Conditional Heteroskedasticity in Asset Returns: A New Approach

Source: Nelson, D. B. (1991) · Econometrica 59(2), 347–370 · doi:10.2307/2938260

TL;DR

Introduces EGARCH (exponential GARCH), which models the logarithm of the conditional

variance as a linear function of past standardized shocks. This fixes three drawbacks Nelson

identifies in GARCH for asset-pricing work: (i) GARCH treats positive and negative shocks

symmetrically and so rules out the leverage effect (Black 1976) by assumption; (ii) GARCH

imposes non-negativity constraints on coefficients that are often violated and restrict the

variance dynamics; (iii) "persistence" of variance shocks is ambiguous in GARCH. By working in

logs, EGARCH guarantees a positive variance for any parameter values, allows an asymmetric

response to the sign of returns, and yields a linear process whose stationarity is easy to check.

What it models

The time-varying conditional variance σ²_t of asset return innovations. Returns are written

ε_t = σ_t z_t with z_t i.i.d., mean 0, variance 1. GARCH (Bollerslev 1986) makes σ²_t a linear

combination, with positive weights, of past squared residuals — capturing volatility clustering

but symmetric in the sign of shocks and constrained to keep σ²_t ≥ 0. EGARCH instead specifies

the log variance, removing both the sign-symmetry and the positivity constraint.

Specification (the equation)

EGARCH models the log conditional variance as an infinite moving average in a function g(·) of

the standardized shocks:

ln(σ²_t) = α_t + Σ_{k≥1} β_k · g(z_{t−k}),

where the asymmetry function is

g(z_t) = θ·z_t + γ·[ |z_t| − E|z_t| ].

By construction {g(z_t)} is zero-mean, i.i.d. The term γ·[|z_t| − E|z_t|] responds to the

magnitude of shocks; the term θ·z_t responds to their sign. g is piecewise linear in z_t:

slope (θ + γ) for z_t > 0 and slope (θ − γ) for z_t < 0. With θ < 0 a negative return

innovation raises future log-variance more than a positive one of the same size — the leverage

effect. Because the left side is ln(σ²_t), the variance σ²_t = exp(α_t + Σ β_k g(z_{t−k})) is

positive for any parameter values, with no inequality restrictions (the β_k may be positive

or negative, so cyclical dynamics are permitted). In applications the infinite MA is replaced by

a parsimonious ARMA representation for ln(σ²_t) (Nelson's eq. 2.3).

Innovations z_t are drawn from the GED (Generalized Error Distribution), with a tail-thickness

parameter v: v = 2 gives the normal, v < 2 gives fatter-than-normal tails (v = 1 is the double

exponential), and v > 2 thinner tails. This lets EGARCH match the fat tails of asset returns

beyond what time-varying variance alone produces.

Estimation

Estimated by (quasi-) maximum likelihood. Nelson fits a model in which the conditional mean of

excess returns R_t is linear in σ²_t (a GARCH-in-mean / risk-premium specification), with

ε_t = R_t − E_{t−1}[R_t] following the EGARCH process and z_t i.i.d. GED(v). The application is to

daily returns on the CRSP Value-Weighted Market Index, 1962–1987. Model selection (information

criterion) chose an ARMA(2,1) parameterization for ln(σ²_t). Because ln(σ²_t) is a linear

process, its strict/covariance stationarity and ergodicity follow from standard linear-process

conditions (e.g., the AR roots lying outside the unit circle), unlike the disputed persistence

norms of GARCH/IGARCH.

What it captures (stylized facts)

is the model's headline contribution and is built in by construction, not by constraint.

Use & extensions

Jagannathan–Runkle 1993), standard for equities where the leverage effect is strong.

(Engle–Ng 1993) was developed to compare such asymmetric models.

Limitations

(forecasts of σ², not ln σ², require care with the exponential).

combined with the exponential), unconditional moments of σ²_t may not be finite, so existence

conditions must be checked.

Key references

Provenance: verified/generated from the paper's full text.