Day 4: Cointegration

Robert W. Walker

2026-07-23

On Reduced and Structural Forms and Systems of Equations

What are structural forms and what are reduced forms? Are we familiar with what the rank and order conditions are?

Systems of Simultaneous Equations

Which beg questions about identification as above. Let’s briefly discuss the system that they present.

Vector AutoRegression

Choose a relevant set of lag lengths and write each variable in the system as a function of lags of itself and other variables to the chosen lengths. The key insight is that this VAR is the reduced form to some more complicated as yet unspecified structural form. But if the goal is to specify how variables related to one another and to use data to discover Granger causality and responses to impulse injected in the system.

On Granger Causation

Granger causation in a panel data setting. This is a rather complex topic because of heterogeneity.

  • An identical causal relationship could exist for each i \in N
  • No causal relationship could exist for any i \in N.
  • Something in between the above two extremes.

Matching the theoretical claim and the empirical analysis here, like everywhere else, is absolutely crucial. Also, in some ways, this is just our earliest ANOVA example but now we will use the lags of the dependent variable to establish the alternative hypotheses.

Implementation

  • Look at stationarity. We want to make certain that this holds.
  • Test a hypothesis, many forms can be implied.

Conditional on stationarity, the procedure goes like this. For example, to test the hypothesis that, for all i, x does not Granger cause y, we choose a lag length, call it k. We then regress lags of y and k lags of x on y in a model with unit specific slopes and unit specific intercepts. We do not include contemporaneous x because Granger causality is all about temporal priority. This is the unrestricted model. The hypothesis implies the restriction that all the coefficients on all of the lagged x are zero for each cross-section and for all k lags of x.

In other words, the unrestricted model is: y_{it} = \rho_{i,1} y_{i,t-1} + \rho_{i,2} y_{i,t-2} + \ldots + \rho_{i,m} y_{i,t-m} + \beta_{i,1} x_{i,t-1} + \ldots + \beta_{i,k} x_{i,t-k} + \epsilon_{it} while the restricted model is y_{it} = \rho_{i,1} y_{i,t-1} + \rho_{i,2} y_{i,t-2} + \ldots + \rho_{i,m} y_{i,t-m} + \epsilon_{it} because the null hypothesis implies that all the \beta are zero for all lags t-1 to t-k lags of x.

Backfilling the Lagged Dependent Variable problem

Let’s analyse this. This will preview dynamic interpretation in panel settings also. The long-run multiplier.

Let’s Talk about simultaneous equations and systems of equations.

Stationarity conditions

We normally restrict autoregressive models to stationary data, and then some constraints on the values of the parameters are required.

General condition for stationarity

Complex roots of 1-\phi_1 z - \phi_2 z^2 - \dots - \phi_pz^p lie outside the unit circle on the complex plane.

  • For p=1: -1<\phi_1<1.
  • For p=2:-1<\phi_2<1\qquad \phi_2+\phi_1 < 1 \qquad \phi_2 -\phi_1 < 1.
  • More complicated conditions hold for p\ge3.
  • Estimation software takes care of this.

Invertibility

  • Any MA( q ) process can be written as an AR( \infty ) process if we impose some constraints on the MA parameters.
  • Then the MA model is called “invertible”.
  • Invertible models have some mathematical properties that make them easier to use in practice.
  • Invertibility of an ARIMA model is equivalent to forecastability of an ETS model.

Invertibility

General condition for invertibility: Complex roots of 1+\theta_1 z + \theta_2 z^2 + \dots + \theta_qz^q lie outside the unit circle on the complex plane.

  • For q=1: -1<\theta_1<1.
  • For q=2: -1<\theta_2<1 \qquad \theta_2+\theta_1 >-1 \qquad \theta_1 -\theta_2 < 1
  • More complicated conditions hold for q\ge3.
  • Estimation software takes care of this.

On Equilibrium

What does equilibrium mean? Equilibration?

Cointegration

The Problem with Differencing

The notion of cointegration is a very important idea that comes out of the work of Hendry, Granger, and Engle.

Start with a static regression model:

Y_t = \beta_0 + \beta_1 X_t + \epsilon_t

If we test for non-stationarity and find that both Y_t and X_t are non-stationary, then we need to difference both series: (1 - L)Y_t and (1 - L)X_t.

We can rewrite the regression model as:

(1 - L)Y_t = \beta_0 + \beta_1(1 - L)X_t + \epsilon_t

Important

The problem: we are no longer able to make any inferences about the long-run equilibrium processes at work. Cointegration and Error Correction Models substantially bridge this gap.

Cointegration and Error Correction

The Core Idea

(1 - L)Y_t = \beta_0 + \beta_1(1 - L)X_t + \epsilon_t

Imagine Y_t and X_t are both non-stationary, and we want to analyze the long-run equilibrium relationship between them.

