Additional Steps to ECM and ARDL

Ednaldo Silva’s blog dated August 21, 2025, contains three sequential equations about the profit rate. My purpose is to demonstrate the connection between his logarithmic 2nd equation and the 3rd equation, referencing the ECM and ARDL econometric models. I used the same econometric reference he cited and adopted some notations of the textbook by Dimitrios Asteriou and Stephen Hall.

1. Long-term identity (balance)

0 = 𝒍𝒏 s(𝘵) – 𝒍𝒏 k(𝘵) – 𝒍𝒏 r(t) ⇒ 0 = e(t), 

(e(t) is the Error-Correction-Term (ECT)).

If 𝒍𝒏 s(𝘵) and 𝒍𝒏 k(𝘵) are both non-stationary (i.e., integrated processes, called I(1)),

then their difference (𝒍𝒏 r(t) = 𝒍𝒏 s(𝘵) – 𝒍𝒏 k(𝘵)) can still be stationary.

→ This would be a cointegration relationship.

(Asteriou and Hall, Equation (17.2))

Meaning (economic):

• Even if profit margins and capital intensity each contain trends, their combination (the profit rate r(t)) could remain stable around a long-term mean.
• If no cointegration is present, then 𝒍𝒏 r(t) would also be non-stationary.

Procedure:

1. Test with integration theory whether 𝒍𝒏 s(𝘵) and 𝒍𝒏 k(𝘵) are both integrated (I(1)).

2. If yes, then from 𝒍𝒏 r(t) = 𝒍𝒏 s(𝘵) – 𝒍𝒏 k(𝘵) it follows that the profit rate r(t) can be not integrated

(called I(0)) → i.e., stationary, despite non-stationary individual variables.

2. Why a dynamic model?

In actual data, the identity does not apply exactly in every period (measurement errors, inertias).

So, we allow short-term deviations around the balance.

3. Start with an ARDL for:

Y(t) = 𝒍𝒏 r(t), with regressors: X₁(t) = 𝒍𝒏 s(𝘵), X₂(t) = 𝒍𝒏 k(𝘵)

𝒍𝒏 r(t) = α₀ + α₁ 𝒍𝒏 r(t-1) + γ s₀ 𝒍𝒏 s(𝘵) + γ k₀ 𝒍𝒏 k(𝘵) + γ s₁ 𝒍𝒏 s(t-1) + γ k₁ 𝒍𝒏 k(t-1) + u(t)

Because: Y(t) = α₀ + α₁ Y(t-1) + γ₀ X(t) + γ₁ X(t-1)

⇒ Y(t) = α₀ + α₁ Y(t-1) + γ₀ X₁(t) + γ₀ X₂(t) + γ₁ X₁(t-1) + γ₁ X₂(t-1),

(Asteriou and Hall, Equation (17.11))

4. Rewrite for changes (ARDL to ECM):

Subtract 𝒍𝒏 r(t−1) from both sides and collect terms:

Δ 𝒍𝒏 r(t) = γ s₀ Δ 𝒍𝒏 s(𝘵) + γ k₀ Δ 𝒍𝒏 k(𝘵) − π[𝒍𝒏 r(t-1) − β₀ − β_s 𝒍𝒏 s(t-1) − β_k 𝒍𝒏 k(t-1)] + u(t)

where: γ s₀ = β, γ k₀ = −γ, π = (1 − α₁) > 0,

β_s = (γ s₀ + γ s₁) / (1 − α₁),

β_k = (γ k₀ + γ k₁) / (1 − α₁),

β₀ = α₀ / (1 − α₁)

(Asteriou and Hall, Equation (17.16))

Because: Δ Y(t) = γ₀ X(t) – (1-a) [Y(t-1) – (α₀ / (1 − α₁)) – ((γ₀ + γ₁) / (1 − α₁)) X(t-1)] + u(t), 

(Asteriou and Hall, Equation (17.17))

5. Build in long-term restrictions (the identity):

From 𝒍𝒏 r(t) = 𝒍𝒏 s(𝘵) − 𝒍𝒏 k(𝘵) follows for the long-term coefficients:

β₀ = 0, β_s = 1, β_k = −1

Because: Y = 𝒍𝒏 r = 𝒍𝒏 s – 𝒍𝒏 k = X₁ – X₂ → 𝒍𝒏 r = 0 ⇒ r = 1

⇒ Y(t) = β₀ + β₁·X(t) = (X₁(t) – X₂(t))

Insert the equation above and convert the bracket:

−π [𝒍𝒏 r(t-1) − (𝒍𝒏 s(t-1) − 𝒍𝒏 k(t-1))] = π [𝒍𝒏 s(t-1) − 𝒍𝒏 k(t-1) − 𝒍𝒏 r(t-1)]

6. Naming of the signs and cleaning up:

Write α = π > 0, β = γ s₀, −γ = γ k₀

Then we obtain the ECM result:

Δ 𝒍𝒏 r(t) = α [𝒍𝒏 s(t-1) − 𝒍𝒏 k(t-1) − 𝒍𝒏 r(t-1)] + β Δ 𝒍𝒏 s(𝘵) − γ Δ 𝒍𝒏 k(𝘵) + u(t)

([𝒍𝒏 s(t-1) − 𝒍𝒏 k(t-1) − 𝒍𝒏 r(t-1)] = ECT(t-1))

Interpretation:

If r(t) is too high, relative to s(𝘵) / k(𝘵), then the bracket is negative

→ at α > 0, Δ 𝒍𝒏 r(t) falls: Return towards balance.

β and γ are short-term effects of the current change in s(𝘵) or k(𝘵).

7. Note:

ARDL + long-term coefficient restriction (= identity) lead to ECM, where α measures the speed of adjustment and the bracket corresponds to the deviation from the long-term ratio.

References

Asteriou, D., & Hall, G. (2016). Applied Econometrics, 3rd Edition. Palgrave Macmillan. Chapter 17 covers ARDL and ECM.

Silva, Ednaldo (2025). “Profit Rate Dynamics via ARDL: A Minimum Specification,” EdgarStat Blog: https://edgarstat.com/blog/profit-rate-dynamics-via-ardl-a-minimal-specification/