I'm working on a project to compare different approaches to time series modeling. In the model selection process, we perform residual analysis for the fitted models. For regression, we need to check the normality assumption in the residuals by performing Shapiro-Walk's test, or plotting QQ-plot, etc. Do we also need to check whether the residuals are normally distributed after fitting a model using HW algorithm?
There is no normality assumption in fitting an exponential smoothing model. Even if maximum likelihood estimation is used with a Gaussian likelihood, the estimates will still be good under almost all residual distributions.
There is also no normality assumption when producing point forecasts from an exponential smoothing model.
However, there is often a normality assumption when producing prediction intervals from an exponential smoothing model. But this assumption is easily removed by using bootstrap prediction intervals.
For example, if you use R, then the following code produces forecasts from a multiplicative Holt-Winters model with no normality assumption for the prediction intervals:
library(forecast) fcast <- hw(x, seasonal="multiplicative", bootstrap=TRUE, simulate=TRUE)
By the way, you don't need to worry about normality of residuals in regression either, except if you produce prediction intervals. As with most statistical models, the estimates are more efficient if the residuals are normally distributed, but they are still consistent and unbiased if the residuals are non-normal.