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Chapter 21 — Error Correction Models (ECM)

In previous chapters:

We now bring these ideas together.

The answer is given by the Error Correction Model (ECM).

This chapter follows and extends the structure in your notes. :contentReference[oaicite:0]{index=0}

Learning Objectives

By the end of this chapter, you should be able to:

21.1 From Cointegration to Adjustment

Suppose two variables are cointegrated:

yt=α+βxt+ety_t = \alpha + \beta x_t + e_t

Then:

et=yt−α−βxte_t = y_t - \alpha - \beta x_t

measures deviations from the long-run equilibrium relationship.

21.2 Why an Adjustment Mechanism Is Needed

If the variables drift apart, economic forces may push them back toward equilibrium.

Examples:

21.3 The Error Correction Model

The ECM combines:

Define:

et−1=yt−1−α−βxt−1e_{t-1} = y_{t-1} - \alpha - \beta x_{t-1}

Then the ECM is:

Δyt=γ0+γ1Δxt+λet−1+ut\Delta y_t = \gamma_0 + \gamma_1 \Delta x_t + \lambda e_{t-1} + u_t

21.4 Interpreting the ECM

Each component has a distinct interpretation.

Short-Run Effect

γ1Δxt\gamma_1 \Delta x_t

captures the immediate effect of changes in xtx_t on changes in yty_t.

Error Correction Term

λet−1\lambda e_{t-1}

captures adjustment toward equilibrium.

21.5 The Sign of the Adjustment Coefficient

The sign of λ\lambda is crucial.

Why?

Suppose:

et−1>0e_{t-1} > 0

meaning:

yt−1>α+βxt−1y_{t-1} > \alpha + \beta x_{t-1}

Then a negative λ\lambda pushes:

Δyt<0\Delta y_t < 0

helping restore equilibrium.

21.6 Speed of Adjustment

The magnitude of λ\lambda measures how quickly adjustment occurs.

Examples:

λ\lambdaInterpretation
-0.1slow adjustment
-0.5moderate adjustment
-1rapid adjustment

21.7 Short-Run vs Long-Run Dynamics

ECM separates two different forces.

This is one of the major strengths of ECM.

21.8 Estimating ECM in Gretl

We now estimate an ECM using the same data as in Chapter 20.

This follows your notes closely.

Step 1: Estimate Long-Run Relationship

Estimate:

ols aus const usa

Step 2: Save Residuals

Save the residuals from the long-run regression:

series ehat = $uhat

These residuals estimate deviations from equilibrium.

Step 3: Construct Differenced Variables

Create first differences:

series d_usa = diff(usa)
series d_aus = diff(aus)

Create lagged residuals:

series ehat_1 = ehat(-1)

Step 4: Estimate the ECM

Estimate:

ols d_aus const d_usa ehat_1

Example Output

Model 2: OLS, using observations 1970:2-2000:4 (T = 123)
Dependent variable: d_aus

             coefficient   std. error   t-ratio   p-value 
  --------------------------------------------------------
  const        0.211535    0.0710333     2.978    0.0035   ***
  d_usa        0.567700    0.0983739     5.771    6.29e-08 ***
  ehat_1      −0.138852    0.0426256    −3.257    0.0015   ***
Δaust^=0.212+0.568Δusat−0.139ehatt−1\widehat{\Delta aus_t} = 0.212 + 0.568 \Delta usa_t - 0.139 ehat_{t-1}
[GRETL Screenshot Placeholder: ECM estimation output]

Interpreting the Output

Coefficient on Δusat\Delta usa_t

0.5680.568

This measures the short-run effect.

Coefficient on ehatt−1ehat_{t-1}

−0.139-0.139

This is the adjustment coefficient.

Since the coefficient is:

there is evidence of stable equilibrium adjustment.

21.9 Why ECM Matters

ECM solves a major problem in time series analysis.

Recall the tension:

ApproachProblem
levels regressionspurious regression
differencingloss of long-run information

ECM combines both.

21.10 Connection to ARDL

Recall the ARDL(1,1) model:

yt=α+ϕyt−1+β0xt+β1xt−1+uty_t = \alpha + \phi y_{t-1} + \beta_0 x_t + \beta_1 x_{t-1} + u_t

This can be rewritten as:

Δyt=γΔxt+λ(yt−1−θxt−1)+ut\Delta y_t = \gamma \Delta x_t + \lambda (y_{t-1} - \theta x_{t-1}) + u_t

21.11 ECM and Cointegration

ECM is meaningful only when cointegration exists.

Thus:

21.12 Residual Diagnostics

As with all dynamic models, residuals should behave like white noise.

After estimating the ECM:

Gretl Menu

From model window:

Tests → Autocorrelation

[GRETL Screenshot Placeholder: ECM residual diagnostics]

21.13 Common Pitfalls

21.14 Looking Ahead

So far, we have focused mainly on bivariate relationships.

In the next part of the book, we move to multivariate systems:

where multiple variables interact dynamically over time.

