Time series analysis and seasonality

Welcome to Time Series Analysis!

Hello there! Welcome to one of the most practical parts of your BA1 – Fundamentals of Business Economics studies. Have you ever wondered how a supermarket knows to order more ice cream in July, or how a toy shop prepares for the December rush? They don't just guess; they use Time Series Analysis.

In this chapter, we are going to learn how to look at historical data (past patterns) to help us predict what might happen in the future. Don't worry if you aren't a "maths person"—we will break this down step-by-step using simple logic and real-world examples.

What is a Time Series?

A Time Series is simply a set of data points recorded at regular intervals over a period of time. This could be daily sales, monthly electricity usage, or annual GDP figures.

The goal of analyzing this data is to identify patterns. If we can see a pattern in the past, we can make a "best guess" (a forecast) about the future. This helps businesses plan their budgets, staffing, and stock levels.

Quick Review: To be a "time series," the data must be collected at equal intervals (e.g., every week, every quarter) and be kept in chronological order.

The Four Components of a Time Series

When we look at a graph of business data, it often looks like a messy, zigzagging line. To make sense of it, we imagine that the line is actually made up of four different "ingredients" mixed together:

1. The Trend (T)
This is the "big picture" or the long-term direction of the data. Is the business generally growing, shrinking, or staying the same over several years? We ignore the small day-to-day ups and downs and look at the underlying movement.

2. Seasonal Variation (S)
These are short-term fluctuations that repeat regularly within a fixed period, usually a year.
Example: Sales of winter coats always peak in Q4 (October–December) and drop in Q2 (April–June). This happens every single year like clockwork.

3. Cyclical Variation (C)
These are long-term "waves" caused by the economy (like booms and recessions). These cycles usually last several years. At the BA1 level, we often group this together with the Trend for simplicity.

4. Random (Residual) Variation (R)
These are unpredictable, "one-off" events that don't follow a pattern.
Example: A sudden snowstorm that closes all shops for two days, or a viral social media post that causes a one-week spike in sales. We cannot predict these!

Key Takeaway: The actual value we see in our books (\( Y \)) is a combination of these factors: \( Trend + Seasonal + Random \).

How to Find the Trend: Moving Averages

If you look at a sales graph, the "zigzag" of seasonal sales makes it hard to see the long-term trend. To fix this, we use a technique called Moving Averages. This "smooths out" the bumps to reveal the trend underneath.

The Logic: If you average out the sales for a whole year, the "highs" of summer and the "lows" of winter cancel each other out, leaving you with a smooth average for that period.

Step-by-Step: Simple Moving Average (Odd Number of Periods)

If we want a 3-period moving average for Sales:

1. Take the first three values (Period 1, 2, and 3) and find their average.
2. Place that average result in the middle (Period 2).
3. Drop the first value, add the fourth value, and find the new average.
4. Place that in the next middle slot (Period 3).
5. Repeat until you run out of data.

Centered Moving Averages (Even Number of Periods)

Don't worry if this seems tricky! When we use an even number (like a 4-quarter year), the "middle" of the first four numbers falls between Period 2 and Period 3. To fix this so the data aligns with an actual period, we take a second moving average of our first results. This is called Centering. You must center your moving averages if the number of periods in a cycle is even (like 4 quarters or 12 months).

Common Mistake: Forgetting to center your averages when dealing with quarterly data. If the question gives you 4 quarters, you will likely need to find a "Moving Average" and then a "Centered Moving Average."

The Two Mathematical Models

To calculate our forecast, we usually use one of two models. The exam will usually tell you which one to use.

1. The Additive Model

In this model, we assume the seasonal factor is a fixed amount of money or units, regardless of how big the trend is.

The Formula: \( Actual Value (Y) = T + S + R \)

To find the Seasonal Variation (\( S \)) if we ignore Randomness:
\( S = Y - T \)

Analogy: Imagine you always sell 100 extra ice creams in summer, whether you are a small shop or a huge chain. It's a flat "plus 100."

2. The Multiplicative Model (Proportional Model)

In this model, we assume the seasonal factor is a percentage or ratio of the trend. This is often more realistic for growing businesses.

The Formula: \( Actual Value (Y) = T \times S \times R \)

To find the Seasonal Variation (\( S \)) if we ignore Randomness:
\( S = Y / T \) (This will give you a decimal like 1.10, meaning sales are 10% above the trend).

Analogy: If your business grows, your summer "boost" grows too. Instead of selling "100 extra," you sell "20% more" than your average.

Calculating Seasonal Variations

Once you have used moving averages to find the Trend (T), you compare it to the Actual Sales (Y) to find the Seasonal Variation (S).

Example (Additive):
Actual Sales in Q4 (\( Y \)): \$500
\nTrend Value (\( T \)): \$420
Seasonal Variation (\( S \)): \( 500 - 420 = +80 \)
This means in Q4, sales are typically \$80 above the trend.

Did you know? Over a full cycle (e.g., 4 quarters), the total of the Seasonal Variations in an Additive Model should always sum to Zero. Some quarters are "plus" and some are "minus," but they cancel each other out over the year!

Forecasting the Future

Now for the most important part: Predicting. To forecast a future period, we follow these steps:

1. Project the Trend: Extend the trend line into the future (usually by adding a constant amount if the trend is linear).
2. Apply the Seasonality: Add (Additive) or Multiply (Multiplicative) the seasonal factor for that specific period.

Example Forecast:
You want to predict Sales for next Summer (Q3).
- Your projected Trend for that period is 1,000 units.
- You know from past data that the Seasonal Variation for Q3 is -150 units.
- Forecast: \( 1,000 + (-150) = 850 \) units.

Key Terms Summary

• Time Series: Data recorded over time in chronological order.
• Trend: The underlying long-term direction (up, down, or flat).
• Seasonal Variation: Short-term regular patterns within a year.
• Moving Average: A technique to "smooth" data and identify the trend.
• Additive Model: \( Y = T + S \). Seasonal variation is a fixed value.
• Multiplicative Model: \( Y = T \times S \). Seasonal variation is a percentage/ratio.

Encouraging Note: You’ve got this! Just remember: Trend is the big picture, Seasonality is the calendar pattern, and Moving Averages are just a way to clean up the data so we can see clearly. Keep practicing the calculations, and they will become second nature!