Introduction

Covariance stationarity is a crucial concept in time series analysis. A covariance stationary time series exhibits stable statistical properties over time, making it amenable to various analytical techniques. In this chapter, we will delve into the essential requirements that a series must meet to be considered covariance stationary. Understanding these requirements is fundamental to effectively analyzing and modeling time series data.

Key Requirements for Covariance Stationarity

1) Constant Mean $(\mu)$: For a series to be covariance stationary, its mean should remain constant over time. This means that the series does not exhibit any upward or downward trend in its average value.

2) Constant Variance $\sigma^2$: The variance of the series should remain constant across all time periods. Fluctuations in variability over time can hinder accurate statistical analysis.

3) Constant Autocovariance ($\gamma_h$): The autocovariance between observations at different time lags should not depend on time. In other words, the relationship between the past and the present should remain consistent throughout the series.

4) No Systematic Seasonality: A covariance stationary series should not exhibit any systematic seasonality or cyclical patterns that repeat at fixed intervals.

Example: Let’s consider an example of stock prices. If the stock prices exhibit a consistent upward trend over time, the series violates the requirement of a constant mean. Similarly, if the variability of stock returns increases during market turmoil, the series fails the constant variance criterion.

For a practical example, imagine a temperature dataset. If the mean temperature over the years remains roughly the same, the series satisfies the constant mean criterion. Additionally, if the variance of daily temperature readings remains stable across seasons, the constant variance requirement is met.

Mathematical Formulation

Mathematically, the requirements for covariance stationarity can be represented as follows:

1) Constant Mean: $E[X_t]=\mu$, where $E[X_t]$ represents the expected value of the series at time $t$, and $\mu$ is a constant.

2) Constant Variance: $Var[X_t]=\sigma^2$, where $Var[X_t]$ represents the variance of the series at time $t$, and $\sigma^2$ is a constant.

3) Constant Autocovariance: $Cov[X_t, X_{t+h}]=\gamma_h$, where $Cov[X_t, X_{t+h}]$ is the autocovariance between time $t$ and time $t+h$, and $\gamma_h$ is a constant that depends only on the lag $h$.