Question

Compute the seasonal indices by 4-year moving averages from the given data of production of paper (in thousand tons):

\[ \begin{array}{lcccccccccc} \text{Year:} & 2001 & 2002 & 2003 & 2004 & 2005 & 2006 & 2007 & 2008 & 2009 & 2010 \\ \text{Index:} & 2450 & 1470 & 2150 & 1800 & 1210 & 1950 & 2300 & 2500 & 2480 & 2680 \\ \end{array} \]

Hint:

In 4-year moving averages, always average pairs to center the values. Use Seasonal Index = (Actual / Average) × 100.

Answer 1

Step 1: Compute 4-year rolling sums by adding four consecutive annual figures. For example, the sum for 2001–2004 is $2450 + 1470 + 2150 + 1800 = 7870$. Subsequent sums are calculated similarly: 2002–2005: $1470 + 2150 + 1800 + 1210 = 7630$; 2003–2006: $2150 + 1800 + 1210 + 1950 = 7110$; 2004–2007: $1800 + 1210 + 1950 + 2300 = 7260$; 2005–2008: $1210 + 1950 + 2300 + 2500 = 7960$; 2006–2009: $1950 + 2300 + 2500 + 2480 = 9230$; 2007–2010: $2300 + 2500 + 2480 + 2680 = 9960$. Step 2: Derive 4-year centered moving averages. This is achieved by averaging two successive 4-year totals and dividing by 4. For instance, the average for the period ending in 2004 is calculated as (7870 + 7630)/8 = 1937.5. This procedure should be applied to all applicable periods to obtain centered values. Step 3: Calculate the seasonal index using the formula: Seasonal Index = (Actual Value / Moving Average) × 100, for the midpoint of the period. For example, for the year 2003, with an actual value of 2150 and an approximate moving average of 1937.5, the Seasonal Index is $(2150 / 1937.5) \times 100 \approx 111$. Repeat this calculation for each year. Step 4: Group the years by quarter or season. Compute the average seasonal index for each group. Subsequently, adjust these average seasonal indices so that their sum equals 400, representing four seasons.