Daniel B. Nelson

Overview

Daniel B. Nelson (1959–1994) was an econometrician at the University of Chicago Graduate School of Business. Despite his early death at age 34, he made foundational contributions to ARCH modelling: the EGARCH specification, the strict-stationarity criterion for GARCH, continuous-time limits of ARCH processes, and the optimal asymptotic filter theory (with Foster). He was co-author with Bollerslev and Engle of the 1994 Handbook of Econometrics chapter on ARCH Models.

Key Contributions

• EGARCH (Nelson 1991). Exponential GARCH: $\ln\sigma^2_t = \omega + \beta\ln\sigma^2_{t-1} + g(z_{t-1})$ where $g(z_t) = \theta z_t + \gamma(|z_t| - E|z_t|)$. Allows asymmetric (leverage) responses and log-linear specification that guarantees positive variance without inequality constraints. The asymmetric news impact curve $\partial\ln\sigma^2/\partial z$ differs for positive and negative shocks.
• Strict stationarity of GARCH(1,1) (Nelson 1990b). GARCH(1,1) is strictly stationary iff $E[\ln(\alpha_1 z^2 + \beta_1)] < 0$. This condition is strictly weaker than covariance stationarity ($\alpha_1+\beta_1 < 1$), establishing that IGARCH ($\alpha_1+\beta_1=1$) processes can be well-defined and strictly stationary even though their unconditional second moments diverge.
• Continuous-time limits (Nelson 1990a; Nelson 1992). GARCH(1,1) converges in distribution to the SDE $d\sigma^2 = \theta(\omega-\sigma^2)dt + \sigma^2 dW$; EGARCH converges to a log-normal diffusion. These results establish structural links between discrete ARCH models and the continuous-time stochastic-volatility literature and underpin the Nelson (1990) diffusion limit used in the Andersen-Bollerslev (1998) R² analysis.
• Optimal asymptotic filter (Nelson-Foster 1994). For estimating an underlying continuous-time diffusion, the minimum-asymptotic-error-variance ARCH filter uses absolute-value residuals (not squared) when the true conditional distribution has heavy tails. This provides formal justification for absolute-value GARCH (TARCH/NARCH) over standard squared-residual GARCH.
• Bivariate EGARCH (Braun-Nelson-Sunier 1992). Extended EGARCH to the bivariate case to estimate time-varying conditional beta (covariance/variance ratio) for individual stocks vs. the market, incorporating leverage effects in both dimensions.

Related

• GARCH and BEKK-GARCH
• Stochastic Volatility
• Tim Bollerslev
• Robert Engle
• Bollerslev-Engle-Nelson (1994) ARCH Models