Engle and Granger brought together two fundamental ideas: co-integration and error correction.

  • The notion of co-integration involves a long-run relationship between two non-stationary variables.

  • Even if Y_t and X_t are both non-stationary, they may still have a linear and stationary relationship characterised by (Y_t - cX_t).

The Engle–Granger Two-Step Process

In their 1987 paper, Engle and Granger outlined the following process for assessing cointegration and, if found, estimating an error correction model:

  1. Are the two variables of interest both non-stationary?
  2. If so, is a linear combination of them non-stationary?
    • Estimate a cointegrating regression: Y_t = \hat{c}_0 + \hat{c}_1 X_t + \epsilon_t
    • Test whether the residuals \hat{\epsilon}_t = Y_t - \hat{c}_0 + \hat{c}_1 X_t are stationary.
  3. If \hat{\epsilon}_t is stationary, estimate an error correction model: \small (1 - L)Y_t = \beta_0 + \beta_1(1 - L)X_t - \alpha(Y_{t-1} - \hat{c}_0 + \hat{c}_1 X_{t-1}) + u_t where \alpha(Y_{t-1} - \hat{c}_0 + \hat{c}_1 X_{t-1}) is the error correction mechanism.
  4. Substitute \hat{\epsilon}_t = Y_t - \hat{c}_0 + \hat{c}_1 X_t and run OLS. This is the Engle–Granger two-step process.

The Adjustment Parameter \alpha

(1 - L)Y_t = \beta_0 + \beta_1(1 - L)X_t - \alpha(Y_{t-1} - \hat{c}_0 + \hat{c}_1 X_{t-1}) + u_t

Note that |\alpha| < 1.0 and \alpha should be negative. Its magnitude tells us the speed of adjustment to shocks.

If \alpha = -0.5, then after a shock, half of it goes away in the first period, and then another half of the remaining shock goes away in each subsequent period.

Interpretation and Specification Notes

In an error correction model such as

(1 - L)Y_t = \beta_0 + \beta_1(1 - L)X_t - \alpha(Y_{t-1} - c_0 - c_1 X_{t-1}) + \epsilon_t

the effect of X on Y involves two components:

  • Short-run effect — captured by \beta_1
  • Adjustment/feedback effect — captured by \alpha, indicating how much of the disequilibrium is corrected each period

Notes on ECM specification

  • There can be other independent variables in the model specification.
  • There can be more than one ECM in the same model.

Alternative (Single-Step) Estimation

The two-step process first obtains \hat{\epsilon}_t, then estimates:

(1 - L)Y_t = \beta_0 + \beta_1(1 - L)X_t - \alpha(Y_{t-1} - \hat{c}_0 - \hat{c}_1 X_{t-1}) + u_t

where the ECM uses residuals from Y_t = c_0 + c_1 X_t + \hat{\epsilon}_t.

Expanding the error correction term algebraically:

\small (1 - L)Y_t = \beta_0 + \beta_1(1 - L)X_t - \alpha Y_{t-1} + \alpha\hat{c}_0 + \alpha\hat{c}_1 X_{t-1} + u_t

\small (1 - L)Y_t = (\beta_0 + \alpha\hat{c}_0) + \beta_1(1 - L)X_t - \alpha Y_{t-1} + \alpha\hat{c}_1 X_{t-1} + u_t

This reparameterisation can be estimated in a single OLS step.

Cointegration and Exogeneity

Three Types of Exogeneity

Any time you have a co-integrated series, people will want to flip the causal arrow on you. It is possible that X influences Y, giving rise to a simultaneity bias. This is addressed through exogeneity testing.

There are three types of exogeneity:

  1. Weak: X_t \rightarrow Y_t but Y_t \not\rightarrow X_t

  2. Strong (Granger non-causality): X_t \rightarrow Y_t but Y_t \not\rightarrow X_t and Y_t \not\rightarrow X_{t-1}. If instead Y_t \rightarrow X_{t-1}, we have Granger causality.

  3. Super: Important for forecasting. Rooted in the Lucas critique — if rational agents anticipate and offset policy innovation, parameters will not be stable over time.

Testing for Weak Exogeneity

The test for weak exogeneity has three parts:

  1. Develop a credible model of X — the independent variable in question.

  2. Add the ECM term \epsilon_{t-1} from the main model to the right-hand side of that X model. If this is statistically insignificant, the first part of the test is passed.

  3. Return to the original ECM and add the residuals from step 1. Run a Hausman test (F-test) to see whether model fit has improved. If it has not, then we have weak exogeneity.

Granger Causality Testing

A variable X Granger causes Y if you can improve your forecast of Y using X over and above a model of Y explained by its own history. (Granger 1969)

Step 1 — Run a univariate model of Y based on its own history:

Y_t = \beta_0 + \beta_1 Y_{t-1} + \ldots + \beta_k Y_{t-k} + \epsilon_t

Step 2 — Add lags of X and re-estimate:

Y_t = \beta_0 + \beta_1 Y_{t-1} + \ldots + \beta_k Y_{t-k} + c_1 X_{t-1} + \ldots + c_k X_{t-k} + \epsilon_t

Conduct an F-test to see whether fit has improved.

