GARCH Family Extensions: Asymmetry, Power, and Multivariate

Plain GARCH treats positive and negative shocks identically: a 2% drop and a 2% rally have exactly the same effect on next-day variance. The data disagree. Since Black (1976) we have known that negative returns raise volatility more than positive returns of the same magnitude. The asymmetry is real, robust across markets, and economically consequential — the volatility skew in options markets has it somewhere in its ancestry.

The variance equation feeds only on $\varepsilon_{t-1}^2$, so the sign of the shock is destroyed at the squaring step — a blind spot hard-wired into the structure of GARCH(1,1). There are others: autocorrelations of squared returns decay far more slowly than GARCH allows (long memory), and correlations between assets move, most violently in crises (the multivariate problem). The extensions that make up the GARCH family grew up around these gaps. This article works through them gap by gap, and every extension has to pass the same test: do the extra parameters earn their keep out of sample? Textbook references are Tsay (2010, Ch. 3, 7) and Francq & Zakoian (2010, Ch. 7-10); original papers are cited where they come up.

The Leverage Effect

Black (1976) observed that stock volatility rises when prices fall, and offered a literal explanation: the price drops, the debt-to-equity ratio rises, the firm gets riskier, volatility follows — hence the name leverage effect. The name survived; the explanation did not entirely. Equity index volatility shows the same effect, and an index has no balance sheet to lever. Volatility feedback, changing risk premia, and behavioral effects have all been proposed to fill the gap, and the debate over mechanism remains open. The empirical fact itself is not in dispute: volatility after a negative return is systematically higher than after a positive return of the same size.

To capture this, the sign has to be smuggled back into the variance equation. The next three models are three ways of doing it.

GJR-GARCH: A Switch for Negative Shocks

Glosten, Jagannathan & Runkle (1993) take the most direct route — add a term that switches on only for negative shocks:

The indicator $\mathbb{1}[\varepsilon_{t-1} < 0]$ is 1 when the previous shock was negative and 0 otherwise, so negative shocks enter the variance recursion with weight $(\alpha + \gamma)$ while positive shocks get only $\alpha$. An estimated $\gamma > 0$ is the leverage effect in numbers. The constraints $\omega > 0$, $\alpha \geq 0$, $\alpha + \gamma \geq 0$, $\beta \geq 0$ keep the variance positive.

The stationarity condition has a detail that is easy to misremember: the persistence parameter of GJR(1,1) is not $\alpha + \gamma + \beta$ but $\alpha + \gamma/2 + \beta$. The 1/2 comes from the fact that under a symmetric innovation distribution, negative shocks occur exactly half the time — the leverage term is only working on average half the time. Persistence below 1 gives covariance stationarity.

Plotting $E[\sigma_{t+1}^2 \mid \varepsilon_t]$ against $\varepsilon_t$ gives the news impact curve, which is now asymmetric: for $\varepsilon_t > 0$ it is $\omega + \alpha \varepsilon_t^2 + \beta \sigma^2$, linear in $\varepsilon_t^2$; for $\varepsilon_t < 0$ the slope steepens to $(\alpha + \gamma)$. The curve has a kink at zero and is steeper on the negative side — bad news has a larger marginal effect on tomorrow’s variance, which is exactly the shape the data demand.

In code the model costs almost nothing extra; in the arch package it is one more argument:

from arch import arch_model


model = arch_model(r, mean='Constant', vol='GARCH', p=1, o=1, q=1, dist='t')
# o=1 adds the leverage term (gamma)
result = model.fit()
print(result.summary())

# Conditional volatility
vol = result.conditional_volatility

# News impact curve: plug in shocks of various sizes
import numpy as np
shocks = np.linspace(-0.05, 0.05, 21)
params = result.params
omega, alpha, gamma, beta = params['omega'], params['alpha[1]'], params['gamma[1]'], params['beta[1]']
long_run_var = omega / (1 - alpha - gamma/2 - beta)
nic = omega + alpha * shocks**2 + gamma * shocks**2 * (shocks < 0) + beta * long_run_var

EGARCH: Move to the Log Scale

Nelson (1991) went a different way: instead of patching the variance, model the log variance directly.

where $g(z) = \theta z + \gamma [|z| - E|z|]$ and $z_t = \varepsilon_t / \sigma_t$ is the standardized residual. The function $g$ is asymmetric in $z$ by construction: positive $z$ contributes $\theta z + \gamma(z - \sqrt{2/\pi})$, negative $z$ contributes $\theta z + \gamma(-z - \sqrt{2/\pi})$ (using $E|z| = \sqrt{2/\pi}$ for the standard normal). The construction separates two kinds of asymmetry. $\theta < 0$ is the sign effect — negative returns raise log variance more than positive ones. $\gamma > 0$ is the size effect — large shocks raise variance more than small ones, regardless of sign. GJR mixes both effects into one $\gamma$; EGARCH gives each its own parameter.

