A time series is said to be “stationary” if it has no trend, exhibits constant variance over time, and has a constant autocorrelation structure over time.

One way to test whether a time series is stationary is to perform an **augmented Dickey-Fuller test**, which uses the following null and alternative hypotheses:

**H _{0}:** The time series is non-stationary. In other words, it has some time-dependent structure and does not have constant variance over time.

**H _{A}:** The time series is stationary.

If the p-value from the test is less than some significance level (e.g. α = .05), then we can reject the null hypothesis and conclude that the time series is stationary.

The following step-by-step example shows how to perform an augmented Dickey-Fuller test in R for a given time series.

**Example: Augmented Dickey-Fuller Test in R**

Suppose we have the following time series data in R:

data <- c(3, 4, 4, 5, 6, 7, 6, 6, 7, 8, 9, 12, 10)

Before we perform an augmented Dickey-Fuller test on the data, we can create a quick plot to visualize the data:

plot(data, type='l')

To perform an augmented Dickey-Fuller test, we can use the adf.test() function from the **tseries **library.

The following code shows how to use this function:

library(tseries) #perform augmented Dickey-Fuller test adf.test(data) Augmented Dickey-Fuller Test data: data Dickey-Fuller = -2.2048, Lag order = 2, p-value = 0.4943 alternative hypothesis: stationary

Here’s how to interpret the most important values in the output:

- Test statistic:
**-2.2048** - P-value:
**0.4943**

Since the p-value is not less than .05, we fail to reject the null hypothesis.

This means the time series is non-stationary. In other words, it has some time-dependent structure and does not have constant variance over time.

**Additional Resources**

How to Perform a Mann-Kendall Trend Test in R

How to Plot a Time Series in R

How to Detrend Data