1.9.5 Failure Modes & Practice

Basic Prerequisites: Econometrics & Time Series (stationarity, structural breaks) and Statistics & Inference (estimation error, validation).

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1. Intuition & Practical Objective

Every previous page was a solution: here is the model, here is the fix, here is the better estimator. This page is the honest audit. Volatility models fail in ways that are systematic, first-principles, and exactly correlated with the moments you care about - they break worst in crises. The job of a practitioner is to know which failure is currently binding and to guard against it.

There are four failure families, and each traces to a violated assumption in the GARCH declaration $a_t={\sigma }_t{\epsilon }_t$:

1. The model is stationary; the world is not - structural breaks inflate estimated persistence toward 1.
2. The model is symmetric; the world is not - leverage means symmetric GARCH understates downside risk.
3. The model is finite-order; the truth is long-memory-ish - high-order estimation is unstable, so you are always approximating.
4. The model treats parameters as known; you estimate them from a handful of crises - estimation error in the tails is largest exactly when it matters.

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2. Mathematical Ground Truth & Derivations

Failure 1 - Structural break vs. genuine persistence. Suppose the true process is GARCH(1,1) with constant parameters but the level ${\alpha }_0$ jumps at some date $\tau$ (a regime shift in the volatility floor). Estimating a single GARCH on the pooled sample, the recursion accumulates variance across the break, and the fitted persistence $\hat{\pi }={\hat{\alpha }}_1+{\hat{\beta }}_1$ is driven toward the IGARCH boundary $\pi =1$. The near-unit-root is an artefact of the break, not evidence of infinite memory (Lamoureux & Lastrapes 1990; Diebold 1986). Practical consequences: the unconditional variance ${\alpha }_0/(1-\pi )$ becomes numerically explosive (denominator $\to 0$), and forecasts never mean-revert.

Diagnostics: (a) CUSUM / Chow tests on the variance equation; (b) fit on sub-samples and compare $\hat{\pi }$; (c) Markov-switching GARCH if the break is recurrent (Regime Detection).

Failure 2 - Symmetry / leverage. Symmetric GARCH imposes $\partial {\sigma }_t^2/\partial a_{t-1}^2={\alpha }_1$ regardless of the sign of $a_{t-1}$. Empirically the derivative is larger on the downside. Consequence: a symmetric model’s downside VaR is biased low and its vol forecast after a crash is too small - the model systematically under-reacts to exactly the events that produce losses. Fix: GJR/EGARCH (03), or add the negative-return term directly to a HAR (04).

Failure 3 - Estimation of high-order models. The ARCH($q$)/GARCH($p,q$) likelihood is flat along the ridge where $\sum {\alpha }_i+\sum {\beta }_j$ is fixed; only the sum is well-identified, not its split. As $p,q$ grow, the positivity constraints multiply, the effective sample per parameter shrinks, and the estimates become unstable (variance blows up, occasionally landing on the constraint boundary). This is why GARCH(1,1) survives every forecast horse-race (Hansen & Lunde 2005). Practical rule: prefer parsimony; test the residual ACF squared, do not add lags by default.

Failure 4 - Tail / estimation error. VaR of a GARCH model at level $p$ is ${\hat{z}}_p\sqrt{{\hat{\sigma }}_{t+1}^2}$, where ${\hat{z}}_p$ uses a distributional assumption. Two errors compound: (i) ${\epsilon }_t$ is not Gaussian - it is fat-tailed and skewed, so Gaussian $z_p$ understates tail risk; (ii) ${\hat{\sigma }}_{t+1}^2$ itself is estimated, and the parameters driving the tail (large-${\alpha }_1$ behaviour) are pinned by the small number of historical crises. Backtests (Christoffersen 1998; Kupiec 1995) are the mandatory audit - see VaR & ES.

Failure 5 - Aggregation and the $\sqrt{h}$ rule. Multi-day VaR via $\sqrt{h}{\sigma }_1$ assumes iid returns. Under GARCH with $\pi <1$, the correct $h$-day variance is ${\sigma }_h^2(\ell )$ (geometric decay), which is smaller than $h{\sigma }_1^2$ when today is elevated - the $\sqrt{h}$ rule is conservative after a shock and anticonservative in calm times. Always use the model’s own term structure (see 06).

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3. Computational Implementation - the break inflates persistence

Standard library only. Builds a GARCH(1,1) series whose volatility floor ${\alpha }_0$ jumps 4× partway through (a structural break, parameters otherwise constant), then fits GARCH(1,1) by MLE on (a) the full sample and (b) the calm first half. The full-sample persistence is pulled toward 1 - the signature of the break-artefact failure.

The true persistence was $0.98$ in both regimes; the full-sample fit reports $0.9913$ - nearly IGARCH - while the clean subsample recovers only $0.9384$. Note that ${\hat{\alpha }}_1=0.070$ in both fits: what the break did was push the entire persistence premium into ${\hat{\beta }}_1$ ($0.921$ vs $0.869$), i.e. into pure level memory, which is exactly the spurious-unit-root channel. A risk system built on the full-sample $\hat{\pi }$ would never let volatility mean-revert.

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4. Failure Modes & First-Principles Breakdowns (checklist)

1. Test for breaks before trusting persistence. Split the sample, run a CUSUM on the standardized residuals $z_t=a_t/{\sigma }_t$, and compare $\hat{\pi }$ across sub-samples. If they differ, model the break, not the memory.
2. Always add an asymmetry term for equity risk. Run the Engle–Ng sign-bias test; if it rejects, symmetric GARCH is wrong on the downside.
3. Backtest the quantile, not the variance. Volatility forecasts can be “accurate” while the VaR is systematically breached (wrong tail model). Use Kupiec/Christoffersen tests.
4. Fat tails in the shock distribution. Replace Gaussian ${\epsilon }_t$ with a standardized Student-$t$ (or GED); the estimated $\nu$ is your fat-tail parameter and moves the tail $z_p$ substantially.
5. Beware the $\sqrt{h}$ shortcut. Use the GARCH/HAR term structure for multi-day horizons.
6. Sub-sample estimation in crises. Tail parameters are estimated from few events; report parameter uncertainty (bootstrap/profile likelihood), and prefer robust, low-order models.
7. Model risk is real and quantitative. Keep a model-risk buffer (Derman 1996) - the loss from using a mis-specified vol model can exceed the apparent precision gain.

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5. References

• Lamoureux, Christopher G. & Lastrapes, William D. (1990): Persistence in Variance, Structural Change, and the GARCH Model, J. Business & Economic Statistics 8(2)
• Diebold, Francis X. (1986): Modeling the Persistence of Conditional Variances
• Hansen, Peter R. & Lunde, Asger (2005): A Forecast Comparison of Volatility Models…, J. Applied Econometrics 20(7)
• Kupiec, Paul (1995) and Christoffersen, Peter (1998): Evaluating Interval Forecasts, International Economic Review 39(4)
• Derman, Emanuel (1996): Model Risk, Quantitative Strategies Research Notes, Goldman Sachs. Verified corpus refs/pillar4.
• Tsay, Ruey S.: Analysis of Financial Time Series (3rd ed., 2010)

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6. Connected Graph Bridges

• Base: Econometrics & Time Series (structural breaks) · Statistics & Inference (estimation error)
• Prior: 03 · Asymmetric Models · 04 · Realized Vol & HAR · Hub: Index
• Continue: 06 · Advanced Extensions
• Applied: VaR & Expected Shortfall · Regime Detection · Backtesting Hygiene