Calculation of Trend values(Yc ) under Least square method and Moving Average method (3 yearly, 4 yearly and 5 yearly moving averages)

Calculation of Trend Values (Yc)

Trend values (Yc) represent the estimated or fitted values of a time series after removing short-term fluctuations. These values are calculated using statistical methods to identify the long-term movement of data. The two most commonly used methods are the Least Squares Method and the Moving Average Method.

(A) Least Squares Method

The Least Squares Method is the most scientific and accurate method of measuring trend. It fits a trend line in such a way that the sum of squared deviations between actual values (Y) and estimated trend values (Yc) is minimum.

Trend Equation

Yc = a + bX

Where:
Yc = Trend value
a = Intercept
b = Slope of the trend line
X = Time variable

Steps for Calculating Trend Values (Yc)

Step 1: Assign Time Values (X)

If the number of years is odd, the middle year is taken as origin (X = 0).
If the number of years is even, origin is taken between the two middle years.

Step 2: Calculate ‘a’ and ‘b’

a = (ΣY) / n

 b = (ΣXY) / ΣX²​

Where n = number of observations

Step 3: Calculate Trend Values (Yc)

Substitute the values of a, b, and X in the trend equation:

Yc = a + (bX)

Merits of Least Squares Method

• Provides exact trend values

• Useful for forecasting

• Widely used in business and economics

(B) Moving Average Method

The Moving Average Method calculates trend values by averaging successive groups of data. It smoothens short-term fluctuations and highlights long-term movement.

1. 3-Yearly Moving Average

This method is used when data shows moderate fluctuations.

Steps for 3-Yearly Moving Average

Step 1: Add values of the first 3 years and divide by 3
Step 2: Move one year forward and repeat the process
Step 3: Place the average against the middle year

Formula

3 – Year Moving Average = (Y1+Y2+Y3) / 3​​

Characteristics

• Simple to calculate

• Trend values correspond directly to a year

• Suitable for short-term trend analysis

2. 4-Yearly Moving Average

This method is used when fluctuations are wider and smoother trend is required.

Steps for 4-Yearly Moving Average

Step 1: Add values of 4 consecutive years and divide by 4
Step 2: Repeat the process by shifting one year forward
Step 3: Since 4 is an even number, centering is required

Centering of Moving Averages

• Take the average of two consecutive 4-year moving averages

• Place the centered value against the corresponding year

Formula

4-Year Moving Average = (Y1+Y2+Y3+Y4) / 4​​

Characteristics

• Produces smoother trend

• More accurate than 3-year average

• Requires centering

3. 5-Yearly Moving Average

This method is used when long-term trend is required and data shows high fluctuations.

Steps for 5-Yearly Moving Average

Step 1: Add values of 5 consecutive years
Step 2: Divide the total by 5
Step 3: Place the average against the middle year

Formula

5-Year Moving Average = (Y1+Y2+Y3+Y4+Y5) / 5​​

Characteristics

• Produces very smooth trend

• Eliminates short-term fluctuations effectively

• Suitable for long-term analysis

Comparison of Least Squares and Moving Average Methods