Minitab is a great tool to conduct an in depth Time Series analysis  Minitab Help
Time Series Analysis
1. Print your modified data set on the answer sheet with your personal dataset changes highlighted in bold type and explain what you did (giving values for F,L, D and student number just below the dataset)
Solution: The modified dataset is shown below:
Year 
quarter 1 
quarter2 
quarter 3 
quarter 4 
1 
66 
155 
100 
40 
2 
103 
187 
154 
77 
3 
144 
265 
233 
124 
4 
199 
289 
253 
186 
5 
222 
340 
289 
215 
6 
241 
365 
308 
276 
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2. You may now wish to modify the dataset into two columns. The first column showing the year and quarter and the second column showing the sales for that period. You may also wish to generate a third column with the numbers 1 through 24 to denote the index number for each sales value.
Solution: As indicated, we present the data in the required format
Year / Quarter 
Sales 
Index 
1/1 
66 
1 
1/2 
155 
2 
1/3 
100 
3 
1/4 
40 
4 
2/1 
103 
5 
2/2 
187 
6 
2/3 
154 
7 
2/4 
77 
8 
3/1 
144 
9 
3/2 
265 
10 
3/3 
233 
11 
3/4 
124 
12 
4/1 
199 
13 
4/2 
289 
14 
4/3 
253 
15 
4/4 
186 
16 
5/1 
222 
17 
5/2 
340 
18 
5/3 
289 
19 
5/4 
215 
20 
6/1 
241 
21 
6/2 
365 
22 
6/3 
308 
23 
6/4 
276 
24 
Seasonal Indexes
3. Generate seasonal indexes for the four quarters using the method outlined in the text in Chapter 184 and/or in the excel spreadsheet for problem 26 of Chapter 18. This is available on mygateway. The seasonal indexes should be formatted to show 6 decimal places.
Solution: We use Minitab we obtain the following seasonal indices. We go to "Stat" > "Time Series" > "Decomposition", we select the Sales variable, and we select the "Multiplicative" model. The outcome, formatted to 6 decimals, is shown below
Year / Quarter 
Sales 
Index 

1/1 
66 
1 
0.908087 
1/2 
155 
2 
1.347777 
1/3 
100 
3 
1.123660 
1/4 
40 
4 
0.620476 
2/1 
103 
5 
0.908087 
2/2 
187 
6 
1.347777 
2/3 
154 
7 
1.123660 
2/4 
77 
8 
0.620476 
3/1 
144 
9 
0.908087 
3/2 
265 
10 
1.347777 
3/3 
233 
11 
1.123660 
3/4 
124 
12 
0.620476 
4/1 
199 
13 
0.908087 
4/2 
289 
14 
1.347777 
4/3 
253 
15 
1.123660 
4/4 
186 
16 
0.620476 
5/1 
222 
17 
0.908087 
5/2 
340 
18 
1.347777 
5/3 
289 
19 
1.123660 
5/4 
215 
20 
0.620476 
6/1 
241 
21 
0.908087 
6/2 
365 
22 
1.347777 
6/3 
308 
23 
1.123660 
6/4 
276 
24 
0.620476 
4. Adjust the seasonal indexes generated so that the 4 indexes add to a total of 4.000000 (to six decimal places).
Solution: The table above already shows the indexes in such a way that they add up to a total of 4.000000.
5. Generate a scatterplot of your modified dataset and also generate a scatterplot of your seasonally adjusted dataset.
Solution: The scatter plot for the modified dataset is shown below:
The seasonally adjusted dataset is shown below:
Now we show both graphs together for a comparative view:
6. Generate a simple linear regression of the form: (seasonally adjusted sales)= b0 +b1(time) where time is from 1 through 24. Note the regression must use seasonally adjusted sales, not the sales of your modified dataset.
Solution: We now perform a Regression Analysis to analyze the trend line associated to the dataset. We use Minitab to obtain
The model is\[{{Y}_{t}}=74.5036+10.1430t\]
7. Prepare deseasonalized forecasts for year 7 quarters 1,2,3 and 4.
Solution: We evaluate the equation at \(t=25,26,26,28\). We obtain:
\[\begin{array}{cc} & t=25\Rightarrow Y=328.0786 \\ & t=26\Rightarrow Y=338.2216 \\ & t=27\Rightarrow Y=348.3646 \\ & t=28\Rightarrow Y=358.5076 \\ \end{array}\]8. Seasonalize your forecasts from #7 above.
Solution: We multiply the forecast from #7 by the seasonal indexes to get:
Deseasonalized 
Indexes 
Seasonalized  
25 
328.0786 
0.908087 
297.9239116 
26 
338.2216 
1.347777 
455.8472934 
27 
348.3646 
1.123660 
391.4433664 
28 
358.5076 
0.620476 
222.4453616 
Time Series Analysis using Regression
9. Return to the dataset as given in Part A, step 2. Generate 3 dummy 01 indicator variables in order to define the individual seasons. Write the definition for seasons 1,2,3, and 4 in terms of the three dummy 01 indicator variables, S1, S2, S3. (Do not use the seasonalized dataset from Part B).
