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Time series

Unit: Quantitative Analysis

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Practice CPA Quantitative Analysis Time series questions with detailed answers and explanations.

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December 2025

 1 Questions
Question 7c
​ ​ ​ ​​A retail company recorded the monthly sales in units of one of its best-selling products over the past 6 months of the year 2025:

MonthSales (units)
June420
July440
August415
September430
October425
November445

Required: 
(i) Forecast the sales for the month of December 2025 using the exponential smoothing method with a smoothing constant α = 0.2.

(ii) Calculate three months moving average forecast for December 2025. 

(iii) Forecast the sales for December 2025 using the 3 months weighted moving averages with weights of 0.5, 0.3 and 0.2 starting with the most recent time period and backwards.  


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August 2025

 2 Questions
Question 1c
​ ​ ​ ​​The quarterly sales of Meshek Traders during the years 2022, 2023 and 2024 were as tabulated below:

Year
Quarterly Sales in Sh.“million” 
Q1
Q2
Q3
Q4
2022
45
78
56
80
2023
44
67
43
86
2024
53
89
81
102

Required: 

(i) The three-quarter moving average. 

(ii) The seasonal index for each quarter assuming the multiplicative model. 

(iii) Derive the deseasonalised sales using the multiplicative model.


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Question 6a
​​Explain THREE uses of time series analysis in business decision making.


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April 2025

 1 Questions
Question 3
​ ​ ​ ​ ​​A company tracked its quarterly sales (in millions) over the past 3 years as shown in the table below:

Quarter
Year
​\(Q_1\)​
​\(Q_2\)​
​\(Q_3\)​
​\(Q_4\)​
1
50
65
80
70
2
55
70
85
75
3
60
75
90
80

Required: 

(a) Obtain a 4-quarter centred moving average for the data. 

(b) Determine the typical seasonal indices for each quarter using the multiplicative model.

(c) Determine the deseasonalised sales for each quarter.

(d) Fit a trend equation to the deseasonalised data using the ordinary least squares (OLS) method.

(e) Forecast the seasonally adjusted sales for each quarter of the coming year using the trend equation in (d) above.


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December 2024

 1 Questions
Question 3a
​ ​​Explain THREE methods of estimating the trend line in time series analysis.


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August 2024

 1 Questions
Question 7a
​ ​ ​ ​ ​ ​​The data below represents the sales made by Yatta Traders for a period of three years:

Sales (Sh.“million”)
Quarters
Year
1
2
3
4
2021
4.4
10.0
15.8
6.4
2022
5.8
10.4
16.4
7.6
2023
6.4
11.6
18.2
8.2

Required: 

(i) The 4-centered moving average trend values. 

(ii) The seasonally adjusted additive indices.


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April 2024

 2 Questions
Question 4a
​ ​​Differentiate between the “multiplicative model” and “additive model” as used in time series


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Question 5a
​ ​ ​ ​​A car hire company has provided the data below showing the number of cars hired out for the last 10 months ending 31 March 2024:

Month
Number of cars
June 2023
110
July 2023
115
August 2023
109
September 2023 
108
October 2023 
106
November 2023
111
December 2023 
107
January 2024
112
February 2024
114
March 2024
110

Required: 

(i) The forecast number of cars for the month of April 2024 using exponential smoothing method with a smoothing constant α = 0.3. 

(ii) The three months moving average forecasts for the month of April 2024. 

(iii) Using suitable computation, advise on the best forecast method.


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December 2023

 1 Questions
Question 2b
​ ​ ​ ​​The following data shows the quarterly sales of cars for Excellent Auto Enterprises for a period of three years:

Sales (Sh.“million”)
Year
Quarter 1
Quarter 2
Quarter 3
Quarter 4
2020
25
20
18
30
2021
33
28
26
38
2022
41
37
34
46

Required: 

(i) The 4-quarter centred moving average trend values. 

(ii) The seasonal multiplicative indices. 

(iii) The deseasonalised sales for each quarter.


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August 2023

 1 Questions
Question 6b
​ ​ ​ ​ ​​The data below shows the sales made by Kuza Limited over a period of 6 years:

Year
2017
2018
2019
2020
2021
2022
Sales (in millions of shillings)
80
78
72
68
70
82

Required: 

(i) The sales forecast for the year 2023 using exponential smoothing (use a smoothing constant of 0.2). 

(ii) The sales forecast for the year 2023 using the ordinary least squares method. 

(iii) Using suitable computations, advise Kuza Ltd. on the preferred forecast method.


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April 2023

 2 Questions
Question 6a
​ ​​In the context of time series analysis, distinguish between the following terms:

(i) “Mean absolute deviation” and “mean squared error”.

(ii) “Additive model” and “multiplicative model”


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Question 6b
​ ​ ​ ​ ​ ​ ​ ​​The quarterly profits of Nyota Limited during the years 2019, 2020, and 2021 were as tabulated below:

Profits in Sh.“million”
Year
Quarter 1 
Quarter 2
Quarter 3
Quarter 4
2019
105
138
116
140
2020
104
122
103
146
2021
113
149
141
162

Required: 

(i) The three-quarter moving averages. 

(ii) The seasonal index for each quarter assuming the multiplicative model. 

(iii) Derive the deseasonalised profits using the multiplicative model.


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December 2022

 1 Questions
Question 6b
​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​​The data below relate to the profits of Soko Yetu Groceries (in thousands of shillings) over a period of four years.

Quarter
Year
1
2
3
4
2019
12
 9
11
14
2020
13
10
17
20
2021
15
13
20
22
2022
16
12
21
-

Required: 

(i) Determine the trend equation using the least squares method. 

(ii) Calculate the seasonal index for each quarter using the multiplicative model.


