Chapter 08

Rigorous

Cointegration and Error Correction

00 · Symbol Glossary

$I(d)$integrated of order d

$X_t$ is $I(d)$ if $d$ differences produce an $I(0)$ series (weakly stationary, for our purposes) and $d-1$ differences do not. Log prices are the usual $I(1)$ candidates; returns are the $I(0)$ candidates.

$\beta$beta — cointegrating vector

A vector such that $\beta^\top Y_t$ is $I(0)$ while $Y_t$ itself is $I(1)$ . In the bivariate case $\beta$ is unique up to scale.

$\alpha$alpha — adjustment speeds

In an error-correction model, $\alpha$ (a vector or matrix) multiplies the lagged equilibrium error. It says who closes the gap, and how fast.

$\Pi$Pi — VECM long-run matrix

In $\Delta Y_t=\Pi Y_{t-1}+\cdots$ , cointegration of rank $r$ is $\Pi=\alpha\beta^\top$ with $\mathrm{rank}(\Pi)=r$ . Johansen tests that rank.

01 · Spurious Regression

A levels VAR on independent nonstationary series can look spectacular: $R^2$ near one, $t$ -statistics in the double digits, and nothing causal happening.

Definition — Spurious Regression (Granger–Newbold)

If $y_t$ and $x_t$ are independent $I(1)$ series, the levels OLS regression $y_t=\hat a+\hat b\, x_t+\hat u_t$ typically yields an $I(1)$ residual, a large $R^2$ , and $t$ -statistics that do not have a $N(0,1)$ limit. Significance is fiction.

❌ $t$-testing one random walk on another in levels

A significant $t$ of an equity index level on an unrelated $I(1)$ series is not evidence of a relationship.

Why it breaks: both series wander; OLS attributes the shared wander to $\hat b$ . The residual inherits a unit root.

Consequence: difference both series, or test whether a candidate residual is $I(0)$ (the cointegration route below). Do not publish levels $t$ -tests of independent $I(1)$ series.

02 · Cointegration

Definition — Bivariate Cointegration

$y_t$ and $x_t$ are both $I(1)$ . They are cointegrated if there exists $\beta\neq 0$ such that $z_t=y_t-\beta x_t$ is $I(0)$ . The pair shares one common stochastic trend; $z_t$ is the equilibrium error.

In a $K$ -vector $Y_t$ that is $I(1)$ , cointegration of rank $r$ means there is a $K\times r$ matrix $\beta$ of rank $r$ with $\beta^\top Y_t$ an $r$ -vector of $I(0)$ series. Then there are $K-r$ independent stochastic trends.

Example — Log spot and log futures

If the basis $s_t-f_t$ is mean-reverting while each log price is $I(1)$ , a cointegrating vector is $(1,-1)$ (or a nearby $\beta$ ). A pairs-trade residual is the same idea: the spread is the candidate $z_t$ . Whether that spread is actually $I(0)$ is an empirical claim, not a slogan.

03 · Engle–Granger Two-Step

Step-by-step — Engle–Granger

Estimate the candidate relation by OLS:

\hat z_t=y_t-\hat a-\hat b\, x_t

Test

\hat z_t

for a unit root (typically ADF). Do not use ordinary Dickey–Fuller critical values as exact — the residual-based test has its own tables (Engle–Granger / Phillips–Ouliaris) or software MacKinnon

p

-values built for cointegration residuals.

If you reject a unit root in

\hat z_t

, treat

(\hat a,\hat b)

as a cointegrating estimate and pass

\hat z_{t-1}

into a second-stage ECM. Step 1 chooses a normalization (which variable is on the left); the cointegrating space is what the test targets, not a single OLS slope by itself.

Superconsistency: under cointegration $\hat b$ converges at rate $T$ rather than the usual $\sqrt{T}$ . That fast rate is what licenses treating $\hat z_t$ as if $\beta$ were known when you pass it to the second stage.

Superconsistency is not a licence to do inference on $\hat b$ itself. The static OLS estimator carries a second-order bias whenever $\Delta x_t$ is correlated with the equilibrium error, and its limiting distribution is not mixed normal, so the reported $t$ -statistic on $\hat b$ has no standard interpretation. If the long-run coefficient is the quantity you want to test — a hedge ratio of exactly $1$ , say — estimate it with DOLS (add leads and lags of $\Delta x_t$ to the static regression) or FM-OLS, which restore asymptotic mixed normality and valid $t$ -tests. A constant and trend in the first step must also match the DGP.

Which variable is on the left?

In a bivariate system the cointegrating vector is unique up to scale, but OLS normalizes on the left-hand variable. If both series are noisy, the two possible regressions need not agree in small samples. That is one reason system estimators exist.

04 · Error Correction

If $y$ and $x$ are cointegrated, the Granger representation theorem says there is an error-correction model (ECM). A bivariate sketch:

\Delta y_t = \alpha_y\, z_{t-1} + \gamma_y\Delta y_{t-1} + \delta_y\Delta x_{t-1} + \varepsilon_{yt}

and a matching equation for $\Delta x_t$ , with $z_{t-1}=y_{t-1}-\beta x_{t-1}$ . The coefficient $\alpha_y$ is the speed at which $y$ closes the gap. If $\alpha_y<0$ when $z=y-\beta x$ , a positive gap pulls $y$ down. At least one of $\alpha_y,\alpha_x$ must be nonzero or the equilibrium does no work — in a bivariate system, often only one series error-corrects while the other adjusts weakly or not at all, depending on which market leads.

A levels VAR that omits $z_{t-1}$ is misspecified under cointegration: the long-run term is missing. A differences-only VAR omits it too, and cannot recover the levels relation.

