Quantitative Data Analysis

Time Series and Stationarity: Time series in statistics is a series of data points ndexed in time order. Most commonly, these data points are considered at successive equally spaced points in time. Thus, we obtain then a sequence of discrete time data. Examples of time series are daily measures of fever,  yearly counts of an epidemic in a country, and the daily closing value of stock market indices. Such data requires special analysis methods (time-domain methods where we are concentrated and frequency-domain methods) in order to extract meaningful statistics and other characteristics (e.g. Granger causality to detect relationship between time-series, a model from the statistical domain or a model from the machine learning domain to predict future values based on previously observed values, the build of a regression model to measure the impact of factors, autocorrelation for measuring short-term memory,

Stationarity in time series refers to a condition where the statistical properties of a dataset, such as mean and variance, remain constant over time. There are two main types of stationarity:

Strict Stationarity: All statistical properties do not change over time for the series.

Weak Stationarity: The mean and variance are constant over time and the covariance between data points depends only on the time lag (the covarince displays the joint variability of two random variables). This is used widely in the description of real-world data (and is what is usually assumed in statisical tests). 

The stationarity is a key component for achieving accurate modeling and forecasting in time series. This has the consequence that stationarity has central role in time series analysis. Because if a series is not stationary and we apply a model for modeling or forecasting the series, parameters are continuously changing and this means that estimating them by averaging over time will not be accurate. This means that the (real) system not necessarily reverting to a long-term mean. When we use a model to describe (and modeling) the process which generated the data, we make such assumption (that data are reverted to the mean), so the conclusions are erroneous and misleading. To conclude, any time series data should be checked for stationarity. There are developed many tests and one of them is the ADF test.

In statistics, an ADF test, is checking if a unit root is present in a time series sample, i.e. the non-stationarity of the series (null hypothesis). Generally, the alternative hypothesis is the stationarity of the series, but we specify that there are three (3) versions of the test (1. Test for unit root, 2. Test for a unit root with constant, 3. Test for a unit root with constant and deterministic time trend.) and the alternative hypothesis depends on the version. Each version results in a different regression model. It is an augmented version of the Dickey–Fuller test for a larger and more complicated set of time series models. 

Null hypothesis (H₀): A Unit Root is present on the time series sample (Non-stationatity of the series).

Aternative hypothesis: (H_a): A Unit Root is not present on the time series sample (Stationarity of the series).

Testing Procedure: It is considered the formula:

 , where p is the lag of the autoregressive process, Δ denotes the difference with the previous value of the process (for example  , where p is the lag of the autoregressive process, Δ denotes the difference with the previous value of the process (for example Δx₅ is x₅-x₄) and alpha (α), beta (β) and theta (θ) are coefficients. If α=0 and β=0, then it is obtained a random walk or if only β=0, then it is obtained a random walk with a drift. To implement the test, we are testing (in the above formula) the value of the coefficient γ. If the null hypothesis is valid, then γ=0 and a unit root is present. Otherwise, if γ is not zero, then there is no unit root and the series are stationary.  It is crucial for the test to determine the lag of the autoregressive process (p) (this can be done with a criterion such as the Bayesian Information Criterion – BIC).

The test is based on a t-test in the coefficient γ of the regression (considered formula).

Sources

Kozhan, R. (2019). Financial Econometrics. Bookboon.

https://en.wikipedia.org/wiki/Augmented_Dickey%E2%80%93Fuller_test