Statistical Forecasting and Time Series Analysis: Assignment Solutions

This assignment solution addresses statistical forecasting techniques using moving averages, exponential smoothing, Holt's method, and Winters method. The solution analyzes a dataset of paper towel sales, comparing the accuracy of 3-span and 10-span moving averages. It also explores exponential smoothing with varying alpha values (0.1, 0.3, 0.5, and 0.7) and compares them to Holt's method, assessing forecast accuracy through variance analysis. The Winters method is then applied to de-seasonalize the data, demonstrating its ability to minimize variations compared to other methods. The assignment concludes by recommending the use of the Winters method when seasonality is observed in the data for more accurate predictive analysis. The solution includes calculations, graphs, and tables to support the analysis and findings.

Running head: STATISTICAL FORECASTING
1
Statistical Forecasting
Student’s Name
Institution Affiliation

STATISTICAL FORECASTING 2
Statistical Forecasting
Question P12-19
a) The actual sales in week 10 was 540 units. However, the projected sales using a span of 3
was 533.33 approximated to 533 units. As a result, there is difference of 7 units between
the actual sales and the predicted sales.
Week Original 3span
MA
10Span
MA
1 603
2 461
3 544 536.00
4 385 463.33
5 579 502.67
6 617 527.00
7 464 553.33
8 529 536.67
9 531 508.00
10 540 533.33 525.30
b) The predicted sales for week 10 while using MA of 3 span was 525. 30 approximated to
525-unit sales.
c) A moving average of a span of 3 is a better predictor of papers sold in week 10. The
predicted value was 533 while a 10 span MA predicts the sales to be 525. The actual sales
were 540 units. As a result, it is apparent that the difference between the actual sale and
the predicted sales is higher (15 units) when sing a 10 span MA compared to using 3span
MA (7 units). Therefore, a 3span moving average is better in making sales prediction
compared to a 10 Span MA. The observation is in line with the Lee, Lee & Lee, (2000)
that a moving average with less number of spans is better in making prediction of
observed values.

STATISTICAL FORECASTING 3
Question P13-51
a) The moving averages graph was a shown below
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
0
500
1000
1500
2000
2500
3000
3500
A line graph of sales MA against time
Sales2 3MA 6MA 12MA
Time in days
Sales
It was noted that a 3MA produced a predicted value of sales that ware closer to the actual
sales compared to 6MA and 12MA. However, 12MA produced a straight line showing
that the effect of seasonality and noise in the market are neutralized when a moving
average with higher points in used in predicting sales.
b) The exponential smoothing formula is a follow:
Ft+1 = Ft +α (At-Ft)
Where Ft+1 is the predicted sales for the current period, Ft is the predicted sales for the
prior period, At is the observed sales for the previous period and α is exponential
smoothing constant.
The exponential smoothing chart at alpha =0.1,0.3,0.5 and 0.7 was as follows

STATISTICAL FORECASTING 4
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
650
1150
1650
2150
2650
3150
Exponenial smoothing chart
Sales2 alpha=0.1 alpha=0.3 alpha=0.5 alpha=0.7
Time (months)
Sales
The results show that when the value of α =0.1, the predicted sales are closer to the actual
sales. When the value of alpha was changed to 0.3, 0.5 and 0.7, the variance between the actual
sales and the predicted sales increased as the value of alpha increases which was observed by
Hyndman, (2008) as well when making prediction using exponential smoothing approach. This
can be seen by comparing sales for various months as shown in the tables below.
Exponential Smoothing table
α=0.1 α=0.3 α=0.5 α=0.7
Date
Actua
l
Sales
Predicte
d sales
Variance
compare
d with
actual
Predicte
d sales
Variance
compared
with
actual
Predicte
d sales
Variance
compare
d with
actual
Predicte
d sales
Variance
compared
with actual
1-Feb 1605 1,6
18 (13)
1,6
39 (34)
1,
714
(
109)
1,
876 (271)
1-Dec 1,894 1,922 (28) 2,101 (207)
2,
303 (409) 2,484 (590)
1-Aug 2,861 2,642 219 2,640 221
2,
552 309 2,341 520
As the value of alpha increases, the predicted value deviates from the observed value with a
larger value. For instance, in August -09, the predicted value is 2,641.633 when α=0.1 and
2,340.755

STATISTICAL FORECASTING 5
When α=0.7. Comparing these values with the observed value of 2,861, 2,641.633 is a better
predictor of the actual value compared to 2,340.755 as the variance of 219.367 at α=0.1 is lower
than 520.245 at α=0.7.
Using Holt’s Method
In this case, the values of alphas used were 0.1,0.3.0.5 and 0.7. In order to get an informative
trend that would enhance comparison of the four variables, a beta value of 0.2 was applied to
generate trends for the predicted variables.
A graph of the plotted values was as shown:
Jan-06
Apr-06
Jul-06
Oct-06
Jan-07
Apr-07
Jul-07
Oct-07
Jan-08
Apr-08
Jul-08
Oct-08
Jan-09
Apr-09
Jul-09
Oct-09
Jan-10
Apr-10
Jul-10
Oct-10
600
1100
1600
2100
2600
3100
3600
4100
A graph of sales against months
Sales2 forecast at apha=0.1 forecast at apha=0.3
forecast at apha=0.5 forecast at apha=0.7
months
Sales
Comparing the results obtained from exponential smoothing the predicted values exhibits a much
smoother trends compared to the exponential smoothing curves. A table showing the predicted
sales for similar observed sales in exponential smoothing were as shown on the table in the next
page. Though some variances between the observed and the actual values increased when using

