Skip to content

Tests for Cointegration

Cointegration tests are used to determine whether a "stable long-term relationship" exists between variables. When variables are cointegrated, it implies that although they may drift apart in the short term, they move together in the long term.

Engle Granger Two Step Test

  • This was one of the first methods developed and is primarily used for 2 variables.
  • The Steps involve:
    1. Estimate long run relationship: Regress \(Y_{t} = \alpha + \beta X_{t} + \epsilon_{t}\) (or include other \(X_{t}\) variables as needed) and obtain the residuals \(\hat{\epsilon}_{t}\).
    2. Test for stationarity of residuals: Use the Augmented Dickey-Fuller (ADF) test. If the residuals are found to be \(I(0)\), then the variables are cointegrated.

Hypothesis Setup:
$$
\begin{align}
H_{0}: & \text{ Residuals are non-stationary } \implies \text{ No Cointegration} \
H_{1}: & \text{ Residuals are stationary } \implies \text{ Cointegrated}
\end{align}
$$

  • Limitation: This test is only suitable for two variables and lacks statistical power if more than two variables are involved in the system.

Johansen Test

Pronounced: "You-han-sen"

  • This is a more comprehensive test that can handle multiple variables at once.
  • It is based on a Vector Autoregression (VAR) model.

  • It examines the rank of the cointegration matrix to determine the actual number of cointegrating relationships (rank \(r\)).

  • Two specific approaches within the test:

    1. Trace Test: Tests the null hypothesis that the number of cointegrating vectors (CV) is \(\le r\) against the alternative that it is \(> r\).
    2. Maximum Eigenvalue Test: Tests the null hypothesis that the number of CV \(= r\) against the alternative that it is \(r+1\).

Phillips-Ouliaris Cointegration Test

  • This is a residual-based test, similar to the #Engle Granger Two Step Test.
  • However, it uses different, more robust test statistics to determine the relationship.
\[ \begin{align} H_{0} &: \text{ No cointegration} \\ H_{1} &: \text{ Cointegration exists} \end{align} \]

It relies on the Phillips-Ouliaris test statistic for the decision.

Durbin-Watson Cointegration Test

  • This is not a direct cointegration test itself. Instead, it helps in testing for spurious relationships (where two unrelated variables show a strong correlation just because of their trends).
  • DW statistic \(\approx 0\): Suggests the possibility of spurious regression, meaning the variables may not be cointegrated.
  • DW \(> 1.5 - 2\): Suggests that a cointegration relationship might actually exist.

Auto-Regressive Distributed Lag (ARDL) Bounds Test

  • This test is used when the underlying series are of mixed integration order (for example, if some are \(I(0)\) and some are \(I(1)\)).
\[ \begin{align} H_{0} &: \text{ No cointegration} \\ H_{1} &: \text{ Cointegration exists} \end{align} \]
  • A major benefit is that it is compatible with various sample sizes, including smaller ones.

Practical Examples

  • House Prices and Rent Prices: These usually move together over long periods despite short-term fluctuations.
  • Commercial Real Estate Prices and Economic Indicators: Property values drift along with broader economic health.
  • Healthcare Spending and GDP: As the economy grows, healthcare spending generally grows with it, maintaining a long-run balance.
  • Drug Prices and R&D Costs: Used to validate cost determination practices by checking if prices and research costs are actually linked in the long run.