Dickey Fuller Test

TL;DR

• Statistical test that evaluates whether a time series is stationary (constant mean and variance) or non-stationary (changing mean and/or variance).
• Sets a null hypothesis that the series is non-stationary and computes a test statistic (t) compared to a critical value.
• Result helps choose forecasting methods: stationary series suit moving-average or autoregressive models; non-stationary series may require exponential smoothing or ARIMA.

Definition

The Dickey-Fuller test is a statistical test used to determine whether a time series is stationary or non-stationary. Stationary time series have a constant mean and variance over time, while non-stationary time series have a mean and variance that change over time.

Explanation

The test is conducted by first establishing a null hypothesis that the time series is non-stationary. The Dickey-Fuller test then calculates a statistic t, described as a measure of the difference between the observed mean and the hypothesized mean of the time series. This t statistic is compared to a critical value; if the calculated t is greater than the critical value, the null hypothesis is rejected, indicating that the time series is stationary. Determining stationarity with this test informs the selection of appropriate forecasting techniques.

Examples

Stationary example

One example of a stationary time series is the monthly average temperature in a particular location. Over time, the monthly average temperature may fluctuate due to seasonal variations, but the overall mean and variance remain constant.

Non-stationary example

An example of a non-stationary time series is the monthly stock price of a particular company. The stock price may fluctuate due to market conditions, but it is also likely to trend upwards or downwards over time, resulting in a changing mean and variance.

Use cases

• Informing the choice of forecasting methods: if a series is found to be stationary, traditional time series techniques such as moving average or autoregressive models can be used.
• If a series is non-stationary, methods such as exponential smoothing or ARIMA models may be needed to account for the changing mean and variance over time.

Notes or pitfalls

• The test requires establishing the null hypothesis that the time series is non-stationary.
• The test statistic t represents the difference between the observed mean and the hypothesized mean and must be compared to a critical value; according to the source, if t is greater than the critical value, the null hypothesis is rejected, indicating stationarity.

Related terms

• Moving average
• Autoregressive models
• Exponential smoothing
• ARIMA