EGARCH models log variance with standardized magnitude and sign news. See an asymmetric forecast, centering conventions and the limits of log-scale predictions.
EGARCH model
An EGARCH model (exponential generalized autoregressive conditional heteroskedasticity) models the logarithm of a return's conditional variance using past standardized return surprises and past log variances. Exponentiating the result keeps each finite variance update positive, while a signed-news term lets equally large positive and negative surprises produce different risk forecasts.
This is a specific extension of volatility modelling: ordinary GARCH reacts to squared surprises and therefore loses their sign. EGARCH can retain it. Nelson's original paper introduced exponential ARCH to address asymmetry, positivity restrictions and the interpretation of persistence.
The news is measured relative to its predicted scale
Write the return and its surprise as
rt=μt+εt,εt=htzt,ht=Var(rt∣Ft−1).
Here μt is the conditional mean, Ft−1 is information available before observing rt, and zt is an iid innovation with mean zero and variance one, independent of that past information. The standardized innovationzt=εt/ht measures surprise in units of the volatility predicted for that period. A 2% surprise is one standard deviation when predicted volatility is 2%, but two standard deviations when it is 1%.
A common asymmetric EGARCH(1,1) convention is
loght+1=ω+βloght+α(∣zt∣−c)+γzt,c=E[∣zt∣].
The magnitude term measures how unusually large the shock is; the signed term distinguishes its direction. With standard normal innovations, c=2/π≈0.797885, making the total news term mean zero. A zero shock does not make that centred term zero: it contributes −αc.
For another innovation distribution, use its expected absolute standardized innovation if the specification is meant to be centred. Some software fixes the normal constant even when fitting another distribution. That remains a valid parameterization, but the magnitude term then need not have mean zero. Changing the centring constant from c to c′ preserves the same recursion if the intercept changes to ω′=ω+α(c′−c).
The official arch EGARCH documentation uses the displayed coefficient roles with the fixed normal constant. Its asymmetric example includes a separate asymmetric order, o=1; the default o=0 omits the signed term. Read the equation and model orders before interpreting a fitted parameter.
A negative coefficient means greater response to negative news—here
Hold the prior variance fixed and compare shocks zt=−a and zt=+a, where a>0. Subtracting the two log-variance updates gives
Thus γ<0 gives the larger variance after a negative shock in this convention. A paper that writes −γzt, or uses γ for the absolute-value term, has a different sign interpretation. A parameter name alone is insufficient.
This models return-sign asymmetry associated with the leverage effect; it does not identify corporate leverage as the economic cause. Nor does a negative γ alone ensure that increasing shock magnitude raises variance on both sides: that also depends on α. The log-news response has slopes γ−α to the left of zero and γ+α to the right. It is continuous, with a kink at zero when α=0.
One-standard-deviation surprises, two forecasts
Consider an illustrative daily decimal-log-return model, with stipulated parameters rather than a fit to market data:
ht=0.0004,ω=0.1log(0.0004),β=0.9,α=0.2,γ=−0.1.
Use standard normal innovations. The prior daily standard deviation is 0.0004=2%, so zt=±1 corresponds to a mean-adjusted return surprise of ±2%. The intercept and carried state sum to log(0.0004).
Observed standardized surprise
Magnitude news 0.2(1−c)
Signed news −0.1zt
Next variance, squared decimal daily returns
Next daily standard deviation
+1
0.0404231
−0.1
0.0003768652
1.9413%
−1
0.0404231
+0.1
0.0004603042
2.1455%
The negative-shock variance is exp(0.2)=1.2214 times the positive-shock variance: about 22.1% higher. The standard-deviation ratio is its square root, exp(0.1)=1.1052, or about 10.52% higher. Variance and volatility percentage changes differ.
Illustrative EGARCH next-variance curve with fixed prior variance 0.0004. A negative one-standard-deviation shock gives variance 0.00046030 and daily standard deviation 2.1455%; a positive shock gives variance 0.00037687 and standard deviation 1.9413%. Their variance ratio is 1.2214. A dashed curve removes the signed term and is symmetric.
Original deterministic model responses, not market observations. The dashed comparison changes only γ to zero. The horizontal axis is standardized news, and the vertical axis is next-period variance in units of 10−4 squared decimal daily returns.
“Larger after negative news” compares the two shocks; it need not mean that every negative shock raises variance above its previous value. The whole recursion, including the carried state and centred magnitude contribution, determines that comparison.
Positive variance is not the same as finite expected variance
For any finite log state, ht+1=exp(loght+1)>0. The coefficients need no nonnegativity restrictions for that positivity result. Stability, estimation and moments impose separate requirements.
With iid unit-variance innovations and centred news, ∣β∣<1 gives the stationary causal log-variance solution. Its mean is
E[loght]=1−βω.
Exponentiating this mean does not give E[ht]. In the example, 0.0004 is the exponentiated mean-log anchor, not the unconditional mean variance. Finite variance-level moments require appropriate exponential moments of the news term. Normal innovations satisfy the relevant conditions here; heavy-tailed innovations may not. In particular, a standardized Student-t innovation can have finite variance while its exponential news factor has infinite expectation. Nelson separates log-process stationarity from these moment conditions in §2 and Appendix I.
The same distinction matters for forecasts. Once rt is observed, ht+1 is determined by the recursion, conditional on the fitted parameters and state. At time t, ht+2 still depends on the unobserved zt+1. Let g(z)=α(∣z∣−c)+γz. With centred news,
where Et means conditioning on Ft, and the second line requires that expectation to be finite. The factor is generally greater than one by convexity. For the example's normal distribution it is approximately 1.0135308. After the negative shock, exponentiating the expected two-step log variance gives 0.0004538857; the actual expected two-step variance is 0.0004600271. Setting future centred news to its mean gets the first calculation, not the second.
The arch forecasting guide describes simulation and residual-bootstrap forecasts for nonlinear variance models such as EGARCH. Its standard analytic EGARCH forecast is one step; do not carry over the ordinary GARCH multi-step recursion. Distribution-specific calculations, such as the Gaussian integration above, can still be derived.
Keep the forecast's information cutoff visible
A forecast for period t+1 may use the return through t, never the target return itself. Historical evaluation must also restrict parameter fitting, conditional-mean estimates and initial variance selection to information available at the forecast origin. A full-sample fitted variance series is not automatically a sequence of historical forecasts.
Check the standardized residuals, remaining dependence in their squares, sign-related misspecification and performance on later data. Allowing asymmetry does not ensure better forecasts, and a conditional variance alone does not specify tail-loss probabilities.
SelfMadeQuant's ARCH and GARCH lesson develops the symmetric baseline, residual diagnostics and quasi-likelihood estimation. The linked page provides a public preview; the full lesson requires membership.
Sources
Daniel B. Nelson, “Conditional Heteroskedasticity in Asset Returns: A New Approach”, Econometrica 59(2), 1991, pp. 347–370: §2, especially equations (2.1)–(2.3) and Theorems 2.1–2.2; Appendix I, especially Theorems A1.1–A1.2. These establish exponential ARCH, signed and centred magnitude news, stationarity and distribution-dependent moments. University-hosted paper.
Kevin Sheppard and contributors, arch.univariate.EGARCH, documentation labelled 7.2.0, accessed 29 October 2026: parameter orders, standardized innovations and the implemented normal-centred variance equation.
Kevin Sheppard and contributors, arch forecasting guide, documentation labelled 7.2.0, accessed 29 October 2026: analytical, simulation and bootstrap methods; forecast origin alignment and use of historical standardized residuals. Numerical examples and figure here are original.
