Stylized Facts of Asset Returns

Definition

Stylized facts of asset returns are empirical regularities that appear robustly across assets, markets, and sample periods. They are the target phenomena that any serious volatility model must reproduce. The canonical list below is documented in daily equity, exchange rate, and interest rate data; the sources are primarily the generalized autoregressive conditional heteroskedasticity (GARCH) and stochastic volatility literature of the 1980s–2000s.

Key Ideas

Returns in levels are approximately serially uncorrelated — consistent with the efficient market hypothesis (EMH).
Returns are not strict white noise: nonlinear functions of returns, especially $r_t^2$ and $|r_t|$ , are strongly autocorrelated.
The conditional variance is time-varying, clustering around high-volatility and low-volatility regimes.
The unconditional distribution of returns has heavier tails than the normal; this excess kurtosis is explained by time-varying volatility rather than a fixed fat-tailed distribution.
Negative return shocks raise future volatility more than positive shocks of equal magnitude — the leverage effect and asymmetric news impact.
These effects are frequency-dependent: they are strong at daily frequency and largely disappear at monthly frequency.

How It Works

1. Linear Independence, Nonlinear Dependence

The foundational diagnostic (Akgiray 1989): fit an autoregressive (AR)(1) to remove the small first-lag autocorrelation ( $\hat\phi_1 \approx 0.18$ – $0.31$ in daily equity data), yielding residuals $\{e_t\}$ . Standard linear independence tests (Ljung-Box, Bartlett, Fisher) fail to reject white noise for $\{e_t\}$ . But $\{e_t^2\}$ and $\{|e_t|\}$ remain significantly autocorrelated at all lags up to 60 days — conclusive evidence of nonlinear dependence. This is the diagnostic argument for GARCH and stochastic volatility (SV) over purely linear models.

2. Fat Tails and Excess Kurtosis

Raw daily returns exhibit excess kurtosis (leptokurtosis) in every sample period and every asset class examined. The key insight from GARCH estimation (Akgiray 1989) is that standardized residuals $e_t / \hat\sigma_t$ pass normality tests once GARCH is fitted — the excess kurtosis of raw returns is entirely explained by time-varying conditional variance, not by a fat-tailed unconditional distribution. In continuous-time SV models (Stein-Stein 1991), the return distribution is a mixture of lognormals — the mixing is over realized average variance — which produces power-law tails:

$S(P,t) \sim P^{-\gamma} \quad \text{as } P \to \infty$

with $\gamma > 2$ for mean-reverting volatility and $\gamma \to 2$ as $t \to \infty$ under a pure random-walk volatility.

3. Volatility Clustering

Large return shocks tend to be followed by further large shocks, and small shocks by small ones. GARCH(1,1) captures this through the recursion

$\sigma_t^2 = \omega + \alpha_1 \varepsilon_{t-1}^2 + \beta_1 \sigma_{t-1}^2$

The autocorrelation function (ACF) of $\{e_t^2\}$ follows the implied GARCH pattern $c_n = (\alpha_1 + \beta_1)^n c_0$ closely — 48 of 60 empirical autocorrelations lie within $\pm 10\%$ of the GARCH-implied values (Akgiray 1989).

4. Persistent Volatility and Near-IGARCH

Estimated GARCH(1,1) parameters on daily equity and exchange rate data consistently yield $\hat\alpha_1 + \hat\beta_1 \approx 0.96$ – $0.99$ , very close to the integrated GARCH (IGARCH) boundary $\alpha_1 + \beta_1 = 1$ at which shocks to variance are permanent. The dominance of $\hat\beta_1 \gg \hat\alpha_1$ means past conditional variance matters far more than the most recent innovation in updating the variance estimate — volatility is sticky.

Formally, IGARCH ( $\alpha_1 + \beta_1 = 1$ ) implies that the unconditional variance does not exist, while conditional variance forecasts diverge at rate $h$ steps ahead. Dickey-Fuller tests reject the exact unit root in 3 of 4 sub-periods in Akgiray (1989), so the process is covariance-stationary but near-integrated. The near-IGARCH finding was anticipated by the co-persistence analysis of Bollerslev-Engle (1993); see GARCH and BEKK-GARCH.

5. Leverage Effect

The leverage effect is the negative correlation between a return shock and subsequent volatility: bad news raises future volatility more than good news of equal magnitude. The name originates from the financial leverage channel — a price drop increases the debt-to-equity ratio, raising equity risk — though the effect operates through multiple channels.

In discrete-time SV, the specification requires care over timing (Yu 2005). Two asymmetric SV (ASV) models:

	ASV1 (Harvey-Shephard 1996)	ASV2 (Jacquier et al. 2004)
Correlation	$\text{corr}(u_t, v_{t+1}) = \rho$	$\text{corr}(u_t, v_t) = \rho$
Timing	return shock → next-period vol shock	same-period
EMH	$E(X_{t+1}\\|X_t) = 0$ ✓	$E(X_{t+1}\\|X_t) \neq 0$ — predictable returns ✗

ASV1 is theoretically correct. Orthogonalising by setting $w_{t+1} = (v_{t+1} - \rho u_t)/\sqrt{1-\rho^2}$ yields:

$\ln\sigma^2_{t+1} = \alpha + \phi\ln\sigma^2_t + \underbrace{\rho\sigma_\eta\sigma_t^{-1}X_t}_{\text{leverage}} + \sigma_\eta\sqrt{1-\rho^2}\,w_{t+1}$

The coefficient on $X_t$ is $\rho\sigma_\eta\sigma_t^{-1}$ : if $\rho < 0$ , a negative return directly raises next-period expected log-variance. The analogous re-parameterisation of ASV2 is intractable — it places $\sigma_{t+1}$ on both sides — so no clean leverage reading exists.

