The notion of cointegration arose out of the concern about spurious or
nonsense regressions in time series. Specifying a relation in terms of levels of the
economic variables, say , often produces
empirical results in which the R For the purpose of illustration we will consider the simple model
in which the error term has no MA part and the cointegrating parameter in
the error correction mechanism (ECM, the part in parentheses) is (1, -a).
In the no growth steady state we have and
the equilibrium shows a long run proportionality between y
- All components of x
_{t}are I(d) and - There exists a vector b = (b
_{1}, b_{2},…,b_{n}) such that bx_{t}is I(d-b), where b > 0.
Note b is called the cointegrating vector. Points to remember: - To make b unique we must normalize on one of the coefficients.
- All variables must be cointegrated of the same order. But, all variables of the same I(d) are not necessarily cointegrated.
- If x
_{t}is nx1 then there may be as many as n-1 cointegrating vectors. The number of cointegrating vectors is called the cointegrating rank.
An interpretation of cointegrated variables is that they share a common stochastic trend. Consider
in which m The term in parentheses must vanish. That is, up to some scalar b
The e - We could add lagged Dr
_{it}to the RHS of both equations without changing the interpretation of the model. - With the added terms we would have a model similar to a vector autoregression (VAR). However, if we were to estimate an unrestricted VAR then we would introduce a misspecification error.
- a
_{s}and a_{L}can be thought of as speed of adjustment parameters. - At least one of a
_{s}and a_{L}must be non-zero.
Let us explore the relationship between the error correction model (ECM) and the VAR.
Suppose we have the simple model y We can write the model as
Using Cramer's Rule
- Both variables have the same characteristic equation, in square brackets. By
multiplying through the equation for y
_{t}, say, by the inverse of [·] you will get an ARMA representation. In order for the AR part to be stationary the roots of {(1- a_{11}L)(1-a_{22}L)-a_{12}a_{21}L^{2}} must lie outside the unit circle. - If the roots of Characteristic equation's (the fraction) polynomial in L (the lag
operator) lie inside the unit circle then both z
_{t}and y_{t}are stationary (see the first point) and cannot be cointegrated. - Even if only 1 root of the characteristic equation lies outside the unit circle then both variables are explosive, so cannot be CI(1,1).
- If both roots are unity then both variables are I(2)and cannot be CI(1,1).
- If a
_{12}= a_{21}= 0 and a_{11}= a_{22}= 1 then both variables are I(1), but do not have any long run relationship, so cannot be CI. - For y
_{t}and z_{t}to CI(1,1) one root must be 1 and the other must be less than 1.
For this particular example we can show
For cointegration either a - If in a VAR the variables are CI(1,1), then an ECM exists.
- If variables are cointegrated and we wish to estimate a VAR then we must impose restrictions on the VAR coefficients.
- Suppose y
_{t}and z_{t}are cointegrated. z_{t}does not Granger cause y_{t}if no lagged values of Dz_{t-i}enter Dy_{t}and if y_{t}does not respond to deviations from long run equilibrium.
2. Estimate the parameters of the long run relationship. For example,
when y The null and alternate hypotheses are 4. If you reject the null in step 3 then estimate the parameters of the ECM
r is stationary. In the earlier section on unit roots we observed that this
was not the case. As an alternative approach one might argue that the series {e
such that u 1. The first step is to see that f 2. Fitting the PPP model to the US against BRD, J and C gives us the following result on the slope coefficient, with standard errors in parentheses:
Under the strictest interpretation of the PPP things don't look good for the modelsince the slope coefficients are clearly different from one. 3. Use the residuals from step 2 to check for unit roots. If f
The numbers in parentheses are 't' statistics. Only for Japan can we reject the null hypothesis. Now construct an ECM for the Japan-US model f You can dowload another example (a WORD file)for money and income that has both the RATS program and the output. You can take a look at the data (an EXCEL file) also. |

© Andrew J. Buck 1999 |