Time Series Stationarity

Study Notes

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.

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 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 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.

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.

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.

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.

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.

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.

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.