The log scale buys genuine convenience. Since $\log \sigma_t^2$ can be any real number and the exponential guarantees $\sigma_t^2 > 0$, no positivity constraints are needed; MLE runs over an unconstrained parameter space rather than crawling along $\geq 0$ boundaries; and stationarity only requires $\sum \beta_i < 1$, with no constraint on the $\alpha$‘s. The convenience has a price. When $\log \sigma_t^2$ wanders deep into negative territory, the exponential amplifies small perturbations into extreme values, so the recursion is numerically fragile; multi-step forecasts can differ substantially from GARCH forecasts because of the Jensen term the log transform introduces; and the model is less intuitive than GARCH or GJR — its coefficients have no direct “weight” reading.

Empirically, EGARCH and GJR-GARCH often fit about equally well in sample and then part ways out of sample. In practice the choice between them tends to be decided by which software package you know, not by any deep theoretical consideration — that is not a jab, it is a description of the field.

APARCH: Let the Data Choose the Power Too

The APARCH (Asymmetric Power ARCH) model of Ding, Granger & Engle (1993) asks a more radical question: why should the object being modeled be the variance (the second power) at all? Let the data pick the power:

where $\delta > 0$ is the power and $\gamma_i$ the asymmetry parameter. The framework swallows the earlier models whole: $\delta = 2$, $\gamma_i = 0$ recovers standard GARCH; $\delta = 1$, $\gamma_i \neq 0$ is a threshold-style asymmetric model on the absolute-value scale (Zakoian’s 1994 TGARCH, a close cousin of GJR); $\delta = 1$, $\gamma_i = 0$ is GARCH in absolute deviations (rare); and general $\delta \neq 2$ amounts to a Box-Cox transformation of volatility.

The estimated $\delta$ is the interesting part: on daily equity returns it usually lands between 1 and 2, closer to 1. That is the data saying that modeling absolute deviations is nearer the right shape than modeling squares. But the intellectual interest never cashed out as forecasting advantage — in practice APARCH rarely beats GJR or EGARCH out of sample. This is the article’s test doing its job for the first time: the extra freedom feasts in sample and starves out of sample.

Forecasting with Asymmetric GARCH

Multi-step forecasting with an asymmetric model has one extra moving part compared with symmetric GARCH: whether the leverage term fires depends on the sign of future shocks, and at forecast time only its probability is available. For GJR(1,1), the $h$-step conditional variance forecast is

where $P(\varepsilon < 0)$ is the unconditional probability of a negative shock — 1/2 for symmetric innovations. As long as $\alpha + \gamma \cdot P(\varepsilon < 0) + \beta < 1$, the forecast converges geometrically to the unconditional variance $\sigma^2$. Note the forecast depends on the distributional assumption: Student-$t$ innovations are symmetric, so $P(\varepsilon < 0)$ is still 1/2 and the convergence rate matches the Gaussian case, but the formula for the unconditional variance changes, so the level of $\sigma^2$ differs.

If what you need is a forecast interval for returns rather than a point forecast of variance, simulate forward: at each step draw $z_t \sim t_\nu$ (or whatever distribution you fitted), compute $\sigma_t^2$ from the recursion, produce $\hat{r}_t = \mu + \sigma_t z_t$, repeat for 10,000 paths, and take the 5th and 95th percentiles of the simulated distribution as the 90% interval. Remember that these intervals ignore parameter uncertainty — treating estimated parameters as truth makes them too narrow; bootstrap or Bayesian methods do better if it matters.

Long Memory: FIGARCH and Its Troubles

Asymmetry handled, on to the second blind spot. Standard GARCH(1,1) implies geometrically decaying autocorrelations in squared returns, while in the data they decay hyperbolically, much more slowly: the autocorrelation function of daily $r_t^2$ looks like $\rho_h \sim h^{-d}$ with $d$ somewhere around 0.3 to 0.5. A geometric model chasing hyperbolic data loses the information sitting in high-order lags.

