ADF Unit Root Test and Engle-Granger Cointegration Test

The Augmented Dickey-Fuller (ADF) test is a unit root testing method used to determine whether time series data is stationary. The Engle-Granger cointegration test analyzes whether two or more non-stationary time series exhibit long-term equilibrium relationships. Both tests are widely applied in econometrics and financial time series analysis.

In MATLAB, the ADF test can be implemented using built-in functions or the Econometrics Toolbox. Key implementation considerations include setting appropriate lag orders and test types (such as including intercept terms or time trends). For the Engle-Granger cointegration test, the core algorithm involves first performing regression on non-stationary series, then conducting unit root testing on the regression residuals. If the residuals are stationary, it indicates cointegration relationship between the series.

Practical implementation notes: ADF testing requires selecting appropriate model specifications based on series characteristics; The Engle-Granger test is sensitive to regression order and may require swapping dependent and independent variables for repeated verification; In MATLAB, the `adftest` function directly performs ADF testing, while cointegration testing typically requires custom implementation of the residual testing pipeline using functions like `regress` for regression and subsequent stationarity checks.

The combined application of these tests effectively reveals intrinsic relationships between non-stationary economic variables and serves as a crucial prerequisite for building Error Correction Models (ECM).