Cointegration analysis in practice

Engle-Granger 2 step, and 1 step, estimation procedures

Cointegration in Single Equations:

Cointegration -  evidence of long-run or equilibrium relationships

With cointegration the residuals from a regression are stationary.

Tested informally and formally for cointegration

                Formal Tests include

 (1) Cointegrating Regression Durbin Watson (CRDW) test

 (2) Cointegrating Regression Dickey Fuller (CRDF) test 

 

Summary of Lecture

                (1) Introduce Granger Representation Theorem.

                                - relates cointegration to Error Correction Models

                (2) Suggest different ways of estimating long run coefficients and short run models

                (3) Multivariate regressions and testing for cointegration.

Cointegration: The usefulness of ECMs

Error correction mechanisms are useful for representing the short run relationships between variables.

Another way of saying we are not always at equilibrium.

The error correction model allows us to return to zero i.e. corrects for deviations from equilibrium.

It relates deviations from equilibrium to changes in the dependent variable i.e. the means of correcting for errors.

The estimation of two variable ECMs

However, are we certain an ECM relationship exists for variables? Does cointegration help?

                                                                y_t = β₀ + β₁x_t + u_t             

Granger Representation Theorem                                                                          

Provided two time series are cointegrated, the short-term disequilibrium relationship between them can always be expressed in the error correction form.

Cointegration and ECMs

Granger Representation Theorem suggests that if we have cointegration then an ECM exists

Dy_t = lagged (Dy_t , Dx_t) – lu_t-1 + e_t

u_t-1              is the disequilibrium error y_t = β₀ + β₁x_t + u_t

l             is the short-run adjustment parameter

This is an important result since it is justification for using ECM.

 If y_t and x_t are cointegrated then the disequilibrium errors u_t will be stationary.

This means there is a force pulling the residual errors towards zero.

Dy_t = lagged (Dy_t , Dx_t) – lu_t-1 + e_t

Notice that all first differenced variables are I(0).

Disequilibrium errors (u_t-1 ) also need to be I(0). 

This is the case when y_t and x_t are cointegrated.

Exact lags are not specified by the Granger Representation Theorem. Specification determined by general to specific approach.

Since Dx_t-1 is an I(0) variable so is Dx_t

Hence it is possible to incorporate unlagged values of D x_t

(but may then need to use Instrumental Variables regression).

Estimating ECMs using Cointegration

How do we obtain an error correction model?

Engle-Granger Two-Step approach

(1) Estimate long run relationship between y_t and x_t

                (2) Incorporate residuals in a short run model

Engle-Granger Two-Step approach

 (1) Estimate long run relationship between y_t and x_t

y_t = β₀ + β₁x_t + u_t

                - When there is cointegration we can be confident that  β₀ and β₁ will not be biased (in large samples). 

As Stock suggested β₀ and  β₁ are consistent.

Also superconsistent. We can ignore dynamic terms.

Now use the residuals from the ‘cointegrating regression’ to test for cointegration (i.e. the existence of a long run equilibrium relationship)

Use the residuals of the estimated long run relationship, test (using DF/ADF statistics) whether or not u is STATIONARY

Note: must use special tabulated critical values for CRDF/CRADF tests.

If the residuals are stationary, then we can conclude that the series are COINTEGRATED

Engle-Granger Two-Step approach

    (2) Incorporate residuals in a short run model

          We take the residuals from the estimated static equation u_t-1 and incorporate them into the short run model.

Dy_t = lagged (Dy_t , Dx_t) – l u_t-1 + e_t

                We consequently estimate this regression.

                We can do so by OLS since all the variables are stationary.

                We should obtain the estimated coefficient l

Engle-Granger Two Step

Problems with the Engle-Granger Two Step

                These are concentrated on the first step.

                 - estimating the static OLS model.

We suggested that OLS estimates of cointegrating regressions will be unbiased in large samples (consistent).

