1. Assignment week 38 Exponential smoothing of monthly observations of the General Index of the Stockholm Stock Exchange. A. Graphical illustration of data First, construct a graph of the original series of monthly values. Stock_Exchange.txt

2. Then construct a graph of the percentage change from month to month. Which smoothing techniques (single, double, Holt-Winters) can be used on the original series, which can be used on the series of percentage change. Original series: Double (Holt’s) (or Holt-Winters’ (Winters’) method) Series of percentage change:Single (or Winters’ without trend)

3. B. Exponential smoothing with predefined smoothing parameters Perform single exponential smoothing on the time series of percentage change (of the General Indices). Set the smoothing parameter, , first to 0.9 and then to 0.1. Variable Change is not in the list, due to the initial missing value  Copy the non- missing values to a new column.

5. Then study the graphs produced and try to understand how the choice of the smoothing parameter affects the forecasted values.  = 0.1 gives very damped predicted values (red curve) wile  = 0.9 gives predicted values highly responding to the recent changes in original series.

6. C. Exponential smoothing with automatic parameter setting Let the program choose an optimal value of the smoothing parameter and calculate forecasts for a two-year period (24 months) after the last observed time-point. Construct a graph for the errors in the one-step-ahead forecasts (residuals) in the whole time series and try to judge upon whether the forecasting methods uses earlier observations in the series in an efficient way.

7. Are the residuals serially correlated – Make a visual judgement. Are the earlier observations used in an efficient way?

8. Use also the autocorrelation function on the residual. (MINITAB-Time Series- Autocorrelation).

9. What do you see in the plot you get? First spike is significantly different from zero, so is also some spikes for larger lags.  Residuals seem to be serially correlated.

10. Exponential smoothing of time series with seasonal variation A. Forecasting the employment in USA Perform an exponential smoothing of the time series of monthly employments figures in USA and calculate forecasts for a two-year period (24 month) after the last observed time-point. Labourforce.txt Time series possesses trend and seasonal variation  Use Winters’ method Seasonal variation do not seem to change with level  Use additive case

12. Then use a suitable model for time series decomposition to make forecasts for the same period (additive or multiplicative).

14. Print out graphs for observed and forecasted values and compare how the seasonal effects are described in each method of forecasting. Which method do you prefer in this case? Observed (and forecasts):

15. Forecasts only: From Winters’ methodFrom Decomposition Make a time series plot with both series of forecasts in the same plot

17. B. Forecasting of monthly mean temperature temperature.txt (title “Stockholm” removed)

18. Use exponential smoothing to make forecasts of monthly mean temperatures in Stockholm. Try single, double (Holt’s method) and Winters’ method. Study the residuals (the errors in one-step-ahead forecasts) and the forecasts for 24 months after the last observed time-point. Are the one-month-ahead and one-year- ahead forecasts realistic? Single exponential smoothing:

19. Double exponential smoothing (Holt’s method): Neither single, nor double exponential smoothing seems to work. Surprising?

20. Winters’ method: Note that we do not have any particularly pronounced trend in data and shifts in level are (if existing) very modest.  Try low values of smoothing parameters for level and trend

21. Residuals become positively correlated Forecasts much better here

22. Compare with an analysis with default values on smoothing parameters: Residuals are much better. Forecasts seem to contain an “artificially” induced trend. We have to keep on trying. Is there a better way for making forecasts than applying exponential smoothing on the original series?