FIGARCH (Baillie, Bollerslev & Mikkelsen, 1996) closes the gap by introducing fractional differencing into the variance recursion; FIGARCH(1,$d$,1) is

The operator $(1 - L)^d$ with $0 < d < 1$ expands into an infinite-order lag polynomial with hyperbolically decaying coefficients — exactly the long memory required. Davidson’s (2004) HYGARCH generalizes it.

But FIGARCH is notorious to estimate: the likelihood theory is non-standard (infinite-variance processes appear in the limit), standard errors are unreliable and bootstrap is often the fallback, and the likelihood surface has multiple local optima. The sober way to think about it: FIGARCH is essentially a parsimonious approximation to a high-order GARCH — say, a GARCH(50,50) with constrained coefficients. If you can fit the high-order GARCH, you do not need FIGARCH; if you cannot (too many parameters), FIGARCH delivers a similar fit with far fewer. It is an engineering answer to a parameter budget, not a new truth.

Long memory and asymmetry can be combined. FIEGARCH (Bollerslev & Mikkelsen, 1996) moves the fractional-differencing operator onto EGARCH’s log-variance recursion; schematically,

capturing long memory and leverage at once. There is evidence in the literature that it beats both standard GARCH and FIGARCH out of sample on daily equity returns, but the conclusion depends on sample and loss function — do not treat it as a law.

Component GARCH: Split the Memory into Fast and Slow

There is a road to slow decay that avoids fractional differencing altogether. The Component GARCH (CGARCH) of Engle & Lee (1999) decomposes volatility into permanent and transitory pieces:

The permanent component $q_t$ moves slowly with persistence $\rho$ (close to 1 for most financial series); the transitory component $s_t$ decays geometrically at rate $\alpha + \beta$. The superposition of two decay rates produces autocorrelations that fade more slowly than any single geometric rate — an approximation to long memory. Half-lives are directly computable: $\log(0.5)/\log(\rho)$ for the permanent component, $\log(0.5)/\log(\alpha + \beta)$ for the transitory. On daily equity returns the typical orders of magnitude are $\rho \approx 0.99$ (half-life around 70 trading days) and $\alpha + \beta \approx 0.9$ (around 7 days) — the exact numbers move with the sample, the magnitudes are stable.

The practical value of the decomposition is separating “the long-run volatility level is shifting” ($q_t$ moving) from “temporary clustering” ($s_t$ moving). For long-horizon forecasting, only the permanent component matters.

GARCH-X: Wiring In Outside Information

The variance equation can also consume exogenous predictors directly:

What goes into $\mathbf{x}_t$ depends on what you have. Implied volatility of the VIX type is the most common choice (high VIX predicts high realized volatility); trading volume tends to coincide with high volatility; scheduled macro announcements (FOMC, CPI) produce volatility spikes; and cross-asset signals — equity volatility spilling into FX — get used too. GARCH-MIDAS (Engle, Ghysels & Sohn, 2013) is the refined version of this road: a short-run GARCH component paired with a long-run MIDAS component driven by low-frequency exogenous variables, useful when monthly or quarterly predictors bear on daily volatility. Notice it is a relative of the component model from the previous section — both are fast-slow decompositions, except here the slow layer is driven by exogenous variables rather than filtered out of the returns themselves.

Multivariate: From CCC to DCC

The third blind spot is the most practical: portfolio risk depends not just on each asset’s variance but on the covariances between them. A fully general multivariate GARCH has $K(K+1)/2$ variance-covariance equations, with a parameter count that explodes in $K$; the naive parameterizations (VECH, BEKK) become infeasible past $K > 5$. The history of multivariate GARCH is a history of haggling with the curse of dimensionality.

The first bargain is CCC-GARCH (Bollerslev, 1990): Constant Conditional Correlation. Each asset follows its own univariate GARCH and the correlation matrix does not move:

where $\sigma_{i,t}^2$ is asset $i$‘s GARCH variance and $\mathbf{R}$ a constant correlation matrix. Estimation is delightfully cheap: fit $K$ univariate GARCHs, then compute $\mathbf{R}$ from the standardized residuals. But the constant-correlation assumption fails precisely when you need the model most. In calm markets, correlations between equity indices sit somewhere around 0.4-0.6; in crisis periods (2008, 2020) they jump to the 0.8-0.95 range. Anyone computing portfolio VaR cannot look away from this: underestimating crisis correlations means underestimating portfolio risk at exactly the moment risk peaks.