However there may be bias in small samples (the samples we use).

 If there is bias in the first step, this will spillover on to the second step.

Typically residuals are only used to test cointegration

One suggestion is that long run parameters should be estimated using  methods unbiased in small samples, the implied residuals derived and then the short run model estimated.

Engle Granger Approach becomes

(1) Use AutoRegressive Distributed Lag (ARDL) method to estimate parameters

                i.e. within a dynamic model

(2) Derive the residuals errors from the long run model

u_t = y_t - β₀ - β₁x_t

(3) Incorporate residuals in the error correction model

Dy_t = lagged (Dy_t , Dx_t) – l u_t-1 + e_t

Alternative suggestion is that short run and long run parameters should be estimated in a single step to avoid bias estimates (in small samples).

Banerjee, Dolado, Hendry and Smith (1986) method

Dy_t = lagged (Dy_t , Dx_t) – lu_t-1 + e_t

Dy_t = lβ₀  + lagged (Dy_t , Dx_t) – l y_t-1 + l β₁x_t-1  + e_t

where   u_t  = y_t - β₀ - β₁x_t

Simulation studies of the properties of this estimator, suggest that in small samples Banerjee et al. approach performs better than Engle-Granger method.

Banerjee, et al. (1986) approach

Dy_t = lβ₀  + lagged (Dy_t , Dx_t) – l y_t-1 + l β₁x_t-1  + e_t

where   u_t  = y_t - β₀ - β₁x_t

Can check cointegration by testing the residuals e_t for stationarity

Although there are two I(1) variables in this equation a linear combination should cointegrate to produce a stationary relationship.

Consequently all variables (or a combination of variables)  will be I(0) and inference can proceed as normal.

Multivariate Cointegration Tests
 

Johansen Approach

We have concentrated on the bivariate case y_t and x_t.

                There can only be one cointegrating relationship between these variables.

Is this the case when there are three variables?

                It may be the case that there is more than one relationship.

Where we have variables y_t , x_t and  z_t.

                Johansen approach not only examines if y_t , x_t and  z_t cointegrated.

                But also if y_t cointegrates with x_t on its own and y_t cointegrates with z_t  on its own. 

                   

Single Equation Approach

Dy_t = lagged (Dy_t , Dx_t) – lu_t-1 + e_t

Soren Johansen Approach

Can test for the number of cointegrating relationships.

Assuming    y_t cointegrates with x_t                 = LR1

                          y_t cointegrates with z_t             = LR2

Short run model becomes

                Dy_t = lagged (Dy_t Dx_t Dz_t ) – l ₁₁LR1_t-1 - l ₁₂LR2_t-1 + e_1t

                Dx_t = lagged (Dy_t Dx_t Dz_t ) – l ₂₁LR1_t-1 - l ₂₂LR2_t-1 + e_2t

                Dz_t = lagged (Dy_t Dx_t Dz_t ) – l ₃₁LR1_t-1 - l ₃₂LR2_t-1 + e_3t

 

Testing for, and estimating, a cointegrating relationship

• Pretest the variables for the their order of integration
• Estimate the Cointegration Regression
• Check whether there is a cointegrating (i.e. long run equilibrium) relationship
• If so, estimate the dynamic error correction model
• Assess model adeuquacy 

Pretest the variables for their order of integration

• By definition, cointegration necessitates that the variables be integrated of the same order
• Use of DF or ADF tests to determine the order of integration
• If variables are I(0) - Standard Time Series Methods
• If variables are integrated of different order (one I(0), one I(0) or I(2) etc) then it is possible to conclude that the two variables are not cointegrated
• If the variables are I(1), or are integrated of the same order, go on

Assess model adequacy and obtain a parsimonious final specification

Assess if the ECM model you have estimated is misspecified using standard diagnostic tests. 

If the model is not misspecified, use a general –to – specific modelling approach to obtain a parsimonious final model.

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