How many error correction (EC) term to add for a single equation model?

Question

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1. What the question is asking (in plain language)

You have a single‑equation ARIMAX (or error‑correction) model

[ y_t = \beta_0+\beta_1x_{1,t}+\dots+\beta_5x_{5,t}+ \text{(error‑correction terms)} + u_t, ]

where the dependent variable (y_t) and all five regressors (x_{i,t}) are integrated of order 1, (I(1)).

The student is told that some of the series move together in the long run (they share a stochastic trend).
The questions are:

If
- (y_t,\,x_{1,t},\,x_{2,t},\,x_{3,t}) share one common stochastic trend, and
- (y_t,\,x_{4,t},\,x_{5,t}) share another common stochastic trend,
how many error‑correction (EC) terms should be put into the single equation for (y_t)?
If, in addition, (x_{2,t}) and (x_{3,t}) also share a third stochastic trend that is different from the two trends above, how many EC terms are now required?

In other words: Given the long‑run relationships, what is the rank of the error‑correction (Π) matrix that is relevant for the equation of (y_t)?

2. Step‑by‑step solution

2.1 Basic concepts

Symbol	Meaning
(k)	Number of variables in the system (here (k = 6): (y) plus five (x)’s).
(\Pi)	Long‑run impact matrix in the vector error‑correction representation (VECM).
(\text{rank}(\Pi)=r)	Number of linearly independent cointegrating vectors (i.e. number of EC terms you could have in a full VECM).
Number of stochastic trends = (k-r).
In a single‑equation ECM for (y) we only need the cointegrating vectors that involve (y). Call this number (r_y). It is not necessarily equal to the full system rank (r).

Thus we have to (i) infer the total number of stochastic trends, (ii) compute the system rank (r = k - (\text{# trends})), and finally (iii) count how many of the (r) cointegrating vectors contain (y).

2.2 Scenario 1

(y, x_1, x_2, x_3) share one trend; (y, x_4, x_5) share another trend.

Identify distinct stochastic trends
Trend A is common to ({y, x_1, x_2, x_3}).
Trend B is common to ({y, x_4, x_5}).

There is no information that these two trends are the same, so we treat them as two independent stochastic trends.
How many trends in total?
Since the two groups overlap only through (y), the two trends are distinct.
[ #\text{trends}=2. ]
System rank (the number of independent cointegrating relations in the full 6‑variable system)
[ r = k - (#\text{trends}) = 6 - 2 = 4. ]

So a full VECM would contain 4 EC terms.
How many of those 4 involve (y)?
- From Trend A we can form a cointegrating vector that includes (y, x_1, x_2, x_3).
- From Trend B we can form a cointegrating vector that includes (y, x_4, x_5).
Any other linear combination of the four vectors that is independent of the two above will not involve (y) (they will be linear combinations of the regressors only). Hence the number of distinct cointegrating relations containing (y) is

[ r_y = 2 . ]
Answer for (1)

You need 2 error‑correction terms in the single‑equation model for (y_t).
The full system rank is 4, but only two of those four EC terms are relevant for the equation of (y).

2.3 Scenario 2

In addition to the relationships above, (x_{2,t}) and (x_{3,t}) share a third stochastic trend.

Add the new trend
Trend C is common only to ({x_2, x_3}) and is independent of Trends A and B.
Total number of distinct stochastic trends

[ #\text{trends}=3 . ]
System rank

[ r = k - (#\text{trends}) = 6 - 3 = 3 . ]

The full VECM would now have three cointegrating vectors.
Which of those three involve (y)?

Vector 1 – from Trend A (contains (y, x_1, x_2, x_3)).
Vector 2 – from Trend B (contains (y, x_4, x_5)).
Vector 3 – from Trend C (contains only (x_2, x_3)).

Only the first two involve the dependent variable (y).

Therefore

[ r_y = 2 . ]
Answer for (2)

You still need 2 error‑correction terms in the equation for (y_t).
The extra stochastic trend adds a third cointegrating relation, but it does not involve (y), so it does not enter the single‑equation ECM.

3. Final answers

Situation	Total stochastic trends	System rank (r)	Cointegrating vectors that contain (y) ((r_y))	# EC terms to include in the single‑equation model
1. Two overlapping groups (A & B)	2	(6-2 = 4)	2	2
2. Add a third trend involving only (x_2, x_3)	3	(6-3 = 3)	2	2

4. Common Mistakes

Mistake	Why it’s wrong	How to avoid it
Confusing the system rank (r) with the number of EC terms needed for a single equation.	The rank tells you how many total cointegrating relations exist in the whole vector of variables. Only those that contain the dependent variable matter for its ECM.	First determine the total rank, then count how many independent cointegrating vectors actually involve the dependent variable.
Counting each variable that shares a trend as a separate EC term.	If several variables share the same stochastic trend, they generate one cointegrating relation, not one per variable.	A group of variables that moves together on one trend contributes one EC term (the linear combination that eliminates the trend).
Assuming overlapping groups automatically give more EC terms.	Overlap (e.g., (y) appears in both groups) does not multiply the number of independent relations; it just indicates that (y) participates in more than one relation.	Identify the distinct stochastic trends; each distinct trend reduces the rank by one, regardless of overlap.
Treating a trend that only involves regressors as relevant for the dependent‑variable equation.	An EC term that contains only regressors cannot be entered into the equation for (y); it would be collinear with the regressors.	After you have the full set of cointegrating vectors, drop any that do not contain the dependent variable when writing the single‑equation ECM.
Leaving out an EC term because it looks “redundant”.	Two EC terms may look similar but can be linearly independent; dropping one can miss a genuine long‑run equilibrium relationship.	Verify linear independence (e.g., by checking the rank of the matrix of cointegrating vectors) before discarding any term.

Keeping these

Original question: How many error correction (EC) term to add for a single equation model? on Cross Validated (Stats Stack Exchange), licensed CC BY-SA.