Common mistake 1 with ARCH / GARCH volatility

Fitting GARCH to price levels instead of returns. Convert first.

Common mistake 2 with ARCH / GARCH volatility

Interpreting GARCH as forecasting the direction of returns. It forecasts their SIZE, not their sign.

Common mistake 3 with ARCH / GARCH volatility

α + β at or above 1 means the model is degenerate - the variance never settles.

Common mistake 4 with ARCH / GARCH volatility

Forecasting variance far ahead and quoting it as certainty. The conditional variance converges to the long-run level at a rate set by alpha plus beta, so a long-horizon GARCH forecast is essentially the unconditional variance with extra steps.

EconStatsCraft › ARCH / GARCH volatility

ARCH / GARCH volatility

GARCH(1,1) by maximum likelihood with Normal or Student-t shocks. Pre-tests for ARCH effects and forecasts the volatility path.

What it needs

Series - Returns, not price levels.

The intuition

Financial returns are close to unpredictable in direction but highly predictable in MAGNITUDE. Big moves cluster: a turbulent day is usually followed by another. GARCH makes today's variance a weighted average of yesterday's surprise (α) and yesterday's variance (β), which is exactly an exponentially weighted memory of past turbulence.

When to use it

Daily or higher-frequency financial returns, risk measurement, Value-at-Risk, option pricing, or any series showing volatility clustering.

Where the estimator comes from

With \varepsilon_t=\sigma_t z_t and z_t i.i.d. mean zero unit variance, GARCH(1,1) sets \sigma_t^2=\omega+\alpha\varepsilon_{t-1}^2+\beta\sigma_{t-1}^2. Recursive substitution shows \sigma_t^2=\frac{\omega}{1-\beta}+\alpha\sum_{j\ge 0}\beta^{\,j}\varepsilon_{t-1-j}^2 - an infinite ARCH with geometrically declining weights. Estimation maximises \sum_t\left[-\tfrac{1}{2}\log\sigma_t^2-\tfrac{\varepsilon_t^2}{2\sigma_t^2}\right] numerically; the standard errors here come from a numerical Hessian.

The model

\varepsilon_t=\sigma_t z_t,\qquad \sigma_t^{2}=\omega+\alpha\,\varepsilon_{t-1}^{2}+\beta\,\sigma_{t-1}^{2}

What it assumes

ARCH effects are actually present - the module pre-tests this with Engle's LM test.
ω > 0, α ≥ 0, β ≥ 0, and α + β < 1 for the variance to be finite.
The model is fitted on RETURNS, not on price levels.
The mean equation is right. GARCH models the variance of the residuals from some mean model, so a badly specified mean - an omitted trend or an ignored autocorrelation - shows up as spurious volatility clustering.
The returns are not dominated by a handful of discrete events. A small number of large jumps produces high estimated persistence that is really the model trying to accommodate outliers.

Statistical properties

Requires ω > 0, α ≥ 0, β ≥ 0 for positive variance, and α + β < 1 for a finite unconditional variance ω/(1−α−β).
Persistence α + β governs how slowly volatility shocks decay; daily equity data typically give 0.95–0.99.
GARCH generates fat tails even with Normal innovations - but usually not fat ENOUGH, which is why Student-t fits better.
Symmetric by construction: it cannot capture the leverage effect, where bad news raises volatility more than good news. EGARCH and GJR-GARCH exist for that.
Quasi-ML is consistent for the variance parameters even when the innovation distribution is misspecified, provided robust standard errors are used.

How to read the output

α: how sharply volatility reacts to a new shock.
β: how long today's volatility remembers yesterday's.
α + β is persistence. Daily financial data typically give 0.95–0.99, meaning volatility shocks fade very slowly.
The half-life tells you how many periods it takes for a volatility shock to decay by half.
Student-t innovations usually fit financial data far better than Normal, because returns have fat tails.
Alpha is the reaction to yesterday's shock and beta the persistence of the existing level. Their sum is how slowly volatility mean-reverts, and 0.99 against 0.95 is the difference between shocks lasting months and lasting weeks.
The long-run variance is omega over (1 - alpha - beta), and it is worth computing: if it implies an annualised volatility unlike anything the asset has shown, the model is fitted to a subsample rather than a process.
Standardised residuals are what to test, not raw residuals. If they still show ARCH effects the variance model has not captured the clustering.

A worked reading

ω = 0.021, α = 0.078, β = 0.902, ν = 8.4. Persistence is 0.980, so a volatility shock has a half-life of about 34 days - turbulence in this market takes over a month to subside. The estimated 8.4 degrees of freedom confirm returns are much fatter-tailed than Normal.

Symptom → diagnosis → what to do

If you see this	It means	Do this
ARCH LM does not reject	No volatility clustering	GARCH is not identified; a constant variance is fine
α + β ≥ 1	Integrated variance	Use IGARCH, or check for a structural break in volatility
Residual squares still autocorrelated	Under-specified	Try GARCH(1,2) or GARCH(2,1)
Standardised residuals still fat-tailed	Normal innovations inadequate	Switch to Student-t
Fitted to price levels	Meaningless - variance of a non-stationary series	Convert to log returns first
Volatility spikes at one date dominate	A structural break or a data error	Check that observation

Common mistakes

Fitting GARCH to price levels instead of returns. Convert first.
Interpreting GARCH as forecasting the direction of returns. It forecasts their SIZE, not their sign.
α + β at or above 1 means the model is degenerate - the variance never settles.
Forecasting variance far ahead and quoting it as certainty. The conditional variance converges to the long-run level at a rate set by alpha plus beta, so a long-horizon GARCH forecast is essentially the unconditional variance with extra steps.

Where to read this properly

Engle (1982) - ARCH
Bollerslev (1986) - GARCH
Nelson (1991) - EGARCH and the leverage effect
Tsay (2010), Analysis of Financial Time Series, 3rd ed., Wiley, chapter 3

Related Time series - one series

Time-varying coefficients (Kalman) · Unit root tests · Correlogram (ACF / PACF) · ARIMA / SARIMA · Rolling-origin cross-validation · Theta method · Unobserved components · STL forecasting · Forecast combination · Markov regime switching · Change frequency (resample) · Decompose & filter · Spectral analysis · Exponential smoothing · Forecast accuracy

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ARCH / GARCH volatility

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

Related Time series - one series

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