Question

Discuss two formal tests for cointegration and why they are considered superior to the Engle

and Granger method.

Answer 1

DEFINITION OF COINTEGRATION

Cointegration tests analyze non-stationary time series— processes that have variances and means that vary over time. In other words, the method allows you to estimate the long-run parameters or equilibrium in systems with unit root variables (Rao, 2007).

Two sets of variables are cointegrated if a linear combination of those variables has a lower order of integration. For example, cointegration exists if a set of I(1) variables can be modeled with linear combinations that are I(0). The order of integration here—I(1)— tells you that a single set of differences can transform the non-stationary variables to stationarity. Although looking at a graph can sometimes tell you if you have an I(1) process, you may need to run a test such as the KPSS test or the Augmented Dickey-Fuller test to figure it out.

Why is cointegration test important.

Cointegration tests analyze non-stationary time series— processes that have variances and means that vary over time. In other words, the method allows you to estimate the long-run parameters or equilibrium in systems with unit root variables (Rao, 2007)

Background

In order to analyze time series with classical methods like ordinary least squares , an assumption is made: The variances and means of the series are constants that are independent of time (i.e. the processes are stationary). Non-stationary time series (or unit root variables) don’t meet this assumption, so the results from any hypothesis test will be biased misleading. These series have to be analyzed with different methods. One of these methods is called cointegration.

More formally, cointegration is where two I(1) time series xt and yt can be described by the stationary process
ut = yt − αxt

Tests for Cointegration

Tests for cointegration identify stable, long-run relationships between sets of variables. However, Rao (2007) notes that if the test fails to find such a relationship, it isn’t proof that one doesn’t exists— it only suggests that one doesn’t exist.

TWO METHOD WE WILL DISCUSS

Phillips–Ouliaris

The Philips-Ouliaris (1990) is a residual-based unit root test. It is an improvement over the Engle-Ganger test; Prior to 1987, tests for cointegration worked on the assumption that regression errors are independent with common variance—which is rarely true in real life (Chaovalitwongse et. al, 2010).

H0: No cointegration exists
H1: Cointegration exists

The Philips-Ouliaris test takes supplementary variability into account (stemming from the fact that residuals are estimates instead of the actual parameter values). The tests is also invariant to normalization of the cointegration relationship (i.e. which variable is counted as the dependent Variable

Johansen test

Johansen’s test is another improvement over the Engle-Granger test. It avoids the issue of choosing a dependent variable as well as issues created when errors are carried from one step to the next. As such, the test can detect multiple cointegrating vectors.

WHEN WE COMPARE TO ENGLE- GRANGER

Engle–Granger

The Engle Granger method first constructs residuals (errors) based on the static regression.The residuals are tested for the presence of unit roots using ADF or a similar test. If the time series is cointegrated, then the residuals will be practically stationary. A major issue with the Engle-Granger method is that choice of the dependent variable may lead to different conclusions (Armstrong, 2001), an issue corrected by more recent tests such as Phillips-Ouliaris and Johansen’s.

H0: No cointegration exists
H1: Cointegration exists

This test is usually performed by software such as MATLAB or STAT (using the egranger command).

CONCLUSION

Many economic theories imply that a linear combination of variables is stationary
although individually they are not. If there is such a stable linear combination among variables, the variables are said to be cointegrated. In existence of cointegration or long-run relationship, the variables have the same stochastic trends and therefore they cannot drift too far apart. In time series analysis of macroeconomic studies, hence, one should check for stationarity and cointegration to avoid losing long term information. There are several methods in examining the cointegration analyses. Engle- Granger and Johansen procedures are the most commonly used among others in the literature. In Engle-Granger procedure, one examines the residuals from long-run equilibrium relationship by ordinary least squares method. The variables are
cointegrated if these residuals do not yield unit root. Johansen procedure, in estimation cointegration relationship, estimates a vector autoregression in first differences and includes the lagged level of the variables in some period t-p.