Cointegration 199
described in an error-correction format described in the following
two equations:(^) ∆=ybti 10 +∆by 11 ti−−+∆cxjtjt+−dy 11 ()−−axt 11 +ett
j
n
i
n
= =
∑∑
1 1
(10.5)
(^) ∆=ybti 20 +∆by 22 ti−−+∆cxjtjt+−dy 21 ()−−axt 12 +ett
j
n
i
n
= =
∑∑
1 1
(10.6)
Equation (10.5) tells us that the changes in yt depend on■ (^) Its own past changes.
■ (^) The past changes in xt.
■ (^) The disequilibrium between xt− 1 and yt− 1 ,, (yt− 1 − axt− 1 ).
The size of the error-correction term, d 1 in equation (10.5), captures
the speed of adjustment of xt and yt to the previous period’s disequi-
librium. Equation (10.6) has a corresponding interpretation for the
error- correction term d 2.
The appropriate lag length is found by experimenting with dif-
ferent lag lengths. For each lag the Akaike information criterion
(AIC) or the Bayesian (or Schwarz) information criterion (BIC)
is calculated and the lag with the lowest value of the criteria is
employed.^11
The value of (yt− 1 – axt− 1 ) is estimated using the residuals from
the cointegrating equation (10.3), zt− 1. This procedure is only legiti-
mate if the variables are cointegrated. The error-correction term,
zt− 1 , will be stationary by definition if and only if the variables are
cointegrated. The remaining terms in the equation (i.e., the lag dif-
ference of each variable) are also stationary because the levels were
assumed nonstationary. This guarantees the stationarity of all the
variables in equations (10.5) and (10.6) and justifies the use of the
OLS estimation method.
empirical illustration of the engle-granger procedure The dividend growth
model of stock price valuation asserts that the fundamental value of a stock
(^11) For a summary of these criteria, see Appendix E.