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Diagnosing Non-IID Standardized Residuals in GARCH Models

Article Quant Q&A · Author: Kondo
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Summary

The document describes a thesis problem: fitting conditional volatility models to financial returns so their standardized residuals are approximately independent and identically distributed with zero mean and unit variance. GARCH(1,1) and GJR-GARCH(1,1) reportedly meet this goal for some series, while several other specifications do not for two ETF return series. The author demeans log returns, estimates parameters on an initial sample, and applies ARCH tests at several lags; those tests reject homoscedasticity for the problem series.

The text raises two methodological questions: whether model fitting or implementation might be at fault, and whether ARCH tests are appropriate for assessing standardized residuals. It supplies no resolution or diagnostic procedure, and its results are the author’s preliminary account rather than a documented model comparison. It does, however, frame residual diagnostics as a separate step from selecting a volatility model and cautions that test choice matters.

Key ideas

  • The modeling objective is standardized residuals that are approximately IID with mean zero and unit variance.
  • GARCH and GJR-GARCH reportedly work for some series but leave residual heteroscedasticity in others.
  • The author tests standardized residuals with ARCH tests at multiple lags.
  • The document questions whether ARCH tests are suitable for assessing residual IID behavior.

Tags

StatisticsVolatilityEquitiesTechnical indicators
Full text
# How to choose a GARCH model which delivers iid standardized residuals?


# How to choose a GARCH model which delivers iid standardized residuals?












For my thesis I first need to examine nine financial time series and fit a conditional volatility model such that the obtained standardized residuals ($z_t = \epsilon_t / \sigma_t$) are approximately iid with mean 0 and variance 1.

Whereas GARCH(1,1) succeeds in delivering iid standardized residuals for five of these series, and GJR-GARCH(1,1) achieves iid standardized residuals for other two series, I've not been able to get iid $z_t$ for the remaining two series using GARCH, GJRGARCH, ThresholdGARCH, EGARCH, NAGARCH and CSGARCH.

When I shared my results with my thesis supervisor, he said I've probably done something wrong since GARCH, GJRGARCH and ThresholdGARCH usually succeed wrt this goal. The problem is I don't understand what I could have done wrong.

The mentioned series are three SPDR ETFs (XLF and XLU). Closing prices can be found here (this is my first question, so I'm sorry if this isn't the way you're supposed to share data):

XLF: http://real-chart.finance.yahoo.com/table.csv?s=XLF&a=11&b=22&c=1998&d=02&e=31&f=2016&g=d&ignore=.csv

XLU: http://real-chart.finance.yahoo.com/table.csv?s=XLU&a=11&b=22&c=1998&d=02&e=31&f=2016&g=d&ignore=.csv

After obtaining log returns and demeaning them, I use the first 1766 observations to estimate all parameters and obtain standardized residuals. I then conduct ARCH tests on standardized residuals (at lags 1, 5, and 10), which for these two series reject homoscedasticity. Therefore I don't obtain iid residuals and can't go on with my analysis.

Any help would be greatly appreciated.

PS: Is there any other test which is more adequate in testing iid residuals? I think I've read somewhere that, while most people still use ARCH tests on residuals, these are supposed to test raw data and should not be used for residuals.

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