Knowledge of ECM Model - 1

Reference: document of Stata

1.The first thing which you need understand is : cointegration
Cointegration analysis provides a framework for estimation, inference, and interpretation when the variables are not covariance stationary.
1.1 What's covarian stationary ?
       An variable is called covarian stationary if its mean and all its autocovariances are finite and do not change over time.
        Instead of being covariance stationary, it appear to be " first - difference stationary"This means that the level of a time series is not stationary but its first difference is. In general, a process whose dth difference is stationary is an integrated process of order d or I(d)
       In the first difference case, a time series can be written:
           xt = x(t-1) + et (1)
        where the et are independently and identically distributed (i.i.d) with man 0 and a finite variance stationary.
1.2
        when we have an equation: yt = a + b xt + et  where et is be the i.i.d assumption
OLS: we can consistently estimate the parameters a and b if and only if E[xtet] = 0
If b = 0 ==> no relationghip between y and x.
Granger and Newbold (1974) had proven : given a large  enough dataset, we can almost always reject the null hypothesis of the test that b=0 even though b is in fact zero.

Engle and Granger (1987) provides more intuition. Redefine yt and xt to be








Because of et is i.i.d, I(1) and  |p|<1 implies that vt and yt+alpha xt are I(0)
and xt, yt are I(1)

2. VECM framework
We use Johansen's result to establish VECM-
2.1 Determine trends in a cointegrating VECM