Stationarity Test

Understanding stationarity

A time series is considered stationary when its statistical properties - such as mean, variance, and autocorrelation - remain constant over time. This property is crucial because:

1. It allows meaningful statistical inference
2. It enables reliable forecasting
3. It supports the application of many time-series models

Types of stationarity

There are two main types of stationarity:

1. Strict (Strong) Stationarity

   • The joint probability distribution remains unchanged when shifted in time
   • All statistical moments must be constant

2. Weak (Covariance) Stationarity

   • Mean remains constant
   • Variance remains finite and constant
   • Autocorrelation function depends only on time lag

Common stationarity tests

Dickey-Fuller Test

The Dickey-Fuller test is one of the most widely used stationarity tests. It tests the null hypothesis that a unit root is present in a time series sample. The basic test equation is:

$\Delta y_t = (\rho-1)y_{t-1} + \epsilon_t$

Where:

• $y_t$ is the time series
• $\rho$ is the coefficient
• $\epsilon_t$ is the error term

Augmented Dickey-Fuller (ADF) Test

The ADF test extends the basic Dickey-Fuller test to handle more complex models:

$\Delta y_t = \alpha + \beta t + (\rho-1)y_{t-1} + \sum_{i=1}^{p} \gamma_i \Delta y_{t-i} + \epsilon_t$

Where:

• $\alpha$ is the drift term
• $\beta t$ is the time trend
• $p$ is the lag order

Applications in financial markets

Stationarity tests are crucial in:

1. Market Analysis

   • Testing mean reversion in price series
   • Validating statistical arbitrage strategies
   • Analyzing spread relationships

2. Risk Management

   • Assessing stability of volatility processes
   • Validating assumptions in Value at Risk (VaR) models
   • Evaluating correlation stability

Making non-stationary data stationary

Common transformations include:

1. Differencing

   • First differences for trend removal
   • Seasonal differencing for periodic patterns

2. Detrending

   • Linear trend removal
   • Exponential moving average smoothing

3. Variance Stabilization

   • Log transformation
   • Box-Cox transformation

Practical considerations

When applying stationarity tests:

1. Sample Size

   • Larger samples provide more reliable results
   • Short series may require alternative approaches

2. Test Selection

   • Different tests have varying power against different alternatives
   • Consider using multiple tests for robustness

3. Economic Context

   • Not all financial series should be stationary
   • Some non-stationarity may be economically meaningful

Implementation in time-series databases

Modern time-series databases often include built-in functionality for:

1. Automated Testing

   • Batch processing of multiple series
   • Real-time stationarity monitoring

2. Data Transformation

   • On-the-fly differencing
   • Automated detrending

3. Result Storage

   • Efficient storage of test statistics
   • Historical test results tracking

The understanding and application of stationarity tests remains fundamental to time-series analysis, particularly in financial markets where stable statistical properties are crucial for modeling and prediction.