Unit 5: Time Series-III (Estimation of Seasonal Component)

Table of Contents

5.1 Introduction to Seasonal Variation

This unit focuses on isolating the Seasonal Component (S). These are regular fluctuations that complete a full cycle within one year (e.g., quarterly, monthly).

The goal is to create a Seasonal Index for each period (e.g., each quarter). This index shows how much that period is, on average, above or below the general trend.

For quarterly data, the sum of the four seasonal indices should be 400% (or 4). For monthly data, the sum should be 1200% (or 12).

5.2 Method of Simple Averages

This is the easiest method, but it's only suitable if there is no significant trend (T) in the data.

  1. Step 1: Arrange the data by year and quarter (or month).
  2. Step 2: Calculate the total for each quarter (e.g., add all Q1 values, all Q2 values, etc.).
  3. Step 3: Calculate the average for each quarter. (e.g., Q1 Total / Number of years).
  4. Step 4: Calculate the "Grand Average" (the average of all the quarterly averages).
  5. Step 5: Calculate the Seasonal Index (SI) for each quarter:
    SI (Quarter) = (Quarterly Average / Grand Average) * 100
This method is not accurate if a trend exists, because the trend will inflate the averages of later quarters/months.

5.3 Ratio to Trend Method

This is a more accurate method that first removes the trend (T) and then finds the seasonal component from what's left. It assumes a Multiplicative Model (Y = T × S × C × I).

  1. Step 1: Find the Trend (T). Use the Method of Least Squares from Unit 4 to get the trend equation (Y-hat = a + bX). Calculate the trend value (T) for every single quarter/month.
  2. Step 2: De-trend the data. For each period, express the original value (Y) as a percentage of the trend value (T).
    (Y / T) * 100 = (S × C × I)

    This step removes the trend, leaving the other components.

  3. Step 3: Average the percentages. Arrange these percentages by quarter/month (just like in the Simple Averages method). Find the average percentage for Q1, Q2, Q3, and Q4. (We assume this averages out the C and I components, leaving only S).
  4. Step 4: Adjust the indices. The four averages from Step 3 are our "raw" seasonal indices. Their sum may not be exactly 400. We must adjust them.
    Adjustment Factor = 400 / (Sum of raw indices)
    Final Seasonal Index = Raw Index * Adjustment Factor

5.4 Ratio to Moving Average Method

This is the most common and reliable method. It uses a moving average to estimate the trend, which is more flexible than a rigid straight line.

  1. Step 1: Estimate Trend (T). Calculate the Centered Moving Average (CMA) for each period (as seen in Unit 4). This CMA is our estimate for the T × C components.
  2. Step 2: Isolate (S × I). For each period, divide the original value (Y) by the CMA.
    (Y / CMA) * 100 = (T × S × C × I) / (T × C) = S × I

    This gives us the Seasonal and Irregular components, expressed as a percentage.

  3. Step 3: Average to find S. Arrange these (S × I) percentages by quarter/month. Calculate the average (or median) for Q1, Q2, Q3, and Q4. This averaging step is assumed to cancel out the random "I" component, leaving just the "S" component.
  4. Step 4: Adjust the indices. Same as the Ratio to Trend method. Sum the four raw indices and adjust them to total 400.
    Adjustment Factor = 400 / (Sum of raw indices)
    Final Seasonal Index = Raw Index * Adjustment Factor
Ratio to Moving Average is generally preferred over Ratio to Trend because the moving average (CMA) is a much better, more flexible estimate of the trend+cycle than a simple straight line (Least Squares).

This method calculates seasonality without first finding the trend. It is more complex and less common.

  1. Step 1: Calculate Link Relatives. For each quarter/month, express its value as a percentage of the previous period's value.
    Link Relative = (Current Period's Value / Previous Period's Value) * 100
  2. Step 2: Average the Link Relatives. Calculate the average Link Relative for each quarter/month (e.g., average of all Q1 Link Relatives, Q2 Link Relatives...).
  3. Step 3: Chain the Averages. "Chain" the averages back to a base (Q1=100) to get Chain Relatives.
    • Chain Relative (Q1) = 100 (by definition)
    • Chain Relative (Q2) = (Avg. Link Relative Q2 * Chain Relative Q1) / 100
    • Chain Relative (Q3) = (Avg. Link Relative Q3 * Chain Relative Q2) / 100
    • Chain Relative (Q4) = (Avg. Link Relative Q4 * Chain Relative Q3) / 100
  4. Step 4: Correct for Trend. The new "Chain Relative for Q1" (calculated from Q4) will not be 100 due to the presence of a trend. The difference is the trend "error" that has accumulated. This error is then distributed among the chain relatives.
  5. Step 5: Adjust to 100. The final corrected indices are adjusted (like in other methods) to have an average of 100.

This method is computationally intensive and is less intuitive than the Ratio to Moving Average method.