it equal to zero, ending up with a GARCH(1,1). Usually, you do not gain so much by raising up the GARCH order, with respect to the GARCH(1,1); instead, you will be worse in efficiency: this is why I agree with @Egodym on the use of selection criteria. $\endgroup$ – simmy Jun 1 '16 at 16:51
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You're right. Hansen and Lunde ran 330 specifications, and found GARCH (1,1) the best fitting volatility model. However, in some cases other specifications can beat the results of GARCH (1,1).

Checking the ACF/PACF of the squared error term is necessary, although, not sufficient condition. Let's assume the following GARCH (m,s) model $$y_t=a_0+a(L)\varepsilon_t^2+b(L)y_t $$ $$\varepsilon_t=v_t\sqrt{a_0+a(L)\varepsilon_t^2+b(L)y_t }$$ where $v_t$ is a white-noise procedure. There're many other things to investigate:

  • Checking the ACF/PACF of the error term. Residuals cannot be autocorrelated. The ARMA model ($y_t=a_0+a(L)\varepsilon_t^2+b(L)y_t $) part of the GARCH model is not correctly specified if the residuals are autocorrelated.
  • Checking the ACF/PACF of the squared error term. You can test whether your procedure is a GARCH procedure or not. Equivalently, one can examine Ljung–Box Q-statistics. It can be used to test for groups of significant coefficients. Rejecting the null hypothesis that the $\varepsilon_t^2$ sequence is serially uncorrelated is equivalent to rejecting the null hypothesis of no ARCH or GARCH errors.
  • Normality of the errors. When you estimate GARCH models with Maximum Likelihood method the default probability density function is usually standard normal in many packages (e.g. in R). You can easily perform a Jarque-Bera normality test on your residuals. (Note that: For most financial assets, the distribution function for the rate of return is fat tailed, like Student's t-distribution.)
  • Information criterion. You can compare how well the estimated models perform using AIC or SBC. It's a fast strategy.

I'd recommend you to check ACFs and normality tests of different specifications. In some cases, there is a trade-off: some models do better at ACFs, some at normality tests. Maybe the residuals of yours are pretty asymmetric. For further improvement you can apply EGARCH, TARCH or ARCH-M.

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