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Analysis of Time Series | MGT 202 Business Statistics

Analysis of Time Series | MGT 202 Business Statistics

TU BBS MGT 202 Unit 7: time series components, trend (semi-average, moving average, least squares), and seasonal indices (simple average, ratio to MA for quarters).

Oct 28, 2026
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This unit covers the meaning and components of a time series, methods of measuring secular trend (with the least squares straight-line trend worked in detail), and seasonal indices by the simple average and ratio-to-moving-average methods, followed by board exam questions listed by exam year.

Time Series and Its Components

A series formed from a set of statistical data arranged in accordance with their time of occurrence is said to be a time series. A time series shows the relation between two variables, one being time.

Components of Time Series (Factors Affecting Time Series)

ComponentExample
Secular trendSize of population, volume of production
Seasonal variationCold drinks and ice-cream in high demand in summer
Cyclical fluctuationRecession, depression
Irregular fluctuationFluctuation due to unusual occurrences such as war, strike, flood, earthquake, etc.

Measurement of Secular Trend

Methods of Measuring Trend

  1. Graphical or free-hand curve method
  2. Semi-average method
  3. Moving average method
  4. Least squares method

Least Squares Method

The equation of the straight-line trend is \(Y_c = a + bx\). To find \(a\) and \(b\), solve the two normal equations:

$$\begin{aligned} \sum Y &= na + b\sum x \quad \text{(i)} \\ \sum xY &= a\sum x + b\sum x^{2} \quad \text{(ii)} \end{aligned}$$

Since the time origin is taken at the middle so that \(\sum x = 0\):

$$\begin{aligned} \sum Y &= na \;\Rightarrow\; a = \frac{\sum Y}{n} \\ \sum xY &= b\sum x^{2} \;\Rightarrow\; b = \frac{\sum xY}{\sum x^{2}} \end{aligned}$$
  • \(Y\) = actual value of \(Y\); \(Y_c\) = computed value of \(Y\) (trend value).
  • \(a\) = constant computed when \(\sum x = 0\).
  • \(b\) = constant giving the change in \(Y\) for a one-unit change in \(x\); if \(b > 0\) the trend is rising.
  • \(x\) = time (coded deviation from the middle period).
  • For an even number of years, code \(x = 2(t - \text{middle value})\) so that \(x\) takes values …, −3, −1, 1, 3, …

Example 1: Trend, short-term fluctuations, monthly increase and forecast (odd years)

Profit (Rs. '000) of a firm, with \(x = t - 2009\):

Year (t)Profit (Y)xx²xYTrend (Yc)Y/Yc × 100Y − Yc
200660−39−18061.0898.23−1.08
200772−24−14466.15108.845.85
200875−11−7571.22105.313.78
20096500076.2985.20−11.29
201080118081.3698.33−1.36
2011852417086.4398.35−1.43
2012973929191.50106.015.50
n = 7∑Y = 534∑x = 0∑x² = 28∑xY = 142
$$\begin{aligned} a &= \frac{534}{7} = 76.29, \qquad b = \frac{142}{28} = 5.07 \\ Y_c &= 76.29 + 5.07x \\ \text{Monthly increase} &= \frac{b}{12} = \frac{5.07}{12} = 0.4225\ (\text{'000}) = \text{Rs. } 422.50 \\ \text{Profit for 2015: } x &= 2015 - 2009 = 6,\quad Y_c = 76.29 + 5.07 \times 6 = 106.71\ (\text{'000}) = \text{Rs. } 106{,}710 \end{aligned}$$

The last two columns give the short-term fluctuations under the multiplicative model (\(Y/Y_c \times 100\)) and the additive model (\(Y - Y_c\)).

Example 2: Sales trend and forecast

Year2006200720082009201020112012
Sales (Y)148207246329378476517

With \(x = t - 2009\): \(\sum Y = 2301\), \(\sum x^2 = 28\), \(\sum xY = 1777\).

$$\begin{aligned} a &= \frac{2301}{7} = 328.714, \qquad b = \frac{1777}{28} = 63.46 \\ Y_c &= 328.714 + 63.46x \\ \text{Sales for 2014: } x &= 5,\quad Y_c = 328.714 + 63.46 \times 5 = 646.014\ \text{(million)} \end{aligned}$$

Since \(b = 63.46 > 0\), it is a rising trend.

Example 3: Monthly data

MonthProfit (Y)x = t − 5x²xYTrend (Yc)
Jan60−416−24060.02
Feb72−39−21665.57
Mar75−24−15071.12
Apr65−11−6576.67
May8000082.22
Jun85118587.77
Jul952419093.32
Aug1033930998.87
Sep105416420104.42
n = 9∑Y = 740∑x = 0∑x² = 60∑xY = 333
$$a = \frac{740}{9} = 82.22, \qquad b = \frac{333}{60} = 5.55, \qquad Y_c = 82.22 + 5.55x$$

Example 4: Even number of years

Fit a straight-line trend and estimate sales (Rs. '000) for 2004 and 2005.

