Components of time series

Welcome to Time Series Analysis!

Hello there! Today, we are going to dive into the world of Time Series. As a future CPA, you won’t just be looking at numbers; you’ll be looking at patterns. Whether you are forecasting sales for a client or budgeting for next year’s expenses, understanding how data changes over time is a superpower. Don't worry if math isn't your favorite subject—we are going to break this down into simple, bite-sized pieces that make sense in the real world.

What is a Time Series?
A Time Series is simply a set of observations (data points) recorded at regular intervals over a period of time. Think of it like your bank statement or your phone’s screen time report—it’s data collected daily, monthly, or yearly.

The Four Components of Time Series

Data can look messy when you first see it on a graph. However, statisticians believe that any time series is actually made up of four "ingredients." We call these the Components of Time Series. To remember them, just think of the acronym: T.S.C.I.

1. The Trend (T)

The Trend is the long-term movement of the data. Is the business generally growing, shrinking, or staying flat over several years? Trends ignore the small daily "wiggles" and look at the big picture.

Example: Think of the use of smartphones over the last 15 years. Even if sales dropped for one specific month, the long-term Trend has been a massive upward climb.

2. Seasonal Variation (S)

Seasonal Variations are short-term fluctuations that repeat themselves regularly within a fixed period, usually one year or less. These are predictable patterns based on the time of year, month, or even day of the week.

Example: Mooncake sales in Hong Kong skyrocket every year during the Mid-Autumn Festival and drop to almost zero afterward. This is a classic Seasonal pattern.

3. Cyclical Variation (C)

Cyclical Variations are long-term "waves" that happen over several years. They are usually linked to the Business Cycle (periods of economic boom and recession). Unlike seasonal patterns, cycles don't have a fixed, predictable length.

Example: The real estate market in Hong Kong often goes through cycles of high prices followed by a few years of cooling down before rising again.

4. Irregular Variation (I)

Irregular Variations (also called Random Variations) are the "surprises." These are unpredictable events that don't follow a pattern. They are caused by one-off occurrences like natural disasters, sudden strikes, or a global pandemic.

Example: A sudden spike in face mask sales in early 2020 was an Irregular event that no statistical model could have predicted based on previous years' data.

Quick Review Box:
• Trend: Long-term direction.
• Seasonal: Repeats within a year (predictable).
• Cyclical: Economic waves (lasts many years).
• Irregular: Unpredictable/Random.

How the Components Work Together: The Models

In your exams, you need to know how these four components are combined to give us the actual value we see (the Observed Value, $Y$). There are two main ways to look at this:

A. The Additive Model

In this model, we assume the components are independent of each other. We simply add them together. The formula is:

\( Y = T + S + C + I \)

Use this model when the Seasonal Variation stays roughly the same size regardless of whether the trend is going up or down. For example, if you sell exactly 500 more units every December, no matter if your annual sales are 1,000 or 10,000.

B. The Multiplicative Model

This model is more common in the real world. It assumes that seasonal changes are a percentage of the trend. The formula is:

\( Y = T \times S \times C \times I \)

Use this model when the Seasonal Variation gets bigger as the trend increases. For example, if you sell 20% more in December. If your trend is 1,000, you sell 200 extra. If your trend grows to 10,000, that 20% "seasonal" bump is now 2,000 extra units!

Did you know?
In many accounting exams, the Cyclical (C) and Trend (T) components are often grouped together as a single "Trend-Cycle" component because they are both long-term movements.

Step-by-Step: Analyzing the Data

When you are given a set of data and asked to find the components, follow these steps:

1. Identify the Trend: Usually done by calculating a Moving Average (this smooths out the seasonal and irregular bumps).
2. Isolate Seasonal Variation: Subtract the Trend from the actual data (in Additive models) or divide the actual data by the Trend (in Multiplicative models).
3. Forecast: Use the Trend and the Seasonal factor to predict future values.

Common Mistake to Avoid:
Don't confuse Seasonal with Cyclical!
• Seasonal = Happens every year at the same time (e.g., Christmas).
• Cyclical = Happens over many years and is tied to the economy (e.g., The 2008 Financial Crisis).

Summary and Key Takeaways

Understanding Time Series is all about "decomposing" or breaking down a total number into its root causes. By knowing what is a long-term Trend versus what is just a temporary Seasonal spike, a CPA can provide much more accurate financial advice.

Key Points to Remember:

• A Time Series is data over time.
• T.S.C.I. stands for Trend, Seasonal, Cyclical, and Irregular.
• Additive Model: \( Y = T + S + C + I \) (Changes are absolute amounts).
• Multiplicative Model: \( Y = T \times S \times C \times I \) (Changes are proportional/percentages).
• Irregular components are the only ones that cannot be predicted.

Don't worry if this seems tricky at first! Once you start looking at graphs of company sales, you'll start seeing these components everywhere. You've got this!