Choosing GARCH Orders Using Residual Diagnostics and Forecast Accuracy
Summary
The document considers how to choose between GARCH specifications when diagnostics and coefficient tests point in different directions. In the example, a GARCH(1,1) with t-distributed errors leaves no detected serial correlation in residuals or squared residuals, although its ARCH coefficient is statistically insignificant. A simpler specification leaves autocorrelation in both, while information criteria favor GARCH(1,1). One response recommends retaining the model that removes residual dependence, then comparing actual forecasts using several loss functions rather than relying only on coefficient significance or information criteria.
Other contributors raise questions about common GARCH orders, higher-order ARCH approximations, and whether GARCH(1,1) generalizes across markets. These claims are not backed by comparative results in the document, and the discussion does not settle a universal order-selection rule. It also distinguishes significance in the mean equation from choosing the conditional variance structure. The practical guidance is therefore to assess residual diagnostics and out-of-sample forecast performance together, with the chosen loss measures matched to the forecasting objective.
Key ideas
- A statistically insignificant ARCH coefficient does not by itself settle which variance model forecasts better.
- Residual and squared-residual autocorrelation can signal structure left unexplained by a model.
- Information criteria may not align with residual diagnostics or forecast performance.
- Compare candidate models using forecasts and multiple loss functions.
- The discussion provides no universal GARCH order rule or evidence that one specification fits every market.
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Full text
# GARCH parameters # GARCH parameters I'm trying to estimate parameters of GARCH(p,q) model. I tried p=1, q=1 with t-distribution errors. Ljung-Box showed no correlation in residuals and squared residual. But the null hypothesis that ARCH-term's coefficient equals 0 was not rejected. So I tried p=0, q=1. Ljung-Box indicated serial correlation in residuals and squared residuals. Moreover, AIC and SC chose the former model. Should I choose GARCH(1,1), though one coefficient is statistically insignificant ? ## Answer by Malick (score 1) https://quant.stackexchange.com/a/21912 I would keep the model with p=1 and q=1 even those the null hypothesis that ARCH-term's coefficient equals 0 was not rejected. The reason is that (generally) the less autocorrelations there are in the resulting serie, the more accurate your forecast will be. Indeed if you estimate a model and leaves some autocorrelation it means it is still possible to improve your model by taking benefit of these autocorrelations to produce more accurate forecast. SIC and AIC may be sometime misleading since they only care on specific statistical properties (likelihood, number of parameters...). Finally to be sure, I would recommend you to produce forecasts and to keep the "best" model based on a bunch of loss functions. ## Answer by Maciel (score 0) https://quant.stackexchange.com/a/26264 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). - 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! ## Answer by Maciel (score 0) https://quant.stackexchange.com/a/26265 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. 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 ARCH are not available. How can we estimate the order of GARCH(m.s)?
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