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August 2022

 1 Questions
Question 1b
​​ The data below relates to the sales of Madi and Sons Electrical Company for the six months ending 30 June 2022:

Month
January
February
March
April
May
June
Sales (Sh.“000”) 
80
76
78
82
72
82

Required: 

Using exponential smoothing with a smoothing constant (α) of 0.25, determine the forecast sales for the month of July 2022.


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April 2022

 1 Questions
Question 7a
​ ​ ​ ​ ​​The table below shows the quarterly profits of Kahawa Limited (in millions of shillings) for the years 2019, 2020 and 2021:

Quarterly profits (Sh."million")
Year
Quarter 1
Quarter 2
Quarter 3
Quarter 4
2019
23
32
27
21
2020
27
35
32
24
2021
31
43
40
29

Required: 

(i) The three-quarter moving average of the profits. 

(ii) The quarterly seasonal variations of the profits using the additive model.

(iii)  Forecast the adjusted profits for the year 2022 given that the actual profits (in Sh."million") in the year 2022 are 35, 50, 47 and 33 for Quarter 1, Quarter 2, Quarter 3 and Quarter 4 respectively.


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December 2021

 2 Questions
Question 1b
​ ​ ​ ​ ​ ​ ​ ​​The following data relates to the profits reported by XYZ Ltd. in each of the months in the year 2020:

Month
Profit (Sh."million")
January
40
February
38
March
39
April
41
May
36
June
41
July
34
August
37
September
35
October
37
November
40
December
41

Required: 

(i) Estimate the trend line using the ordinary least squares method. 

(ii) Estimate the profit reported in March of the year 2021.


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Question 1a
​​Explain the following terms as used in time series analysis:

(i) Cyclical variations.

(ii) Random variations.

(iii) Seasonal variations.

(iv) Trend. 


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September 2021

 1 Questions
Question 1b
​ ​ ​​The following data presents the quarterly imports of electrical products by a certain electronics company for the given four-year period:

Imports in Sh."Million"
Year
Quarter 1
Quarter 2
Quarter 3
Quarter 4
2017
2018
2019
2020
43
50
62
73
71
82
89
95
  83
  93
101
120
59
75
81
89

Required: 
(i) Using 4-quarter moving averages, calculate the seasonal variations from the above data. 

(ii) Derive the deseasonalised data for the imports using the additive model.


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May 2021

 1 Questions
Question 2b
​​Enumerate two advantages and two disadvantages of the ordinary least squares method of forecasting.


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November 2020

 1 Questions
Question 4c
​ ​ ​​The data given below shows the profits in shillings million made by an economic sector in your country during the various quarters of the given years.

       Profits in quarters
Year
2016
2017
2018
2019
Q1
83
105
140
168
Q2
260
383
430
503
Q3
215
248
323
340
Q4
393
553
588
755

Required:
(i).  3 quarter moving average of the series.
(ii). The deseasonalised profit of the economic sector using the additive model.


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November 2019

 1 Questions
Question 3a
​​ In the context of time series analysis, describe three differences between "additive" and "multiplicative" models.


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November 2018

 2 Questions
Question 4a
​ ​ ​ ​​The data below represent the number of students enrolled in a certain college over a four year period:

        Number of students enrolled
                        Quarter
Year
1
2
3
4
2015
70
100
80
60
2016
50
40
120
80
2017
60
70
70
40
2018
90
100
130
-

Required:
(a) The adjusted seasonal component for each of the four quarters, using the multiplicative model. 
(b) Estimate the enrollment of students in each of the four quarters of year 2019 using the simple least squares method. 


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Question 4b
​Estimate the enrollment of students in each of the four quarters of year 2019 using the simple least squares method. 


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December 2017

 1 Questions
Question 1b
​​Highlight the four components of a time series.


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November 2016

 1 Questions
Question 3b
​ ​ ​ ​ ​ ​​The following data show quarterly production of oranges by a certain large scale farmer in thousands of kilogrammes:

Year
Quarter 1
Quarter 2
Quarter 3
Quarter 4
2012
250
200
180
300
2013
330
280
260
380
2014
410
370
340
460
2015
478
-
-
-

Required:
(i).  The adjusted seasonal component for the four quarters using the additive model.
(ii). The deseasonalised production data for each quarter.
(iii). Explain the significance of the deseasonalised data.


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May 2016

 1 Questions
Question 2c
The following data were obtained from the records of Kiwandani Limited for the year 2015:



Month
January
February
March
April
May
June 		Total
overhead cost (y)
(Sh.)
16,250
15,000
15,000
14,500
15,250
15,750 		Director labour
hours (x)

1,056
736
840
800
880
1,008

Required:
(i) The least squares regression function relating direct labour hours and total overhead cost.
(ii) The coefficient of determination. Comment on your result.


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November 2015

 1 Questions
Question 2c
The data below show the number of cars imported by a certain car dealer over a four-year period:

Year
2011
2012
2013
2014 		Quarter 1
20
21
23
27 		Quarter 2
32
42
39
39 		Quarter 3
62
75
77
92 		Quarter 4
29
31
48
53

Required:
(i) The trend equation, using the least squares method.

(ii) Average seasonal index for each quarter using the multiplicative model.

(iii) Year 2015 seasonally adjusted import forecasts for each quarter.


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Pilot September 2015

 2 Questions
Question 5b
​​The data below represent the sales made by Pengo Traders for a period of three years:

Sales (Sh. "000,000")
Quarter
Year
2012
2013
2014		1
2.2
2.9
3.2		2
5
5.2
5.8		3
7.9
8.2
9.1		4
3.2
3.8
4.1

Required:
(i) The centred moving average trend values.
(ii) The seasonal additive indices.
(iii) The deseasonalised time series.


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Question 5c
​​Highlight the four components of a time series.


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