❌ Calling any lagged spread an ‘error-correction term’

Including $y_{t-1}-\hat\beta x_{t-1}$ in a differences regression when the residual is still $I(1)$ is not error correction.

Why it breaks: the ECM representation assumes cointegration. Without it you have mixed a spurious lagged levels combination into $\Delta y_t$ .

Consequence: test cointegration first. Then estimate $\alpha$ .

05 · Johansen, Briefly

For $K>2$ , or when you do not want to choose a left-hand variable, the Johansen procedure estimates a VECM

\Delta Y_t = \Pi Y_{t-1} + \sum_{i=1}^{p-1} \Gamma_i \Delta Y_{t-i} + \varepsilon_t

where $\Pi=\sum_{i=1}^{p}A_i-I$ collects the long-run behaviour of the levels VAR. Everything hinges on $\mathrm{rank}(\Pi)=r$ :

$r$	Meaning	Correct model
$0$	$\Pi=0$ , no cointegration	VAR in differences
$0<r<K$	$\Pi=\alpha\beta^\top$ , $r$ relations, $K-r$ common trends	VECM
$K$	$Y_t$ was already $I(0)$	VAR in levels

The two statistics are built from the ordered squared canonical correlations $\hat\lambda_1\geq\cdots\geq\hat\lambda_K$ of a reduced-rank regression. Stated, not derived:

\mathrm{LR}_{\text{trace}}(r) = -T\sum_{i=r+1}^{K}\log(1-\hat\lambda_i), \qquad \mathrm{LR}_{\max}(r) = -T\log(1-\hat\lambda_{r+1})

They test different hypotheses. Trace takes $H_0:\mathrm{rank}\leq r$ against the unrestricted alternative $\mathrm{rank}=K$ . Maximum-eigenvalue takes $H_0:\mathrm{rank}=r$ against exactly one more, $\mathrm{rank}=r+1$ . In practice you run them sequentially from $r=0$ upward and stop at the first non-rejection — which means the overall error rate is not the nominal one you read off each step, and the two statistics can disagree about $r$ .

❌ Comparing a trace statistic to a $\chi^2$ table

The Johansen statistics are likelihood ratios, but their null distributions are not $\chi^2$ .

Why it breaks: under the null there are $K-r$ unit roots in the system. The limits are functionals of a $(K-r)$ -dimensional Brownian motion — the multivariate cousin of the Dickey–Fuller distribution of Chapter 04. The tabulated values depend on $K-r$ and on which of Johansen’s five deterministic specifications you chose (no terms; constant restricted to the cointegrating relation; unrestricted constant; trend restricted to the relation; unrestricted trend).

Consequence: use the Johansen–MacKinnon–Haug–Michelis tables that match your $K-r$ and your deterministic case, and state which case you ran. A degrees-of-freedom count is not available here.

What Johansen estimates is a space, not a vector

The factorization $\Pi=\alpha\beta^\top$ is not unique. For any nonsingular $r\times r$ matrix $M$ , $(\alpha M)(\beta M^{-\top})^\top=\alpha\beta^\top$ gives the identical $\Pi$ . Only the cointegrating space — the column span of $\beta$ — is identified by the data; individual columns need $r^2$ normalization restrictions before they can be read as economic relations. For $r=1$ this collapses to the “unique up to scale” statement made earlier in this chapter. For $r\geq 2$ , a printed $\hat\beta$ is a software normalization choice, and interpreting one of its columns as the long-run relation without imposing and testing restrictions is an error.

This course does not derive those statistics or quote their critical values. Treat Johansen as a system rank test plus a jointly estimated $(\alpha,\beta)$ . Use it when Engle–Granger’s single-equation normalization is the wrong tool.

06 · Exercises

EXERCISE 8.1

Independent random walks; the residual inherits a unit root.

Spurious regression: $\hat u_t$ is $I(1)$ , OLS $t$ -statistics do not have a standard normal limit, and $R^2$ can look large. Differencing, or a cointegration test on the residual, is the check.

You regress one independent random walk on another in levels and get $t=8$ and $R^2=0.7$ . What went wrong, and what should you look at next?

EXERCISE 8.2

$z_t$ should be $I(0)$ ; $\beta$ is unique up to scale.

$z_t=s_t-0.5\, f_t$ . Any nonzero scale of $(1,-0.5)$ spans the same cointegrating space.

If $s_t$ and $f_t$ are $I(1)$ and $s_t-0.5\, f_t$ is $I(0)$ , write a cointegrating residual $z_t$ .

EXERCISE 8.3

The lagged gap belongs in the $\Delta$ equations.

An error-correction term $\alpha(y_{t-1}-\beta x_{t-1})$ (plus short-run $\Delta$ lags). A plain VAR in differences leaves the long-run relation out.

In an ECM for cointegrated $(y,x)$ , what extra regressor appears next to lagged differences?

07 · Chapter Summary

Idea	Content
$I(1)$	levels nonstationary; $\Delta$ stationary
Spurious regression	independent $I(1)$ series look related in levels OLS
Cointegration	$\beta^\top Y_t$ is $I(0)$
Engle–Granger	OLS levels, then a residual unit-root test with the right tables
$\hat b$	$T$ -consistent, but no standard $t$ ; use DOLS or FM-OLS to test it
ECM	$\Delta Y$ depends on the lagged equilibrium error
Johansen	rank of $\Pi=\alpha\beta^\top$ ; trace vs max-eigenvalue test different $H_0$
$\beta$	only the cointegrating space is identified

Next: Chapter 09 — Spectral Analysis and Linear Filters, the same second-order structure read in the frequency domain.