I'm using the SPSS Statistics TSMODEL procedure (Analyze>Forecasting>Create Model in the menus) to fit exponential smoothing models to time series data and produce forecasts. I'm attempting to understand or manually recreate the production of the fitted and forecasted values. I'm looking at the section Initialization of Exponential Smoothing where it discusses initialization of the backcasting process used to produce initial state values for producing forecasts and it seems that the formulas don't allow me to accurately reproduce the results from the procedure. Are there errors in these formulas?
Resolving the problem
Yes, there are some errors in this section of the TSMODEL Algorithms. Specifically, in each of the models with trend components, the slope coefficients produced by the linear regressions of Y on time mentioned in the formulas must be multiplied by -1 in backcasting. Also, the formula for the vector of seasonal states for the Winters multiplicative model is incorrect. It should state that the value of S for each period is the sum of the intercept and slope coefficients for that period's regression divided by the sum of the means of the intercept and slope coefficients over all periods.