This paper compares several statistical models for monthly stock return volatility using U.S. data from 1834 to 1925. The focus is on the importance of nonlinearities in stock return behavior that are not captured by conventional ARCH or GARCH models, and the nonstationarity of stock volatility over the entire period. The authors use a variety of methods, including a two-step estimator, GARCH, exponential GARCH (EGARCH), Hamilton's two-state regime-switching model, nonparametric kernel estimation, and a nonparametric flexible Fourier form estimator. They find that the nonparametric estimates, particularly the kernel and Fourier estimates, show different behavior from the parametric estimates during periods of stock price declines. The paper also analyzes the behavior of these models during significant episodes of stock volatility, such as banking crises and other major events. Finally, the authors test for covariance stationarity in the stock return series, finding evidence of a lack of homogeneity in the variance over the entire sample period.This paper compares several statistical models for monthly stock return volatility using U.S. data from 1834 to 1925. The focus is on the importance of nonlinearities in stock return behavior that are not captured by conventional ARCH or GARCH models, and the nonstationarity of stock volatility over the entire period. The authors use a variety of methods, including a two-step estimator, GARCH, exponential GARCH (EGARCH), Hamilton's two-state regime-switching model, nonparametric kernel estimation, and a nonparametric flexible Fourier form estimator. They find that the nonparametric estimates, particularly the kernel and Fourier estimates, show different behavior from the parametric estimates during periods of stock price declines. The paper also analyzes the behavior of these models during significant episodes of stock volatility, such as banking crises and other major events. Finally, the authors test for covariance stationarity in the stock return series, finding evidence of a lack of homogeneity in the variance over the entire sample period.