Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation
Source: Engle, R. F. (1982) · Econometrica 50(4), 987–1007 · DOI: 10.2307/1912773
TL;DR
Introduces ARCH (autoregressive conditional heteroscedasticity), the first model in which the conditional variance changes over time and depends on past squared shocks, while the unconditional variance stays constant. ARCH formalizes volatility clustering — large changes follow large changes — and launched the entire field of conditional-volatility modeling. The empirical application estimates time-varying uncertainty in U.K. inflation (1958–1977); Engle shared the 2003 Nobel for this work.
What it models
Classical regression assumes a constant one-period forecast variance V(yₜ | yₜ₋₁) = σ². Engle relaxes this: the conditional variance is a function of the recent past, so even when the conditional mean is well modeled, the uncertainty of the forecast itself moves through calm and turbulent periods and is predictable.
Today's conditional variance rises after recent large (squared) shocks. Nonnegativity (αᵢ ≥ 0) and stationarity (Σαᵢ < 1) constraints are required. The empirical model uses a fourth-order ARCH with linearly declining weights to economize on parameters (eq. 38):
Maximum likelihood via a simple scoring iteration (alternating steps on the variance parameters α and the mean parameters β). OLS remains consistent but is inefficient; the relative efficiency gain can be arbitrarily large. To test for ARCH, Engle proposes a Lagrange multiplier test based on the autocorrelation of the squared OLS residuals (a TR² statistic), which is the standard ARCH-LM test still used today.
What it captures
Volatility clustering / time-varying uncertainty. In the U.K. inflation application (quarterly CPI vs. manual wage rates, sample 1958-II to 1977-II), the LM test for first-order ARCH was insignificant but the fourth-order test was highly significant (χ²₄ = 15.2).
The estimated one-step-ahead forecast variances hₜ range from 23×10⁻⁶ to 481×10⁻⁶, i.e. forecast standard deviation of inflation rising from ≈0.6% in the late 1960s to ≈1.5% in the chaotic mid-1970s (average hₜ since 1974 = 230×10⁻⁶ vs. 42×10⁻⁶ in the last four years of the sixties).
The ARCH model produces residual outliers that are far less clustered in time than OLS outliers — better-calibrated conditional confidence intervals.
Use & extensions
The foundation of all conditional-volatility modeling: GARCH (Bollerslev 1986), EGARCH (Nelson 1991), GJR/threshold GARCH, stochastic volatility, realized volatility.
Volatility forecasting, Value-at-Risk, and option pricing all build on the ARCH idea that variance is time-varying and forecastable.
Limitations
Pure ARCH needs many lags to capture persistence; GARCH adds lagged variance terms to do this parsimoniously.
Symmetric in shocks — no leverage effect (negative returns raising volatility more than positive); addressed by EGARCH/GJR-GARCH.
Conditional normality is assumed; asset returns typically need fatter-tailed innovations.
Key references
Engle, R. (1982) — Autoregressive Conditional Heteroscedasticity… — Econometrica
Bollerslev, T. (1986) — Generalized Autoregressive Conditional Heteroskedasticity — Journal of Econometrics
Nelson, D. (1991) — Conditional Heteroskedasticity in Asset Returns (EGARCH) — Econometrica
Provenance: verified/generated from the paper's full text.
