Johansen test

The Johansen Test is a statistical method used to determine the presence and number of cointegration relationships between multiple time series. Developed by Søren Johansen in 1988, this test is widely used in econometrics to assess the long-term equilibrium relationships among integrated variables.

Background

In time series analysis, cointegration refers to a situation where two or more non-stationary series are linked by a stable, long-term relationship. While individual series may exhibit trends or random walks, a linear combination of them can be stationary. The Johansen Test provides a framework for identifying such relationships, which is crucial for modeling and forecasting economic and financial data.

Methodology

The Johansen Test is based on the Vector Autoregressive (VAR) model and involves the following steps:

1. Model Specification: Consider a VAR model of order \( p \) for \( k \) time series variables.

2. Formulation of the Vector Error Correction Model (VECM): The VAR model is reformulated into a VECM to capture both short-term dynamics and long-term relationships.

3. Estimation of the Cointegration Rank: The test involves estimating the rank of the cointegration matrix \( \Pi \), which indicates the number of cointegrating vectors.

4. Test Statistics: Two test statistics are used:

  - Trace Test: Tests the null hypothesis that the number of cointegrating vectors is \( r \) against the alternative that it is greater than \( r \).
  - Maximum Eigenvalue Test: Tests the null hypothesis that the number of cointegrating vectors is \( r \) against the alternative of \( r+1 \).

5. Critical Values: The test statistics are compared against critical values to determine the number of cointegrating relationships.

Assumptions

The Johansen Test assumes that the time series are:

- Integrated of the same order, typically I(1). - The error terms are normally distributed. - The system is stable and the parameters are constant over time.

Applications

The Johansen Test is extensively used in:

- Econometrics for modeling relationships between economic indicators. - Finance for analyzing the long-term relationships between asset prices. - Macroeconomics for studying the interactions between macroeconomic variables.

Limitations

While powerful, the Johansen Test has limitations:

- Sensitive to the choice of lag length in the VAR model. - Requires large sample sizes for reliable results. - Assumes linear relationships, which may not capture all dynamics in the data.

Also see

- Cointegration - Vector Autoregression - Granger Causality - Augmented Dickey-Fuller Test