The Holt-Winters method has been widely used to forecast a seasonal time series in application fields as a nonparametric forecasting technique. In this paper, we investigate the asymptotic forecast errors of the Holt-Winters method. For that purpose, we show that the nonlinear least squares estimates of the smoothing parameters included in the smoothing algorithm hold strong convergence properties under suitable conditions. Then we show the mean squared errors and the limiting distributions of the forecast errors for some stochastic processes. Finally, numerical studies are performed to evaluate the forecasting performance of the Holt-Winters method.