Time Series Stationarity

Stationarity in Time Series

Trend Stationarity

• A time series is said to be trend stationary if it can be made stationary by removing a deterministic trend component.
• The trend can be linear or non-linear.
• Removing the trend component does not affect the underlying structure of the time series.
• Example: A time series with a linear upward trend can be made stationary by subtracting the trend from the data.

Mean Reverting

• A time series is said to be mean reverting if it tends to return to its historical mean over time.
• Mean reverting series exhibit a tendency to revert to their mean, but may not always do so.
• Mean reverting series are often used in finance to model asset prices and returns.
• Example: Stock prices tend to revert to their historical mean over time, but may fluctuate in the short term.

Seasonal Decomposition

• Seasonal decomposition is a technique used to separate a time series into its trend, seasonal, and residual components.
• The goal is to identify and isolate the patterns and anomalies in the data.
• The decomposition can be additive or multiplicative, depending on the nature of the data.
• Example: A time series of daily temperatures can be decomposed into its trend, seasonal, and residual components to identify patterns and anomalies.

Autocorrelation

• Autocorrelation refers to the correlation between a time series and lagged versions of itself.
• Autocorrelation is a measure of how well a time series is correlated with its past values.
• Autocorrelation is used to identify patterns and structures in the data.
• Example: A time series with high autocorrelation at lag 1 means that the current value is highly correlated with the previous value.

Augmented Dickey Fuller (ADF) Test

• The ADF test is a statistical test used to determine if a time series is stationary or not.
• The test is based on the idea that a stationary time series should have no unit root.
• The test involves estimating the following regression equation: Δy_t = βy_(t-1) + ε_t
• The null hypothesis is that the time series is non-stationary (β = 0), and the alternative hypothesis is that the time series is stationary (β < 0).
• The test produces a test statistic and a p-value, which can be used to determine the significance of the result.

Stationarity in Time Series

Trend Stationarity

• Time series is trend stationary if it can be made stationary by removing a deterministic trend component, which can be linear or non-linear.
• Removing the trend component does not affect the underlying structure of the time series.
• Example: Subtracting the trend from a time series with a linear upward trend makes it stationary.

Mean Reverting

• Time series is mean reverting if it tends to return to its historical mean over time.
• Mean reverting series exhibit a tendency to revert to their mean, but may not always do so.
• Often used in finance to model asset prices and returns.
• Example: Stock prices tend to revert to their historical mean over time, but may fluctuate in the short term.

Seasonal Decomposition

• Technique used to separate a time series into its trend, seasonal, and residual components.
• Goal is to identify and isolate patterns and anomalies in the data.
• Decomposition can be additive or multiplicative, depending on the nature of the data.
• Example: Decomposing daily temperatures into trend, seasonal, and residual components to identify patterns and anomalies.

Autocorrelation

• Refers to the correlation between a time series and lagged versions of itself.
• Measure of how well a time series is correlated with its past values.
• Used to identify patterns and structures in the data.
• Example: High autocorrelation at lag 1 means the current value is highly correlated with the previous value.

Augmented Dickey Fuller (ADF) Test

• Statistical test used to determine if a time series is stationary or not.
• Test is based on the idea that a stationary time series should have no unit root.
• Involves estimating the regression equation: Δy_t = βy_(t-1) + ε_t.
• Null hypothesis: Time series is non-stationary (β = 0).
• Alternative hypothesis: Time series is stationary (β < 0).
• Test produces a test statistic and a p-value, which determine the significance of the result.