Co-integration Test

Co-integration TestDr Gargi SanatiCourse: AEMPGDM, 2014-16NIBM{Generalizing from the univariate case can be misleading.

From the standard Box-Jenkins approach to univariate ARMA modeling, stationarity is an essential assumption. Without it, the underlying theory and estimation techniques become invalid.

If the goal of a VAR analysis is to determine relationships among the original variables, differencing loses information. In this context, Sims, Stock, and Watson advise against differencing, even in the presence of unit roots.

In the presence of cointegration, simple differencing is a model misspecification, since long-term information appears in the levels

Money stock, interest rates, income, and prices (common models of money demand)Investment, income, and consumption (common models of productivity)Consumption and long-term income expectation (Permanent Income Hypothesis)Exchange rates and prices in foreign and domestic markets (Purchasing Power Parity)Spot and forward currency exchange rates and interest rates (Covered Interest Rate Parity)Interest rates of different maturities (Term Structure Expectations Hypothesis)Interest rates and inflation (Fisher Equation)

Examples of variables that are commonly described with a cointegrated VAR model include:

Example of Co-integrated Series

The concept of co-integration, as introduced by Granger (1981), uses an important property of I(1) variables viz., there can be linear combinations of these variables that are I (0).

In case there indeed exist such linear combinations, then the variables are said to be co-integrated.

Suppose that there are two I (1) variables, yt and xt then yt and xt are said to be co-integrated if there exists a such that yt - xt is I (0).Co-integration: Concept Engel Granger Co-integration TestJohansen and Juselius Co-integration TestTwo different Techniques

Engle-Grangers residual-based test

The residual-based test is the earlier test for cointegration , as proposed by Engle and Granger (1987). In this procedure once all the variables, say K in number have been found to be I (1).

Steps:Run OLS regression in the single equation model The standard unit root tests are applied to the residuals

Now, if the residual is found to have no unit root by the ADF and /or PP tests, then we conclude that the variables are cointegrated; Otherwise the conclusion is that the variables are not cointegrated.

Limitation: Residual based single equation co-integration method is not able to provide the number of co-integrating vectors in the case of more than one co-integrating vector.

Johansen and Juselius Co-integration TestThe J-J test is a system method of the vector autoregressive framework (VAR) which considers all the variables as endogenous. It helps to avoid arbitrary choice of the endogenous variable in the model specification. Xt= + 1Xt-1 + 2 Xt-2 + + p-1 Xt-p+1 + Xt-p + Dt + t;With constant and deterministic trendThe J-J test is focused on estimating the long run matrix () the equilibrium is defined by Xt-p=0. For multiple co-integrating vectors, we have 1137.61 [0.005]**r=222.89 [0.026]*r=2r>214.72 [0.064]r=313.99 [0.053]r=3r=40.73 [0.393]r=40.73 [0.393]

In practice, cases 1 and 5 are rarely used. You should use case 1 only if you know that all series have zero mean. Case 5 may provide a good fit in-sample but will produce implausible forecasts out-of-sample. As a rough guide, use case 2 if none of the series appear to have a trend. For trending series, use case 3 if you believe all trends are stochastic; if you believe some of the series are trend stationary, use case 4.

Application in EviewsThe rate at which series "correct" from disequilibrium is represented by a vector of adjustment speeds, which are incorporated into the VAR model at time t through a multiplicative error-correction term yt1.

Cointegrated variables are generally unstable in their levels, but exhibit mean-reverting "spreads" (generalized by the cointegrating relation) that force the variables to move around common stochastic trends. Cointegration is also distinguished from the short-term synchronies of positive covariance, which only measures the tendency to move together at each time step. Modification of the VAR model to include cointegrated variables balances the short-term dynamics of the system with long-term tendencies.

Error Correction MechanismGranger Causality