Understanding Deseasonalised Time-Series Data Components

Time-series data often contains several components that represent different patterns over time. These components are typically identified through a process called time series decomposition. The common components are:

• Trend (T): The long-term underlying direction of the series (upward, downward, or stable).
• Cyclical (C): Oscillations around the trend, usually lasting longer than a year and often related to business cycles.
• Seasonal (S): Regular patterns that repeat over a fixed period, typically a year or less (e.g., quarterly, monthly).
• Irregular (I): Random or unpredictable fluctuations that remain after accounting for the other components.

Time Series Decomposition Models

There are two main models for time series decomposition:

• Additive Model: \(Y_t = T_t + C_t + S_t + I_t\)
• Multiplicative Model: \(Y_t = T_t \times C_t \times S_t \times I_t\)

The question mentions Trend (T), Cyclical (C), and Irregular (I) components and asks about deseasonalised time-series data. Deseasonalising data means removing the Seasonal (S) component. The structure of the provided options suggests a multiplicative relationship between the components, which is typical for many economic and business time series where the magnitude of seasonal variations grows with the level of the series.

Deseasonalising in the Multiplicative Model

In the multiplicative model, the original time series (\(Y_t\)) is expressed as the product of its components:

\(Y_t = T_t \times C_t \times S_t \times I_t\)

To deseasonalise the data, we remove the seasonal component (\(S_t\)). The deseasonalised series (\(D_t\)) is obtained by dividing the original series by the seasonal component:

\(D_t = \frac{Y_t}{S_t} = \frac{T_t \times C_t \times S_t \times I_t}{S_t}\)

Assuming \(S_t \neq 0\), the deseasonalised series is:

\(D_t = T_t \times C_t \times I_t\)

This shows that the deseasonalised time-series data, in a multiplicative model, consists of the product of the Trend (T), Cyclical (C), and Irregular (I) components.

Analyzing the Options

Let's look at the given options in light of the deseasonalised multiplicative model \(D_t = T_t \times C_t \times I_t\).

• Option 1: \(\frac{T.I}{C}\times100\) - This does not match the \(T \times C \times I\) structure.
• Option 2: \(\frac{C.I}{T}\times100\) - This does not match the \(T \times C \times I\) structure.
• Option 3: (T. C. I) \(\times\) 100 - This matches the structure \(T \times C \times I\), multiplied by 100. The multiplication by 100 is commonly done when expressing components as indices or percentages, but the core relationship for deseasonalised data in a multiplicative model is the product of T, C, and I.
• Option 4: \(\frac{T.C}{I}\times100\) - This does not match the \(T \times C \times I\) structure.

Therefore, the expression that represents the deseasonalised time-series data with only trend (T), cyclical (C), and irregular (I) components in a multiplicative context is (T. C. I), possibly scaled by 100 for presentation as an index.

Additional Information on Time Series Decomposition and Deseasonalising

While the multiplicative model is common, especially for economic data, the additive model is also used, particularly when the seasonal variations are relatively constant in magnitude regardless of the level of the series.

In the additive model:

\(Y_t = T_t + C_t + S_t + I_t\)

To deseasonalise data in the additive model, the seasonal component is subtracted:

\(D_t = Y_t - S_t = T_t + C_t + I_t\)

In this case, the deseasonalised series consists of the sum of Trend, Cyclical, and Irregular components. However, the question and options suggest a multiplicative relationship by listing T.C.I as a product structure.

Deseasonalising time-series data is an important step in forecasting and analysis, as it helps reveal the underlying trend and cyclical patterns that might be obscured by strong seasonal variations. It allows for better comparison of data points across different seasons.