L38 Tests for Cointegration
- Determine whether "stable long-term relationship" exists
- Cointegrated \(\implies\) although they may drift, they move together in the long term
Engle Granger Two Step Test¶
- One of the first methods (2 variables)
- Steps
- Estimate long run relationship: Regress \(Y_{t} = \alpha + \beta X_{t} +\epsilon_{t}\) and obtain residuals1
- Test for stationarity of residuals: ADF. If residuals are \(I(0)\) then the residuals are cointegrated
\[
\begin{align}
H_{0}: & \text{Residuals are non-stationary} \implies \text{Cointegrated} \\
H_{1}: & \text{Resdiausl are stationary} \implies \text{Cointegrated}
\end{align}
\]
- Limitation: only suitable for two variables
- lacks power if more than two variables are involved
Johansen Test¶
"You-han-sen"
- More comprehensive test that can handle multiple variables
- Based on VAR model
-
Examines the rank of the cointegration matrix (to determine the number of cointegrating relationships)
-
Two approaches
- Trace Test: number of CV \(\le r\) against \(\geq r\)
- Maximum Eigenvalue Test: number of CV \(=r\) against \(r+1\)
\[
\begin{align}
H_{1} & \\
H_{2} &
\end{align}
\]
Phillips-Ouliaris Cointegration Test¶
- Residual-based like #Engle Granger Two Step Test
- Uses different tests statistics (more robust)
\[
\begin{align}
H_{0} &: \text{No cointegration} \\
H_{1} &: \text{Cointegration exists}
\end{align}
\]
Phillips-Ouliaris test statistic
Durbin-Watson Cointegration Test¶
- Not a direct cointegration test. Helps in testing for spurious relationships2
- DW static \(\approx 0\) \(\implies\) possibility of spurious regression \(\implies\) may not be cointegrated
- DW \(\gt 1.5-2\) \(\implies\) Cointegration relationship might exist
Auto-Regressive Distributed Lag (ARDL) Bounds Test¶
- When underlying series are of mixed integration order (some \(I(0)\) and some \(I(1)\))
\[
\begin{align}
H_{0} &: \text{No cointegration} \\
H_{1} &: \text{Cointegration exists}
\end{align}
\]
- compatible with various sample sizes
Practical Examples¶
- House Prices and Rent Prices
- Commercial Real Estate Prices and Economic Indicators
- Healthcare Spending and GDP
- spending grows with economy
- Drug Prices and R&D Costs
- validate the cost determination practices