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.
- Plot the time series data on a graph.
- 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.
- 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).
- Step 2: Calculate the arithmetic mean (average) for each part.
- Step 3: Plot these two averages on the graph at the mid-point of their respective time periods.
- 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.
| Year | Value (Y) | 3-Year Moving Total | 3-Year Moving Average (Trend, T) |
|---|
| 2018 | 10 | - | - |
| 2019 | 12 | 10 + 12 + 15 = 37 | 37 / 3 = 12.33 |
| 2020 | 15 | 12 + 15 + 13 = 40 | 40 / 3 = 13.33 |
| 2021 | 13 | 15 + 13 + 17 = 45 | 45 / 3 = 15.00 |
| 2022 | 17 | - | - |
4-Quarter Moving Average (Even Period)
This is common for quarterly data and has an extra step to solve the "centering problem."
- Step 1: Calculate 4-Quarter Moving Totals. (e.g., Q1+Q2+Q3+Q4). This value is placed between Q2 and Q3.
- Step 2: Calculate 4-Quarter Moving Averages. (Total / 4). This value is also between quarters.
- Step 3: Center the Averages. Take the average of two adjacent moving averages. This "centers" the value, aligning it with an actual quarter.
- Pros:
- Simple to understand and compute.
- Very flexible and can isolate the trend effectively if the period is chosen correctly.
- Cons:
- Data is lost at the ends (e.g., in a 3-year MA, you can't find the trend for the first or last year).
- Can be influenced by extreme random variations.
4.5 Method of Least Squares (Fitting Mathematical Curves)
This is the most mathematical and objective method. It finds the "line of best fit" by minimizing the sum of the squared differences between the actual data (Y) and the trend line (Y-hat).
The most common trend is a straight line (linear trend):
Y-hat = a + bX
Where Y-hat is the trend value, 'a' is the Y-intercept, 'b' is the slope (the amount of growth per time period), and 'X' is the time variable.
Time Variable (X)
To make calculations easier, we "code" the time variable 't' (e.g., 2018, 2019...) into a new variable 'X' such that ΣX = 0.
- Odd Number of Years (e.g., 5 years: 2018-2022):
Code the middle year as 0.
t = [2018, 2019, 2020, 2021, 2022]
X = [-2, -1, 0, 1, 2] -> ΣX = 0
- Even Number of Years (e.g., 6 years: 2018-2023):
Code the two middle years as -1 and +1, using half-units.
t = [2018, 2019, 2020, 2021, 2022, 2023]
X = [-5, -3, -1, 1, 3, 5] -> ΣX = 0
Solving for 'a' and 'b' (when ΣX = 0)
The normal equations from Unit 3 (for regression) are:
(I) ΣY = na + b(ΣX)
(II) ΣXY = a(ΣX) + b(ΣX²)
But since we cleverly made ΣX = 0, they simplify to:
(I) ΣY = na => a = ΣY / n = Y-bar (The mean of Y)
(II) ΣXY = b(ΣX²) => b = ΣXY / ΣX²
Procedure for Exam:
- Create a table with columns: Year (t), Coded Time (X), Value (Y), XY, and X².
- Calculate the sums at the bottom: ΣY, ΣXY, and ΣX².
- Find 'a' and 'b' using the simple formulas:
a = ΣY / n and b = ΣXY / ΣX².
- Write the final trend equation: Y-hat = a + bX.
- You can now find the trend value for any year by plugging in its 'X' value.
- Pros: Mathematical, objective, and gives a unique trend line. The line can be easily extrapolated for forecasting.
- Cons: Assumes the trend follows a specific mathematical form (like a line). If the true trend is complex, this method will give a poor fit.