# ECM (error correction model): interpretation of a I(1) differentiated time serie as its “variation around its long-run trend”

I am reading Greene's "Econometric Analysis" book, and more specifically the chapter on cointegration. I do not understand his interpretation of the components of an error correction model (page 1003 from the 7th edition, chapter 21.3.2 "Error correction and VAR representations").

Here is the theoretical context: suppose the two variables $y_t$ and $z_t$ are I(1) and cointegrated and that the cointegrating vector is $[1,−θ]$. Then all three variables, $\Delta{y_t} = y_t − y_{t−1}$, $\Delta{z_t}$, and $(y_t − θz_t)$ are I(0) and the error correction model can be written as follows :

$${\Delta}y_t = \mathbf{x'}_tβ + γ ({\Delta}z_t) + λ(y_{t−1} − θz_{t−1}) + ε_t$$

Greene interprets this equation as the variation in $y_t$ around its long-run trend in terms of a set of I(0) exogenous factors $x_t$, the variation of $z_t$ around its long-run trend, and the error correction $(y_t−θz_t)$, which is the equilibrium error in the model of cointegration.

I bugged on the interpretation of ${\Delta}y_t$ being the "variation of $y_t$ around their long-term trend"(same goes for and ${\Delta}z_t$ and $z_t$). As I am trying to get a good intuition behind the error correction model and each of its components, can someone help me understand this phrase?

I had the idea of writing $y_t$ and $z_t$ as trend stationary series such as $y_t= \alpha + \beta t + ε_t$ so that $\Delta y_t=\beta + ε_t + ε_{t-1}$. Then $\Delta y_t$ would indeed represent a variation around $y_t$'s trend. However, if we write $y_t$ as $y_t= \alpha + \beta w_t + ε_t$ where $w_t$ is a I(1) serie (such as a random walk), then I do not understand why $\Delta y_t$ is the variation in $y_t$ around its long-run trend (in this example we would have $y_t=\beta \Delta w_t + ε_t + ε_{t-1}$).

• Perhaps Greene means "variation of $y_t$ around its stochastic trend", i.e. around a random walk. – Richard Hardy Mar 29 '18 at 14:39