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Interpreting Conflicting ARCH Tests and Persistent GARCH Fits

Article Quant Q&A · Author: Paul
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Summary

The document describes a volatility-modeling question involving monthly returns on a Brazilian equity index. After fitting an ARIMA model, the author finds mixed evidence in its residuals: plots and autocorrelation-based checks on squared residuals do not show clear dependence, while a Lagrange Multiplier test indicates ARCH effects across several lags. The author also sees possible volatility clustering by eye and finds that information-criterion-selected GARCH-family models have a beta estimate near one, producing slowly declining conditional volatility forecasts.

The post asks whether these results imply that GARCH should not be used, but it contains no answer, model estimates beyond that description, or out-of-sample evaluation. It therefore illustrates a diagnostic disagreement rather than establishing a rule for selecting or rejecting a volatility model. Its interpretation is limited by the unspecified sample size, model specification, test settings, and forecast performance. Readers would need to examine those details and the stability of the estimates before drawing conclusions about conditional heteroscedasticity or persistence.

Key ideas

  • The author reports conflicting evidence about ARCH effects in residuals from an ARIMA model of Brazilian equity-index returns.
  • Squared-residual plots and autocorrelation-based tests appear inconclusive, while a Lagrange Multiplier test flags dependence across several lags.
  • Selected GARCH-family fits have beta estimates near one and produce slowly declining volatility forecasts.
  • The document raises, but does not answer, whether the mixed diagnostics justify using GARCH.

Tags

EquitiesVolatilityStatisticsMachine learning
Full text
# What's the interpretation behind this GARCH modeling?


# What's the interpretation behind this GARCH modeling?












I have an ARIMA model for monthly returns of the brazilian stock market index. Then I test the residuals of the model for ARCH effects. The ACF/PACF of squared residuals show that there are no significant autocorrelations. Portmanteu and McLeod-Li tests also show that there are no heteroscedasticity. Nevertheless, Lagrange Multipler test appear to show there is heteroscedasticity for lags 1 to 8 and the plot of squared residuals itself look a little bit like there might be some volatility clustering. When I fit GARCH models (GARCH, gjrGARCH, AVGarch, TGARCH...), the ones with smallest BIC are (0,1) models, for which the beta is very close to 1 and so the conditional sd forecasts decrease very slowly. This appears to be caused by the fact that no GARCH model should be used, is that correct?

Below: the squared residuals plot, ACF and PACF of squared residuals.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.