Year199819992000200120022003
Sales (Y)405254525053
x = 2(t − 2000.5)−5−3−1135

\(\sum Y = 301\), \(\sum x^2 = 70\), \(\sum xY = 57\).

$$\begin{aligned} a &= \frac{301}{6} = 50.167, \qquad b = \frac{57}{70} = 0.8142 \\ Y_c &= 50.167 + 0.8142x \\ \text{2004: } x &= 2(2004 - 2000.5) = 7,\quad Y_c = 50.167 + 0.8142 \times 7 = 55.866\ (\text{'000}) = \text{Rs. } 55{,}866 \\ \text{2005: } x &= 2(2005 - 2000.5) = 9,\quad Y_c = 50.167 + 0.8142 \times 9 = 57.495\ (\text{'000}) = \text{Rs. } 57{,}495 \end{aligned}$$

Seasonal Index

Seasonal indices are computed by the simple average method or, under the multiplicative model, by the ratio-to-moving-average method.

Simple Average Method

$$\text{Seasonal Index (SI)} = \frac{\text{Average of the season}}{\text{Average of averages}} \times 100, \qquad \text{Average} = \frac{\text{Sum of observations}}{\text{Number of observations}}$$

Example 5: Seasonal data

YearSpringSummerFallWinter
201281075
201391076
2014101176
2015101287
2016111398
Total48563832
Average9.611.27.66.4
SI110.34128.7487.3673.56

Average of averages = \((9.6 + 11.2 + 7.6 + 6.4)/4 = 8.7\); e.g. SI (Spring) = \(9.6/8.7 \times 100 = 110.34\). Summer is seasonally high.

Example 6: Quarterly data with missing quarters

Quarter2008200920102011TotalAverageSI
I—3.53.54.011.03.66793.18
II3.94.13.94.616.54.125104.82
III3.43.73.73.814.63.65092.74
IV3.64.84.5—12.94.300109.26

Average of averages = \((3.667 + 4.125 + 3.65 + 4.3)/4 = 3.9355\). Quarter IV is seasonally high.

Ratio-to-Moving-Average Method (Quarterly Data)

  1. Compute 4-quarterly moving totals and moving averages.
  2. Centre them (average of two successive moving averages) to get the trend \(Y_c\).
  3. Trend-eliminated value = \(Y / Y_c \times 100\).
  4. Average the trend-eliminated values for each quarter; if their sum ≠ 400, multiply by the adjustment factor \(k = 400 / \text{sum of unadjusted SI}\).

Example 7

YearQuarterYCentred moving average (Yc)Y/Yc × 100
2008Q156——
Q270——
Q38066.625120.075
Q46067.12589.385
2009Q15768.25083.516
Q27369.250105.415
Q38669.875123.077
Q46270.37588.099
2010Q16071.00084.507
Q27471.875102.957
Q39073.000123.288
Q46574.87586.811
2011Q16675.62587.273
Q28375.500109.934
Q387——
Q467——

4-quarterly moving totals: 266, 267, 270, 276, 278, 281, 282, 286, 289, 295, 304, 301, 303 (moving averages = totals ÷ 4).

Q1Q2Q3Q4Total
Total of ratios255.296318.306366.440264.295
Average (unadjusted SI)85.099106.102122.14788.098401.446
Adjusted SI84.792105.720121.70787.781400

Since the sum of unadjusted SI = 401.446 ≠ 400, adjustment factor \(k = 400/401.446 = 0.996398\). Quarter III is seasonally high.

Important Questions

2078 Exam

  1. What are the components of a time series? [2 marks]

2079 Exam

  1. The year of origin of the following trend line equation of sales (in million rupees) is 2015: \(Y = 50 + 2.5X\). Estimate the sales for the year 2022. [2 marks]

  2. The following table reveals the sales of a business house from the year 2016 to 2020: [10 marks]

    Year20162017201820192020
    Sales (in Rs. '000)50516075100

    Fit a trend line equation to the given data. Calculate the trend values and estimate the sales for the year 2024.

2080 Exam

  1. The year of origin of the following trend line equation of production (in tonnes) is 2017: \(y = 40 + 1.5x\). Estimate the production for the year 2033. [2 marks]

  2. Below is given the annual production of sugar (in thousand tons) of a factory: [15 marks]

    Year2011201220132014201520162017
    Production77889485919890
    1. Fit a straight line trend by the method of least squares.
    2. Obtain the trend values.
    3. Plot the given figures on a graph and show the trend line.
    4. What is the monthly increase in production?
    5. Estimate the production of sugar for the year 2020.

2081 Exam

  1. The following time series data show the profit (million Rs.) of XYZ company from fiscal year 2015 to 2023: [15 marks]

    Year201520162017201820192020202120222023
    Profit (million Rs.)151820222523273230
    1. Fit a straight line trend to these data.
    2. Calculate the trend values and short-term fluctuations.
    3. Plot the actual data as well as the trend values on graph paper.
    4. Estimate the profit for 2024.
    5. What is the monthly increment of the profit?

2082 Exam

  1. The following table shows the profit (Rs. millions) of ABC company: [10 marks]

    Year201520162017201820192020202120222023
    Profit162025283027293225

    Fit a straight line trend by the least squares method. Calculate the trend values. Estimate the profit for the year 2024.

2083 Exam

  1. The trend line equation of sales (in million rupees) is \(Y = 50 + 2.5X\). Estimate the sales for the year 2022 when the year of origin is assumed to be 2015. [2 marks]

  2. Fit a straight line trend by the method of least squares and estimate the sales for 2004 and 2005. [10 marks]

    Year199819992000200120022003
    Sales (in Rs. '000)405254525053

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