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.

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. In contrast, 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.

To conduct the Dickey-Fuller test, a null hypothesis is first established, which states that the time series is non-stationary. The test then calculates the statistic t, which is a measure of the difference between the observed mean and the hypothesized mean of the time series. If the calculated t is greater than the critical value, then the null hypothesis is rejected, indicating that the time series is stationary.

The Dickey-Fuller test is useful in identifying whether a time series is stationary or non-stationary, as this information can inform the choice of forecasting methods. For example, if a time series is found to be stationary, then traditional time series analysis techniques such as moving average or autoregressive models can be used for forecasting. On the other hand, if a time series is non-stationary, then more advanced methods such as exponential smoothing or ARIMA models may be needed to properly account for the changing mean and variance over time.

In summary, the Dickey-Fuller test is a statistical tool for determining the stationarity of a time series. It does this by calculating the difference between the observed mean and hypothesized mean of the time series, and comparing this value to a critical value. The test is useful in informing the choice of forecasting methods, and can help analysts make more accurate predictions about future values in the time series.