Key Takeaways

Concept Check

Basic

  1. What is an Error Correction Model (ECM)?

  2. What is the role of the error correction term?

  3. What does the coefficient λ\lambda represent?

Intuition

  1. Why do cointegrated variables require an adjustment mechanism?

  2. What does it mean for deviations from equilibrium to be temporary?

  3. Explain the “rubber band” analogy in the context of ECM.

Structure

  1. What are the two main components of an ECM?

  2. What does Δxt\Delta x_t capture?

  3. What does et−1e_{t-1} capture?

Short-Run vs Long-Run

  1. What is the difference between:

  1. Why is it important to distinguish between the two?

Stability

  1. Why must the adjustment coefficient λ\lambda be negative?

  2. What happens if λ>0\lambda > 0?

Challenge

  1. Suppose λ=−0.9\lambda = -0.9.

Interpretation & Practice

  1. A model shows:

  1. The error correction term is:

  1. The error correction term is positive.

    • What does this imply?

  2. A model shows:

  1. Why is ECM only valid when variables are cointegrated?

  2. If variables are not cointegrated, what happens to the error correction term?

  1. How is ECM related to ARDL models?

Economic Interpretation

  1. Suppose consumption and income are cointegrated.

Challenge

  1. A model fits well in differences but ignores the error correction term.

Numerical Practice

ECM Interpretation

  1. Suppose:

Δyt=0.5Δxt−0.4et−1\Delta y_t = 0.5 \Delta x_t - 0.4 e_{t-1}

Adjustment Speed

  1. If:

λ=−0.2\lambda = -0.2

  1. If:

λ=−0.8\lambda = -0.8

Sign Interpretation

  1. Suppose:

λ=0.3\lambda = 0.3

Model Interpretation

  1. Suppose ECM estimation gives:

Diagnostics

  1. Residuals from ECM show autocorrelation.

Challenge

  1. Suppose:

  1. Suppose:

  1. You estimate an ECM for exchange rate and price level.

You find:

Appendix 21A — Deriving the ECM from an ARDL Model

This appendix shows how ECM arises naturally from an ARDL model.

A.1 Start with ARDL(1,1)

Consider:

yt=α+ϕyt−1+β0xt+β1xt−1+uty_t = \alpha + \phi y_{t-1} + \beta_0 x_t + \beta_1 x_{t-1} + u_t

A.2 Subtract yt−1y_{t-1}

Subtract yt−1y_{t-1} from both sides:

yt−yt−1=α+ϕyt−1+β0xt+β1xt−1−yt−1+uty_t - y_{t-1} = \alpha + \phi y_{t-1} + \beta_0 x_t + \beta_1 x_{t-1} - y_{t-1} + u_t

Since:

Δyt=yt−yt−1\Delta y_t = y_t - y_{t-1}

we obtain:

Δyt=α+(ϕ−1)yt−1+β0xt+β1xt−1+ut\Delta y_t = \alpha + (\phi - 1)y_{t-1} + \beta_0 x_t + \beta_1 x_{t-1} + u_t

A.3 Introduce Δxt\Delta x_t

Add and subtract β0xt−1\beta_0 x_{t-1}:

Δyt=α+(ϕ−1)yt−1+β0(xt−xt−1)+(β0+β1)xt−1+ut\Delta y_t = \alpha + (\phi - 1)y_{t-1} + \beta_0(x_t - x_{t-1}) + (\beta_0 + \beta_1)x_{t-1} + u_t

Recognize:

Δxt=xt−xt−1\Delta x_t = x_t - x_{t-1}

Therefore:

Δyt=α+β0Δxt+(ϕ−1)yt−1+(β0+β1)xt−1+ut\Delta y_t = \alpha + \beta_0 \Delta x_t + (\phi - 1)y_{t-1} + (\beta_0 + \beta_1)x_{t-1} + u_t

A.4 Rearranging

Factor the level terms:

Δyt=α+β0Δxt+(ϕ−1)[yt−1−β0+β11−ϕxt−1]+ut\Delta y_t = \alpha + \beta_0 \Delta x_t + (\phi - 1) \left[ y_{t-1} - \frac{\beta_0 + \beta_1}{1-\phi}x_{t-1} \right] + u_t

Define:

λ=ϕ−1\lambda = \phi - 1

and:

θ=β0+β11−ϕ\theta = \frac{\beta_0 + \beta_1}{1-\phi}

Then:

Δyt=α+β0Δxt+λ(yt−1−θxt−1)+ut\Delta y_t = \alpha + \beta_0 \Delta x_t + \lambda (y_{t-1} - \theta x_{t-1}) + u_t

A.5 Final Interpretation

The ECM contains:

ComponentInterpretation
Δxt\Delta x_tshort-run effect
yt−1−θxt−1y_{t-1} - \theta x_{t-1}disequilibrium term
λ\lambdaspeed of adjustment
Applied Time Series
Chapter 20 — Cointegration and Long-Run Relationships
Applied Time Series
Part VI Capstone — Spurious Regression, Cointegration, and Dynamic Relationships