ERROR-CORRECTION COEFFICIENT

When I met with fellow researchers at conferences (in Malaysia), I always receive a question: must the error-correction coefficient be negative?  Recently, some have asked me: Can the error-correction coefficient be less than -1 (e.g. -1.47 or -1.82)?

I will try to answer the first one here.

Let say we have the following long-run equation that links Y to X, where both are non-stationary (and both are expressed in natural logarithm):



                                                   Yt = a + bXt + et                                             (1)
  

where a + bXtis the expected value of Yt.  Since, with the presence of cointegration, a + bXt is also considered as the LONG RUN value of Yt.   The positive etthus indicates that the actual (realized) value of Y has deviated away from et; that is, it is above its long run value.  Meanwhile, The negative etindicates that the actual Y has falled below its long run value.  The presence of cointegration or long run relation suggests that any deviation of Y from the long run value will be corrected (otherwise, why should there be a long run after all).  In other words, in the long run, Yt = a + bXt.  Accordingly, as Granger states, the presence of cointegration implies and is implied by an error-correction mechanism, upon which the dynamics of Y can be modelled using an error-correction modelling (ECM) as (assuming that X is at least weakly exogenous):



                                      ∆Yt = α + λet-1 + f(lagged  ∆Y, lagged ∆X) + vt                    (2)

Now, if e at time t-1 is positive (that is, Y is above its long run value), how should Y at time t corrects itself?  It is obvious that Y should adjust downward.  The same argument when e at time t-1 is negative (or Y below its long run value).  With positive e, Y adjusts downward the next period.  With negative e, Y adjusts upward the next period. So, it is clear that the relation between Y and the deviation from the long run is negative.  So, the coefficient of the error-correction term (λ) must be negative for the long run equation Yt = a + bXt be restored.  Hope it is clear.

When X is also viewed to be potentially endogenous (as in VAR/VECM analysis), then we can also write the dynamics of X as:


                                      ∆Xt = α + θet-1 + f(lagged  ∆Y, lagged ∆X) + wt                 (3)


The presence of cointegration requires that at least one of the two error-correction coefficients, that is in (2) and (3), must be significant.  Of course, both can be significant.  This will indicate whether Y, X or both bear the burden in making adjustment such that Yt = a + bXt.

Try to ponder this: given the long-run equation in (1), where the error-correction term is extracted, and insignificant lambda in (2), what should be the sign of theta?  For sure, theta in (3) must be significant for the long run to be re-established, but what is its expected sign?  Try to use the same argument on the restoration of the long run equation as above, you will get the answer.  For sure, it needs not be negative.  It can be positive....


I will try to answer the second question later.  Here just some hints.  Think of the stationary process.  Think of convergence to the long run.  Does it needs to converge to the long run mean (or long run equation) from one direction?  Can it be in cyclic or oscillating pattern?  If it can be, what should be the magnitude of the error-correction term?  Does it need to be constrained to less than 0 but more or equal to -1.  More specifically, given the following AR(1) process:





                                                                           Yt = α + θYt-1 + vt                                     (4)


what does stationarity mean, as pertaining to theta or the autoregressive coefficient?