DCC-GARCH (Engle, 2002; Engle & Sheppard, 2001) replaces $\mathbf{R}$ with a time-varying $\mathbf{R}_t$ and gives correlations their own GARCH-like recursion:

where $\mathbf{z}_t = (\varepsilon_{1t}/\sigma_{1t}, \ldots, \varepsilon_{Kt}/\sigma_{Kt})'$ are the standardized residuals and $\bar{\mathbf{Q}}$ is their unconditional covariance. Normalizing back to a correlation matrix:

The parameters $a$ and $b$ govern the correlation dynamics, and estimation is two-step QML: fit the univariate GARCHs first, then estimate $(a, b)$. The whole apparatus scales linearly in $K$, which is why it can go to size. Forecasting $\mathbf{R}_{T+h}$ is just iterating the recursion forward; the forecast converges to the long-run correlation matrix, and correlation shocks have half-life $\log(0.5)/\log(a+b)$. Typical estimates of $a + b$ on daily equity correlations run 0.95 to 0.99, putting the half-life anywhere from a couple of weeks to roughly three months depending on the sample.

DCC has a known wart: when $\mathbf{Q}_t$ is near-singular, finite-sample correlations can escape $(-1, 1)$; the cDCC and ADCC variants exist to patch this. As for the other multivariate specifications, knowing the name and the niche is enough: VECH is the full parameterization of the covariance matrix, with too many parameters; BEKK (Engle & Kroner, 1995) guarantees a positive-definite $\mathbf{H}_t$ by construction but is computationally heavy; RiskMetrics is exponential smoothing of squared returns, useful as a cheap benchmark; GO-GARCH (van der Weide, 2002) goes the factor route, reducing dimension through a small number of latent volatility factors. Managing risk on $K = 50$ assets, DCC is the practical default; researching new correlation dynamics, GO-GARCH or a stochastic volatility model may serve better.

Which One, Then

Time to gather the article’s verdicts. Single asset, unsure whether asymmetry matters: fit GARCH(1,1) with Student-$t$ innovations as the baseline, then GJR(1,1) and EGARCH(1,1) and compare AIC/BIC — if the leverage coefficient ($\hat{\gamma}$ in GJR, $\hat{\theta}$ in EGARCH) is significant, the asymmetry stays. Persistence $\hat{\alpha} + \hat{\gamma}/2 + \hat{\beta}$ pressing against 1 suggests IGARCH or FIGARCH. Before signing off, check two things: standardized residuals should be near iid (Ljung-Box on $\hat{z}_t^2$ should not reject), and always run an out-of-sample comparison under MSE or QLIKE loss, because in-sample fit barely discriminates within this family. Multiple assets with correlations in play: CCC to start, DCC as the workhorse, ADCC if asymmetric correlations matter too. On software, R’s rugarch is the most flexible, Python’s arch covers most variants, MATLAB has the garch toolbox.

A few fitting hazards deserve naming. Parameter estimates near the boundary ($\alpha + \beta$ pinned at 1, $\alpha$ pinned at 0) are unreliable — change constraints or starting values; the optimizer can settle in local optima — try several starting points; and forecast intervals that ignore parameter uncertainty come out too narrow, as noted earlier.

Last, restraint. If you catch yourself reaching for FIGARCH-FIEGARCH-MIDAS with Student-$t$ innovations and exogenous regressors, stop and ask whether the added complexity is improving out-of-sample forecasts. The empirical literature answers with unusual consistency (Hansen & Lunde’s 2005 grand horse race is the reference point): GJR-GARCH(1,1) with Student-$t$ is hard to beat. More complex models occasionally win by a small margin in forecast loss — not by enough to pay for themselves in most applications. For multivariate work DCC is the workhorse; and once the data frequency is high enough to carry intraday information, realized-volatility models (HAR, HEAVY) frequently overtake the entire GARCH family.

Keep the family’s ceiling in mind too. GARCH is reduced-form: it reproduces stylized facts and offers no structural economic mechanism. To learn why volatility is high — information flow, leverage, liquidity, risk premia — you need a different class of model. And for options, GARCH-implied volatility is a serviceable input, but it does not match observed option prices as well as dedicated pricing models (stochastic vol, local vol, jump-diffusion). GARCH is a time-series tool, not a derivatives-pricing tool.

References

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