Step 3 — If fit has improved, then X Granger causes Y.

Non-integrated ECM?

de Boef and Keele [referenced in BFHP on p. 90]

Multiple Cointegrating Vectors: Johansen

The VECM or Vector Error Correction Model

Engle–Granger is limited to a single cointegrating vector. Johansen (1988, 1991) provides a full-system maximum-likelihood approach.

Start with a k-dimensional VAR(p) in levels, then reparametrise as a Vector Error-Correction Model (VECM):

\Delta \mathbf{X}_t = \boldsymbol{\Pi}\,\mathbf{X}_{t-1} + \sum_{i=1}^{p-1}\boldsymbol{\Gamma}_i\,\Delta\mathbf{X}_{t-i} + \boldsymbol{\varepsilon}_t, \qquad \boldsymbol{\varepsilon}_t \sim \mathcal{N}(\mathbf{0},\boldsymbol{\Sigma})

The long-run impact matrix \boldsymbol{\Pi} carries all information about cointegration via its rank r:

\boldsymbol{\Pi} = \boldsymbol{\alpha}\boldsymbol{\beta}^\top

Symbol Dimension Interpretation
\boldsymbol{\beta} k \times r Cointegrating vectors (long-run relationships)
\boldsymbol{\alpha} k \times r Adjustment (loading) coefficients

Rank, Cointegration, and the Π Decomposition

The rank of \boldsymbol{\Pi} determines the number of cointegrating relationships:

Rank r Implication
r = 0 No cointegration; \boldsymbol{\Pi} = \mathbf{0} (pure VAR in differences)
0 < r < k Exactly r linearly independent cointegrating vectors
r = k \boldsymbol{\Pi} is full rank; all k series are I(0)

Key properties of the decomposition \boldsymbol{\Pi} = \boldsymbol{\alpha}\boldsymbol{\beta}^\top:

  • Each column of \boldsymbol{\beta} defines a stationary linear combination \boldsymbol{\beta}_i^\top \mathbf{X}_{t-1} \sim I(0)
  • Each row of \boldsymbol{\alpha} measures the speed at which variable j corrects toward equilibrium
  • Both \boldsymbol{\alpha} and \boldsymbol{\beta} are identified only up to an invertible r \times r rotation — normalisation restrictions are required

The Two Test Statistics

Johansen’s approach solves a reduced-rank regression via canonical correlation between \Delta\mathbf{X}_t and \mathbf{X}_{t-1} (partialling out the lagged differences). This yields ordered eigenvalues \hat{\lambda}_1 \geq \hat{\lambda}_2 \geq \cdots \geq \hat{\lambda}_k.

Trace Statistic

Tests H_0: \operatorname{rank}(\boldsymbol{\Pi}) \leq r against H_1: \operatorname{rank}(\boldsymbol{\Pi}) > r

\lambda_{\text{trace}}(r) = -T \sum_{i=r+1}^{k} \ln\!\left(1 - \hat{\lambda}_i\right)

Maximum Eigenvalue Statistic

Tests H_0: \operatorname{rank} = r against H_1: \operatorname{rank} = r+1

\lambda_{\max}(r, r+1) = -T\ln\!\left(1 - \hat{\lambda}_{r+1}\right)

  • Both statistics follow non-standard distributions (functionals of Brownian motion); critical values depend on k - r and the deterministic specification (Osterwald-Lenum 1992; MacKinnon–Haug–Michelis 1999)
  • Test sequentially: begin at r = 0 and increment until H_0 is not rejected

Deterministic Terms and Practical Considerations

Johansen (1994) defines five canonical models based on where constants and trends enter the VECM:

Case Constant in \boldsymbol{\beta}? Trend in \boldsymbol{\beta}? Typical application
I No No Artificial; series have zero mean
II Yes No Series have non-zero mean, no trend
III Unrestricted No Most common; series are trended, cointegrating relations are not
IV Yes Yes Cointegrating relations have linear trends
V Unrestricted Unrestricted Quadratic trends in levels (rare)

Further practical steps:

  1. Lag selection — use AIC/BIC/HQ on the unrestricted VAR before testing
  2. Identification — normalise \boldsymbol{\beta} (e.g., [\mathbf{I}_r \mid \mathbf{B}]) and impose economic theory restrictions; test overidentifying restrictions via likelihood-ratio tests
  3. Weak exogeneity — rows of \boldsymbol{\alpha} equal to zero imply the corresponding variable does not respond to disequilibrium
  4. Robustness — trace and \lambda_{\max} may disagree; check sensitivity to lag length and deterministic specification