Empirically (Yu 2005, S&P 500 1980–1987): Bayes factor ASV1/ASV2 $\approx 4{,}837$ — decisive. $\hat\rho \approx -0.32$ to $-0.39$ ; ASV2 understates $|\hat\rho|$ by $\approx 20\%$ .

In the GARCH family, the leverage effect is captured by Exponential GARCH (EGARCH) (Nelson 1991) and GJR-GARCH (Glosten-Jagannathan-Runkle 1993); see GARCH and BEKK-GARCH §News Impact Curve.

6. Asymmetric News Impact

The news impact curve (NIC; Engle and Ng 1993) plots $\sigma_t^2$ as a function of the single most recent innovation $\varepsilon_{t-1}$ , holding lagged variance fixed at its unconditional level:

Standard GARCH(1,1): $\text{NIC}(\varepsilon) \propto \varepsilon^2$ — symmetric parabola; positive and negative shocks of equal size have identical effects.
EGARCH: $\ln h_t = \omega + \beta\ln h_{t-1} + \gamma z_{t-1} + \alpha(|z_{t-1}| - E|z|)$ — asymmetric: negative $z_{t-1}$ raises $h_t$ more.
Power-GARCH (Ding, Granger, Engle 1993): $h_t^\delta = \omega + \alpha|\varepsilon_{t-1}|^{2\delta} + \beta h_{t-1}^\delta$ , estimated $\hat\delta = 1.43$ on daily S&P 500 (1928–1991), rejecting both $\delta = 1$ (absolute value) and $\delta = 2$ (variance).

Asymmetric news impact means the variance—return relationship is not symmetric in $\varepsilon$ : the left branch of the NIC lies above the right branch.

7. Volatility Smile

When the volatility of log-returns is stochastic rather than constant, Black-Scholes implied volatility is U-shaped in log-moneyness $c_1 = \ln(P/Ke^{-rT})$ :

$\text{implied variance} \approx E[AV] + \varepsilon, \quad \varepsilon \approx \tfrac{1}{2}\frac{\text{Var}(AV)}{E[AV]}\!\left(-\!\left(\tfrac{c_1}{2}\right)^2 + \left(\tfrac{c_1}{c_2}\right)^2 - 1\right)$

where $AV = t^{-1}\int_0^t \sigma^2(s)\,ds$ is realized average variance and $c_2 = \sqrt{E[AV]\cdot T}$ (Ball-Roma 1994). The correction $\varepsilon$ is quadratic in $c_1$ , minimized at-the-money (ATM) and rising for out-of-the-money (OTM)/in-the-money (ITM) options — the formal derivation of the smile. The smile becomes a skew (downward sloping) once the leverage effect is incorporated (Heston 1993 correlated SV).

8. Frequency Dependence

GARCH and volatility clustering are daily phenomena. Akgiray (1989) finds that monthly returns are approximately independent and identically distributed (i.i.d.) normal: no significant autocorrelation in $\{e_t^2\}$ at the monthly frequency. The explanation is a central limit theorem (CLT) for sums of dependent daily returns — the nonlinear short-run dependence averages out over longer horizons, leaving near-Gaussian monthly increments.

9. Co-Persistence in Variance

Even when individual asset variances are persistent (near-IGARCH), a linear combination of assets may have a stationary conditional variance — the variance analogue of cointegration (Bollerslev-Engle 1993). The co-persistent portfolio eliminates all explosive eigenvectors of the GARCH coefficient matrix.

Empirically: daily Deutsche Mark (DM) and British Pound (BP) vs. US dollar (USD) are both near-IGARCH, but the DM/BP bilateral rate strongly rejects IGARCH ( $t$ -stat on $\hat\alpha+\hat\beta=1$ equals $6.03$ ). The co-persistent vector $\hat\gamma \approx (1,-1)$ arises naturally from estimation — volatility persistence in USD exchange rates is dollar-specific news, not a property of European cross rates.

Why It Matters

A model that ignores any of these facts will systematically misprice risk. GARCH on raw returns without asymmetry understates downside risk; SV without leverage understates put premia.
The near-IGARCH finding has direct implications for long-horizon risk: if volatility is truly integrated, conditional variance forecasts grow without bound.
The frequency-dependence result (GARCH vanishes monthly) is practically important: variance forecasting at quarterly or annual horizons may not need GARCH at all.
The leverage effect specifically explains the volatility skew observed in equity options markets: the premium on downside puts relative to equivalent calls.

Open Questions

How much of the near-IGARCH finding is spurious IGARCH from ignored structural breaks? Hillebrand (2004) shows regime changes cause $\hat\alpha+\hat\beta \to 1$ even in stationary processes; see Structural Break Testing.
Whether the leverage effect is primarily financial (debt-equity channel) or statistical (volatility feedback) remains debated.
Distinguishing GARCH from SV in finite samples is hard: both fit $\{e_t^2\}$ autocorrelations well, and non-nested tests have low power.