Johansen's Approach: Maximum Likelihood Estimation of Vector Error Correction Models

Johansen's Approach is a statistical methodology used to estimate Vector Error Correction Models (VECMs) and test for multiple cointegration relationships among nonstationary and stationary variables.

Historical Context

Johansen’s Approach, developed by Søren Johansen in 1988, revolutionized the field of econometrics by providing a comprehensive framework for analyzing nonstationary time series data with multiple cointegrating relationships. It builds on earlier work in cointegration and the development of Vector Error Correction Models (VECMs).

Types/Categories

• Trace Test: Evaluates the rank of the cointegration matrix.
• Maximum Eigenvalue Test: Assesses the largest eigenvalue for the cointegration matrix.
• VECM with Deterministic Trends: Includes deterministic components like linear trends in the model.

Key Events

• 1988: Introduction of Johansen’s Approach in his seminal paper.
• 1991: Further refinement and extension of the methodology.
• 2000s: Widespread adoption in applied economics and finance.

Detailed Explanation

Johansen’s Approach is used to estimate VECMs, allowing for both stationary and nonstationary data. It involves:

1. Testing for the number of cointegrating vectors using Trace and Maximum Eigenvalue tests.
2. Estimating the VECM parameters via maximum likelihood.

Mathematical Formulation

Consider a \( k \)-dimensional time series \( \mathbf{Y}_t \). The VECM form is:

$$ \Delta \mathbf{Y}_t = \Pi \mathbf{Y}_{t-1} + \sum_{i=1}^{p-1} \Gamma_i \Delta \mathbf{Y}_{t-i} + \mathbf{u}_t $$

where \( \Pi \) is the long-term impact matrix indicating cointegrating relationships.

Importance and Applicability

Johansen’s Approach is crucial in:

• Econometrics: To test for long-term relationships among economic variables.
• Finance: For modeling relationships between asset prices.
• Macroeconomic Policy: For understanding the dynamic adjustments of economic policy variables.

Examples

1. Stock Prices and Interest Rates: Testing if long-term relationships exist between stock prices and interest rates.
2. GDP and Inflation: Analyzing the long-term cointegration between GDP and inflation rates.

Considerations

• Model Specification: Ensure correct lag length and inclusion of deterministic components.
• Stationarity Tests: Perform unit root tests before applying Johansen’s Approach.

Related Terms with Definitions

• Cointegration: A statistical property of a collection of time series variables that indicate a long-term equilibrium relationship.
• Vector Error Correction Model (VECM): A multivariate time series model that captures both short-term deviations and long-term equilibrium relationships.

Comparisons

• Engle-Granger Approach vs Johansen’s Approach: Engle-Granger is a two-step method, while Johansen’s Approach is more comprehensive and can handle multiple cointegrating vectors.

Interesting Facts

• Wide Usage: Johansen’s Approach is widely used in macroeconomic policy analysis, finance, and econometric research.

Famous Quotes

• “Cointegration is a revolution in time series analysis.” - Clive Granger, Nobel Laureate.

Proverbs and Clichés

• Proverb: “Everything is connected in the long run.”
• Cliché: “Finding the needle in the haystack of data.”

Jargon and Slang

• Endogenous Variables: Variables whose values are determined by other variables in the model.
• Eigenvalue: A scalar value indicating the magnitude of an eigenvector in linear algebra.

FAQs

Q1: What is the primary advantage of Johansen’s Approach? A1: It allows for the estimation and testing of multiple cointegration relationships in a system of equations.

Q2: Can Johansen’s Approach handle non-stationary variables? A2: Yes, it is designed specifically for systems with both non-stationary and stationary variables.

Q3: Is there any software available for Johansen’s Approach? A3: Yes, many econometrics software packages like EViews, R, and STATA provide functionalities for Johansen’s Approach.

References

1. Johansen, Søren. “Statistical Analysis of Cointegration Vectors.” Journal of Economic Dynamics and Control, 1988.
2. Johansen, Søren. “Likelihood-Based Inference in Cointegrated Vector Autoregressive Models.” Oxford University Press, 1995.

Summary

Johansen’s Approach is a robust, maximum likelihood-based method for estimating and testing Vector Error Correction Models with multiple cointegration relationships. Its application spans across economics, finance, and other fields dealing with time series data, providing a comprehensive framework to uncover long-term equilibrium relationships. The methodology continues to be pivotal in both academic research and practical econometric analysis.