Unit 4: Time Series-II (Estimation of Trend)

4.1 Introduction to Trend Estimation

The first step in classical decomposition is to isolate the Secular Trend (T). This involves "smoothing" the data to remove the short-term fluctuations (S, C, and I). This unit covers four methods to do this.

4.2 Free Hand Curve Method

This is the simplest and most subjective method.

1. Plot the time series data on a graph.
2. Draw a smooth, free-hand curve or line through the data that you feel best represents the long-term trend, ignoring the short-term "wiggles".

• Pros:
  • Very simple and quick.
  • Can be used for any type of trend (linear or non-linear).
• Cons:
  • Highly subjective. Two different people will draw two different trend lines.
  • Not mathematical; cannot be used for reliable forecasting.

4.3 Method of Semi-Averages

This is a simple mathematical method that is more objective than the free-hand curve.

1. Step 1: Divide the time series data into two equal parts.
   Odd Number of Years: If there is an odd number of data points (e.g., 7 years), omit the middle value and split the remaining data in two (e.g., first 3 years, last 3 years).
2. Step 2: Calculate the arithmetic mean (average) for each part.
3. Step 3: Plot these two averages on the graph at the mid-point of their respective time periods.
4. Step 4: Draw a straight line connecting these two points. This line is the trend line.

• Pros:
  • More objective than the free-hand method.
  • Easy to calculate.
• Cons:
  • Assumes the trend is linear. It cannot be used for non-linear trends.
  • It is still somewhat crude, as it's only based on two points.

4.4 Method of Moving Averages

This is a very common and effective method for smoothing a time series. It works by replacing each data point with the average of itself and its neighbors.

The "period" of the moving average (e.g., 3-year, 5-year, 4-quarter) is chosen to match the length of the seasonal/cyclical variation, as the averaging "cancels out" these fluctuations.

3-Year Moving Average (Odd Period)

For each year (starting from the 2nd), the trend value is the average of that year, the year before, and the year after.