Conditional Heteroscedasticity in Time Series of Stock Returns: Evidence and Forecasts

Summary

One of the first rigorous empirical applications of generalized autoregressive conditional heteroscedasticity (GARCH) to US equity returns. Using Center for Research in Security Prices (CRSP) daily returns on the value-weighted index (1963–1986, $T = 6{,}030$), Akgiray establishes that daily returns are white noise but not strict white noise — the $\{|e_t|\}$ and $\{e_t^2\}$ series remain significantly autocorrelated at lags up to 60 days after first-order autoregressive (AR(1)) pre-filtering. GARCH(1,1) fits better than any autoregressive conditional heteroscedasticity (ARCH) model of order $p$ by log-likelihood, and its out-of-sample monthly variance forecasts dominate historical averages, exponentially weighted moving averages (EWMA), and ARCH on all four error metrics.

Key Claims

• Daily returns on CRSP value-weighted index show significant first-lag autocorrelation ($\hat{\phi}_1 = 0.18$–$0.31$ across sub-periods) but near-zero autocorrelation at longer lags.
• After AR(1) transformation, residuals $\{e_t\}$ appear white noise by standard linear tests (Fisher, Bartlett, Ljung-Box on levels). But $\{|e_t|\}$ and $\{e_t^2\}$ are strongly autocorrelated at all lags up to 60 — conclusive evidence of nonlinear (non-strict-white-noise) dependence.
• Excess kurtosis and non-normality are present in all sub-periods; this is explained by time-varying conditional variance rather than a fixed fat-tailed unconditional distribution.
• GARCH(1,1) strictly dominates ARCH(p) by log-likelihood in every period — no likelihood-ratio (LR) test is needed to declare it superior.
• GARCH(1,1) estimates: $\alpha_1 \approx 0.05$–$0.09$, $\beta \approx 0.76$–$0.92$, $\alpha_1 + \beta \approx 0.96$–$0.99$ in all sub-periods. Dickey-Fuller rejects $\alpha_1 + \beta = 1$ in 3 of 4 periods → stationary but near-integrated GARCH (IGARCH).
• $\beta \gg \alpha_1$ in every period: past conditional variance dominates new innovations in updating the variance → persistent volatility is the key empirical stylized fact.
• Standardized residuals $(e_t - \mu_t)/\sqrt{v_t}$ pass normality tests; the GARCH model fully removes excess kurtosis (residual kurtosis $\leq 0.37$). This shows GARCH accounts for all the leptokurtosis.
• The autocorrelation function (ACF) of $\{e_t^2\}$ follows the GARCH(1,1) recursion $c_n = (\alpha_1 + \beta)\cdot c_{n-1}$ closely (48/60 autocorrelations within $\pm 10\%$ of the implied values).
• Out-of-sample forecasting (24 monthly variance forecasts per period, rolling estimation): GARCH beats historical mean, EWMA, and ARCH on mean error (ME), root mean squared error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) in all four periods. The superiority is largest in high-volatility periods (1969–74, 1975–80).
• Minimum MAPE $> 30\%$ across all models — variance forecasting is inherently difficult.
• GARCH effects vanish at the weekly/monthly level: monthly returns are approximately independent and identically distributed (iid) normal, reflecting a central limit theorem (CLT) for sums of dependent daily returns.
• Including day-of-week (Monday dummy) makes negligible difference — the nonlinear dependence structure is not a weekday artifact.

Concepts Introduced or Extended

• GARCH and BEKK-GARCH

Entities Mentioned

• Vedat Akgiray
• Robert Engle

Quotes

"GARCH(1, 1) processes fit to data very satisfactorily. Various out-of-sample forecasts of monthly return variances are generated and compared statistically. Forecasts based on the GARCH model are found to be superior." (abstract)

"A central limit theorem for sums of dependent (daily) returns may be manifesting itself in these findings about weekly and monthly series." (p. 79)

My Take

This paper's primary contribution is the volatility forecasting comparison, which gives GARCH a concrete practical use case beyond goodness-of-fit. The near-IGARCH finding ($\alpha_1 + \beta \approx 0.97$–$0.99$) foreshadows the co-persistence work of Bollerslev-Engle (1993). The sub-period analysis is honest about structural nonstationarity but the choice of 6-year windows is admittedly arbitrary. The demonstration that AR(1) residuals are white noise but $\{e_t^2\}$ are still correlated is a clean pedagogical argument for why linear models are insufficient.