Solution: We define the seasons 1, 2, 3 and 4 using the dummy variables \({{S}_{1}},{{S}_{2}},{{S}_{3}}\) as shown below
· Season 1: \({{S}_{1}}=1,{{S}_{2}}=0,{{S}_{3}}=0\)
· Season 2: \({{S}_{1}}=0,{{S}_{2}}=1,{{S}_{3}}=0\)
· Season 3: \({{S}_{1}}=0,{{S}_{2}}=0,{{S}_{3}}=1\)
· Season 4: \({{S}_{1}}=0,{{S}_{2}}=0,{{S}_{3}}=0\)
The table with the data is
Sales 
Time ($t$) 
${{S}_{1}}$ 
${{S}_{2}}$ 
${{S}_{3}}$ 
66 
1 
1 
0 
0 
155 
1 
0 
1 
0 
100 
1 
0 
0 
1 
40 
1 
0 
0 
0 
103 
2 
1 
0 
0 
187 
2 
0 
1 
0 
154 
2 
0 
0 
1 
77 
2 
0 
0 
0 
144 
3 
1 
0 
0 
265 
3 
0 
1 
0 
233 
3 
0 
0 
1 
124 
3 
0 
0 
0 
199 
4 
1 
0 
0 
289 
4 
0 
1 
0 
253 
4 
0 
0 
1 
186 
4 
0 
0 
0 
222 
5 
1 
0 
0 
340 
5 
0 
1 
0 
289 
5 
0 
0 
1 
215 
5 
0 
0 
0 
241 
6 
1 
0 
0 
365 
6 
0 
1 
0 
308 
6 
0 
0 
1 
276 
6 
0 
0 
0 
10. Generate a multiple regression of the form sales= b0 +b1(time)+ b2(S1) +b3(S2) +b4(S3). Comment on the suitability of the regression model by analyzing the F statistic, the 4 pvalues for the coefficients b1,b2,b3,b4. Also review the residuals, outliers, residual plots, etc. Interpret the meaning of coefficients b2,b3 and b4.
Solution: The output we get from Minitab is
Overall, the model is significant ( p = 0.000 ).
The pvalues are all 0.000 except for \({{S}_{1}}\)'s pvalue, which is 0.296. This means that the coefficient of the dummy variable \({{S}_{1}}\) is not significantly different from zero. The coefficients \({{\beta }_{2}},{{\beta }_{3}},{{\beta }_{4}}\) represent the average marginal amount of sales of the seasons 1, 2 and 3, with respect to season 4.
The summary for the residual is shown below:
There's a few outliers summarized in the table below.
11. Calculate the Durbin Watson statistic. Is autocorrelation (positive or negative) present?
Solution: The DurbinWatson statistics is 1.64206. Since \({{d}_{L}}=1.04,\text{ }{{d}_{U}}=1.77\), we neither reject nor accept the null hypothesis of noautocorrelation. The residuals visually have a very mild autocorrelation trend.
12. Prepare forecasts for year 7 quarters 1,2,3 and 4.
Solution: We use Minitab to obtain the following predictions:
Exponential Smoothing
13. Use double exponential smoothing (easiest in Minitab). Run with alpha =.2 and a value of .1 for trend. Note that in Minitab under double exponential smoothing the term “level” is used for alpha and “trend” for trend. Generate forecasts for year 7 quarters 1,2,3 and 4.
Solution: The output generated by Minitab is
14. Suppose sales data for year 7 quarters 1,2,3 and 4 are 262, 392, 321, and 305 respectively. Compare your forecasts from 8,12, and 13 by using mean absolute error. Which method is the best? Now compare your forecasts from 8,12, and 13 by using mean squared error. Which method is the best? Finally, compare your forecasts from 8,12, and 13 by using mean absolute percentage error. Which method is the best?
Solution: We'll put all the forecasts in an Excel table:
Season 
Observed 
Time Series 
Multiple Regression 
D. Exponential Smoothing 
7/1 
262 
297.923912 
311.03 
323.73 
7/2 
392 
455.847293 
415.36 
333.2 
7/3 
321 
391.443366 
371.36 
342.671 
7/4 
305 
222.445362 
301.52 
352.142 
The errors are recorded in the following table
Time Series 
Multiple Regression 
D. Exponential Smoothing  
Abs. Mean Error 
63.1923025 
31.5575 
47.33575 
Mean Squared Error 
4286.13512 
1374.467625 
2490.008326 
Mean Abs. % Error 
19.752765 
10.37559531 
15.19213812 
Remember that your answers should be confined to the first two sheets of paper with supporting computer printout attached in back of the two pages. Your answers must be on the first two pages and must be in sufficient detail. An answer of “see appendix page x” will receive no credit.
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