The Johansen Multivariate Cointegration Test

In time series econometrics, stationarity is a fundamental concept. Most conventional statistical tests, such as linear regression, assume that the underlying data is stationarymeaning its mean and variance do not change over time. However, many economic and financial variables exhibit trends and are non-stationary. Standard regression on these non-stationary variables can lead to "spurious regression," where results indicate a relationship that does not actually exist.

To address this, the concept of cointegration was developed. Two or more non-stationary series are said to be cointegrated if they share a long-run, equilibrium relationship. While they may wander apart in the short term, a linear combination of them remains stationary. The Johansen test is a multivariate approach to determine this cointegration among several time series variables.

Historical Context and Theoretical Foundation

The Johansen methodology, developed by Sren Johansen in 1988 and 1991, expands upon earlier work by Clive Granger and Robert Engle. While the Engle-Granger two-step method is useful for analyzing bivariate (two-variable) systems, it has limitations when applied to systems with more than two variables. Specifically, the Engle-Granger method is sensitive to the choice of which variable is normalized to 1 in the cointegrating vector, and it cannot distinguish between multiple possible cointegrating relationships.

Johansen's approach utilizes a Vector Autoregressive (VAR) framework. By rearranging the VAR model into a Vector Error Correction Model (VECM), the test allows researchers to determine the rank of the cointegration space. This rank indicates exactly how many linearly independent cointegrating relationships exist among the variables.

The Mathematical Framework

The Johansen test begins by specifying a VAR model of order p, denoted as VAR(p), for a vector of k non-stationary variables, Y_t.

To test for cointegration, this model is transformed into a Vector Error Correction Model (VECM):

The critical component of this equation is the matrix Π (Pi). The rank of matrix Π determines the long-run properties of the system.

• Rank(Π) = 0: The variables are not cointegrated. There is no long-run relationship, and the VECM reduces to a VAR in differences.
• 0 < Rank(Π) < k: The variables are cointegrated. There are r (where r is the rank) linearly independent cointegrating vectors.
• Rank(Π) = k: The vector of variables is stationary. All variables are I(0), meaning cointegration is not relevant as differencing was not required.

The Two Likelihood Ratio Tests

To determine the rank r, the Johansen procedure provides two distinct likelihood ratio (LR) tests. Both tests follow the Chi-square distribution asymptotically.

The Trace Statistic

The Trace Statistic tests the null hypothesis that the number of cointegrating vectors is less than or equal to r (against the alternative that there are more than r).

Mathematically, it tests H₀: rank(Π) ≤ r versus H₁: rank(Π) > r.

The test proceeds sequentially: starting with r = 0, if the null is rejected, the test continues for r = 1, and so on, until the null hypothesis cannot be rejected. The statistic calculates the sum of the smallest squared canonical correlations.

The Maximum Eigenvalue Statistic

The Maximum Eigenvalue Statistic tests the null hypothesis that the number of cointegrating vectors is exactly r (against the alternative that it is r + 1).

It tests H₀: rank(Π) = r versus H₁: rank(Π) = r + 1.

This test looks at one canonical correlation at a time. It is often considered more powerful for determining the exact number of cointegrating vectors because it does not accumulate previous eigenvalues, unlike the Trace test. However, the Trace test is generally regarded as more robust in finite samples.

Implementation Steps

Conducting the Johansen test requires a structured approach to ensure the reliability of the results.

1. Unit Root Testing

Before performing the Johansen test, one must confirm that the variables are integrated of the same order, typically I(1). If the variables are integrated of different orders (e.g., one is I(1) and one is I(2)), the standard Johansen framework cannot be directly applied. Augmented Dickey-Fuller (ADF) or Phillips-Perron tests are commonly used for this preliminary step.

2. Lag Length Selection

Because the Johansen test relies on a VAR framework, selecting the correct lag length (p) is crucial. Too few lags can leave residual autocorrelation, biasing the results. Too many lags consume degrees of freedom and reduce the power of the test.

Information criteria such as the Akaike Information Criterion (AIC), Schwarz Bayesian Criterion (BIC), or the Hannan-Quinn Criterion (HQ) are typically used to determine the optimal lag length in the unrestricted VAR model.

3. Deterministic Trend Specification

The user must specify the deterministic trend assumptions for the VECM. This is often the most confusing part of the procedure. The general models are:

• Case 1: No deterministic trend in the data, no intercept in the cointegrating equation (CE).
• Case 2: No deterministic trend in the data, intercept in the CE.
• Case 3: Linear deterministic trend in the data, intercept in the CE.
• Case 4: Linear deterministic trend in the data, intercept and trend in the CE.
• Case 5: Quadratic deterministic trend in the data, intercept and trend in the CE.

The choice generally depends on whether the raw data appears to drift upwards or downwards over time. Economic time series usually require either Case 2 or Case 3.

Interpreting the Results

For example, if testing the Trace statistic:

• Hypothesis H₀: r = 0 (no cointegration): The statistic is 150.3. Critical value (5%) is 80.1. Result: Reject H0. There is at least one cointegrating vector.
• Hypothesis H₀: r ≤ 1: The statistic is 45.2. Critical value is 45.0. Result: Reject H0 (just barely). There is at least a second cointegrating vector.
• Hypothesis H₀: r ≤ 2: The statistic is 15.1. Critical value is 30.0. Result: Fail to reject H0.

In this scenario, the conclusion is that there are two cointegrating equations among the variables.

The Cointegrating Vectors

Once the rank r is determined, the estimated eigenvectors represent the cointegrating relationships. The first eigenvector is usually normalized so that the coefficient of the first variable is 1. These normalized coefficients describe the long-run equilibrium weights.

For instance, if the normalized vector is (1, -2.5, 0.5) for variables (Stock Price, Earnings, Interest Rates), the equilibrium relationship implies that Stock Price = 2.5 * Earnings - 0.5 * Interest Rate in the long run. Any deviation from this is the "equilibrium error," which drives the short-term adjustments.

Advantages and Limitations

The Johansen test remains the standard for multivariate cointegration analysis due to several key advantages:

• Multivariate Capacity: It avoids the arbitrary normalization problem (choosing which variable is dependent) inherent in the Engle-Granger approach. It treats all variables as endogenous (jointly determined).
• Multiple Relationships: It identifies multiple cointegrating vectors, which is essential for complex systems where different subsets of variables may move together over different horizons.
• Framework for VECM: The results are directly used to estimate the VECM, allowing economists to study both long-run equilibrium and short-term dynamics (speed of adjustment) simultaneously.

However, there are limitations to keep in mind:

• Small Sample Bias: The maximum likelihood estimation used in Johansens procedure can be biased in small samples. The critical values are derived from asymptotic distributions and may not be perfectly accurate for small datasets.
• Parameter Uncertainty: The test relies on the correct specification of the lag length and deterministic trend components. Misspecification here can lead to incorrect inference regarding the rank.
• Structural Breaks: The standard Johansen test assumes structural parameters are constant over time. If there is a structural break in the data (e.g., a financial crisis or policy change), the standard test may fail to detect cointegration. Modified versions using dummy variables are required in such cases.

Conclusion

The Johansen multivariate cointegration test is a powerful tool for econometricians and financial analysts. By moving beyond bivariate analysis and utilizing a full system approach, it provides a robust method for identifying the underlying long-run forces that bind economic variables together. Whether analyzing the relationship between interest rates, inflation, and output, or constructing statistical arbitrage strategies in stock markets, understanding the Vector Error Correction Model and the Johansen methodology is essential for rigorous time series analysis.