STATISTICAL FORECASTING 6
Holt’s method compared to exponential smoothing, majority of the predicted values were better
in holt’s method compared to exponential smoothing.
Holt’s method table
α=0.1 α=0.3 α=0.5 α=0.7
Date Actua
l Sales
Predicte
d sales
Variance
compare
d with
actual
Predicte
d sales
Variance
compare
d with
actual
Predicte
d sales
Variance
compare
d with
actual
Predicte
d sales
Variance
compared
with actual
Feb-
10 1,605 2,211 (606) 1,884 (279) 1,568 37 1,496 109
Dec-
10 1,894 2,693 (799) 2,695 (801) 2,320 (426) 1,998 (104)
Aug-
09 2,861 2,168 693 2,504 357 2,784 77 2,842 19
Question b
Using Winters constants
The trend for the data was shown in the graph below.
Jan-06
Apr-06
Jul-06
Oct-06
Jan-07
Apr-07
Jul-07
Oct-07
Jan-08
Apr-08
Jul-08
Oct-08
Jan-09
Apr-09
Jul-09
Oct-09
Jan-10
Apr-10
Jul-10
Oct-10
0
500
1000
1500
2000
2500
3000
3500
Trend of sales per months
Time ( Months)
Sales
The trend shows that there is seasonality in the data. As a result, winters method can be applied
to de-seasonalize the data. A season takes a period of 12 months. This therefore means that the
seasonality factor that will be used is 12.

STATISTICAL FORECASTING 7
De-seasonalized trends
Feb-07
May-07
Aug-07
Nov-07
Feb-08
May-08
Aug-08
Nov-08
Feb-09
May-09
Aug-09
Nov-09
Feb-10
May-10
Aug-10
Nov-10
0
1000
2000
3000
4000
5000
6000
Trend of sales against time
Sales2 Predicted value @ alpha=0.1
Predicted value @ alpha=0.3 Predicted value @ alpha=0.5
Predicted value @ alpha=0.7
Months
sales
The trend shows that using Winter’s method, variation in sales observed in the using
exponential smoothing and Holt’s method are minimized. When the data is de-seasonalized, the
impact of seasonality is eliminated and the projections made are more accurate as the noise in the
market is eliminated. Brown (2004) noted that when projecting performance of a product in the
market, it is important to not only consider the short-term trend but also the long-term trend as it
helps in making an appropriate conclusion regarding the market. The best alpha value in making
the prediction was 0.3 and 0.7. It was noted the predicted sales moved in the same direction with
actual sales and the variations were minimal.
When developing the Walters model, a similar value of beta =0.2 and gamma=0.3 were
applied in coming up with the predictors used to determine the trends. The equations used in are
included in the excel file. Also, the season was noted to repeat after 12 months and as a result, no
prediction was made for observations made in first year. Predictions started from the second year
so that the impact of seasonality in the previous year could be applied in making predictions for
the following year (Albright, Winston, Zappe, & Albright, (2009).

STATISTICAL FORECASTING 8
Conclusion
When making prediction on the expected values, moving average, exponential
smoothing, Holt’s method and Winters method can be used to determine predicted values. The
moving average approach predicts values based on the spans that the analyst deems as
appropriate. However, in the event that trends and seasonality exist in the data, they cannot be
captured in the analysis. Exponential smoothing entails the use of a factor with an aim of
smoothening the trend of predicted values. However, the process does not eliminate the impact
seasonality has on predicted values. Holter’s method endeavors to streamline the trend in the
data and uses betas and alphas. The disadvantage in using this method is that seasonality cannot
be eliminated while making prediction. Winters method on the other hand endeavors to
incorporate the impact of trend and seasonality when making prediction. It utilizes alpha, beta
and gamma to eliminate seasonality while making predictions. As a result, predicted values using
Winters method are more accurate compared to the values predicted using moving average,
exponential smoothing or Holter’s method. It is therefore recommended that where seasonality is
observed in the data, Winter’s method should be use in making predictive analysis.

STATISTICAL FORECASTING 9
References
Albright, S. C., Winston, W. L., Zappe, C. J., & Albright, S. C. (2009). Data analysis and
decision making. Toronto, Ont: Nelson Education Ltd.
Brown, R. G. (2004). Smoothing, forecasting and prediction of discrete time series. Mineola,
NY: Dover Publ.
Hyndman, R. J. (2008). Forecasting with exponential smoothing: The state space approach.
Berlin: Springer.
Lee, C. F., Lee, A. C., & Lee, J. C. (2000). Statistics for business and financial economics.
Singapore: World Scientific Publishing.