Introduction

In time series analysis, understanding and accounting for seasonality is crucial for accurate forecasting and decision-making. Seasonality refers to the recurring patterns or fluctuations that occur within a specific time period, such as daily, monthly, or yearly cycles. Regression analysis serves as a powerful tool to model and analyze these seasonal patterns. This chapter delves into the application of regression analysis to effectively model seasonality within non-stationary time series data.

Regression Analysis for Seasonal Modeling

Regression analysis involves exploring the relationship between a dependent variable and one or more independent variables. In the context of time series, it allows us to uncover how the dependent variable changes as the independent variables, often time itself, vary. When dealing with seasonality, time is often a key independent variable.

To model seasonality using regression analysis, we can employ various approaches:

1) Dummy Variables: One common technique is to use dummy variables to capture the effects of different seasons. For example, if we’re analyzing monthly sales data, we can create 11 dummy variables (January to November) to capture the seasonal variation. February, for instance, would have a dummy variable with a value of 1 for February observations and 0 otherwise.

2) Trigonometric Functions: Trigonometric functions, such as sine and cosine functions, can also be used to model seasonality. These functions can represent periodic patterns effectively. For instance, when analyzing daily temperature variations, a sine function could capture the recurring temperature fluctuations.

3) Moving Averages: Regression can be combined with moving averages to smoothen the data and identify underlying seasonal patterns. By fitting a regression line to the moving averages, we can estimate the seasonal components more accurately.

Example: Suppose we have monthly sales data for a retail store. We want to model the seasonality to understand if sales tend to increase during certain months. We decide to use dummy variables to capture this effect. We create 11 dummy variables (January to November) and use multiple linear regression to relate sales to these dummy variables. The regression equation might look like:

$Sales=\beta_0 + \beta_1 \text{Dummy}_{\text{Jan}} + \beta_2 \text{Dummy}_{\text{Feb}}$ + $\ldots + \beta_{11} \text{Dummy}_{\text{Nov}} +\varepsilon$

Where

• $\beta_0$ represents the intercept,
• $\beta_1$ to $\beta_{11}$ are the coefficients associated with each dummy variable, and
• $\varepsilon$ is the error term