Question

The data below is for 20 periods of sales in $millions.

t Sales(Y)
1 30
2 32
3 38
4 38
5 34
6 40
7 46
8 42
9 42
10 44
11 42
12 50
13 58
14 54
15 50
16 58
17 62
18 66
19 66
20 62

1. Plot the time seriseries data. Forecast for the 21st period using simple exponential smoothing with
alpha 0.2 and 0.5 and plot the actual 20 period values with forecasted values. Also calculate the
errors and plot the errors for the two forecasts. Compare the two forecasts using MSE and
RMSE. Comment on the error plots (about stationarity). Do you see systematic underestimation?
How do you measure it and visualize it?

Answer 1

To forecast sales using simple exponential smoothing, plot the actual sales data and calculate forecasts with alpha values of 0.2 and 0.5. Compare forecasts with MSE and RMSE, and examine error plots for stationarity and systematic underestimation.

Step-by-step explanation:

To forecast the 21st period sales using simple exponential smoothing with α (alpha) values of 0.2 and 0.5, you first need to plot the time series data for the previous 20 periods.

The forecasts are generated using the exponential smoothing formula: Forecastt+1 = α×Actualt + (1-α)×Forecastt. Starting forecasts can be estimated as the first data point.

You then calculate the error for each forecast by subtracting the forecasted value from the actual sale value and plot these errors.

To compare the two forecasts using Mean Squared Error (MSE) and Root Mean Squared Error (RMSE), you would calculate MSE by averaging the squares of the errors and RMSE by taking the square root of MSE.

Looking at the error plots can tell you about the stationarity of the time series data and whether there's a systematic underestimation, which can be visualized by plotting the errors and assessing if they're predominantly positive or negative.

For systematic underestimation, you'd measure it by observing if the residual errors are consistently negative, indicating that the forecast is generally less than the actual sales.

Underestimation or overestimation biases can also be seen through mean error or bias calculations.