Cointegration

A single integrated series accumulates its shocks and drifts, but a linear combination of several integrated series can cancel the common trends and revert to a level. That combination is cointegration, and its stationary spread is the object a statistical arbitrage strategy trades. This post builds the error-correction representation from a vector autoregression, splits it into the cointegrating vectors and the adjustment speeds, and derives the two tests that certify the rank [1], [2].

#Cointegration and error correction

::: definition [def

] A series is $I(0)$ when it is stationary with a positive finite long-run variance, and $I(d)$ when its $d$th difference is $I(0)$ but its $(d-1)$th is not. An $I(1)$ series has a unit root, accumulates shocks, and has variance growing linearly in $t$. :::

::: definition [def

] An $I(1)$ vector $y_t\in\mathbb{R}^n$ is cointegrated with rank $r$ when there is an $n\times r$ matrix $\beta$ of full rank with $\beta^{\top}y_t$ stationary. Each column of $\beta$ is a cointegrating vector, a linear combination that cancels the common trends and leaves a stationary spread. :::

The link to a vector autoregression is exact. Let $y_t=\sum_{i=1}^p A_i y_{t-i}+\varepsilon_t$ with white noise $\varepsilon_t$ and $A(z)=I_n-\sum_i A_i z^i$.

::: theorem [thm

] The order-$p$ autoregression reparametrises identically as

If $y_t$ is $I(1)$ then $\operatorname{rank}\Pi=r<n$ and $\Pi=\alpha\beta^{\top}$ with $\alpha,\beta$ of full rank $r$; the columns of $\beta$ are the cointegrating vectors and the rows of $\alpha$ the speeds at which each equation corrects past disequilibrium. When $r=n$ the level is already stationary; when $r=0$ no stationary combination exists and the system is a pure unit-root vector. :::

::: proof Add and subtract to telescope the levels into differences. From $y_t=\sum_{i=1}^p A_i y_{t-i}+\varepsilon_t$ subtract $y_{t-1}$ and write each $y_{t-i}=y_{t-1}-\sum_{m=1}^{i-1}\Delta y_{t-m}$. Collecting the coefficient of $y_{t-1}$ gives $\sum_{i=1}^p A_i-I_n=-A(1)=\Pi$, and collecting the coefficient of $\Delta y_{t-i}$ gives $-\sum_{j=i+1}^{p}A_j=\Gamma_i$, which is Equation (1). Since $\Delta y_t$, the $\Delta y_{t-i}$, and $\varepsilon_t$ are stationary, the term $\Pi y_{t-1}$ must be stationary though $y_{t-1}$ is $I(1)$, so $\Pi$ has reduced rank $r=\operatorname{rank}\Pi<n$. Take the rank factorisation $\Pi=\alpha\beta^{\top}$ with $\alpha,\beta$ of full column rank $r$; left-multiplying $\Pi y_{t-1}=\alpha(\beta^{\top}y_{t-1})$ by a left inverse of $\alpha$ shows $\beta^{\top}y_{t-1}$ is stationary, so the columns of $\beta$ are the $r$ cointegrating vectors. :::

::: remark This is the necessary direction. The converse, that a rank-$r$ $\Pi$ makes $y_t$ exactly $I(1)$ with $r$ cointegrating relations and $n-r$ common trends, is the Granger representation theorem and needs $\alpha_{\perp}^{\top}\Gamma\beta_{\perp}$ nonsingular, with $\Gamma=I_n-\sum_{i=1}^{p-1}\Gamma_i$ and $\alpha_{\perp},\beta_{\perp}$ the orthogonal complements of $\alpha,\beta$, which rules out $I(2)$ behaviour. :::

The factorisation separates levels from dynamics. The columns of $\beta$ are the stationary combinations, $\beta^{\top}y_t$ the disequilibrium each carries, and $\alpha$ the rate at which each equation pulls back toward it. The directions orthogonal to $\alpha$ carry what does not revert.

::: proposition [prop

] Let $\alpha_\perp$ be a full-rank $n\times(n-r)$ orthogonal complement of $\alpha$, so $\alpha_\perp^{\top}\alpha=0$. Then $\alpha_\perp^{\top}y_t$ carries the $n-r$ common stochastic trends of the system and is not mean reverting. :::

::: proof Premultiply Equation (1) by $\alpha_\perp^{\top}$. The error-correction term drops because $\alpha_\perp^{\top}\alpha=0$, leaving $\alpha_\perp^{\top}\Delta y_t=\sum_i\alpha_\perp^{\top}\Gamma_i\Delta y_{t-i}+\alpha_\perp^{\top}\varepsilon_t$, a stationary series with no level feedback. Cumulating, $\alpha_\perp^{\top}y_t$ is a partial sum of those stationary increments, and under the Granger representation condition its long-run variance is nonsingular, so it is $I(1)$ and supplies the $n-r$ trends that the $r$ stationary combinations $\beta^{\top}y_t$ do not span [3]. :::

#Engle-Granger

The simplest estimator fixes one spread by a regression in levels, and the unit-root theory of the previous post makes it converge unusually fast.

::: theorem [thm

] Let $y_{0t}=\beta^{\top}y_{1t}+u_t$ with $y_{1t}$ a $k$-vector $I(1)$ process and $u_t$ stationary. The ordinary least squares estimator is super-consistent, $T(\hat\beta-\beta)=O_p(1)$, converging at rate $T$ rather than $T^{1/2}$. :::

::: proof $\hat\beta-\beta=\bigl(\sum y_{1t}y_{1t}^{\top}\bigr)^{-1}\sum y_{1t}u_t$. The normalised design satisfies $T^{-2}\sum y_{1t}y_{1t}^{\top}\cvW\int_0^1 BB^{\top}$ for a $k$-vector Brownian motion $B$, the matrix analogue of the first functional in the unit-root post, while $T^{-1}\sum y_{1t}u_t=O_p(1)$. The ratio gives $T(\hat\beta-\beta)\cvW\bigl(\int BB^{\top}\bigr)^{-1}\!\int B\,\dd B_u$, an $O_p(1)$ limit, so the error is of order $T^{-1}$. :::

::: remark Cointegration is tested by a Dickey-Fuller statistic on the residual $\hat u_t$. Because $\beta$ is estimated, the null limit is not the Dickey-Fuller law but a residual functional whose quantiles depend on $k$, the Engle-Granger, equivalently Phillips-Ouliaris, critical values [4]. :::

Engle-Granger cointegration test
  input   I(1) series y0 and k-vector of I(1) regressors y1
  1  regress y0 on y1 by least squares, leaving residual u_hat
  2  apply the augmented Dickey-Fuller statistic to u_hat
     with no deterministic term
  3  reject no cointegration when the statistic is below the
     Phillips-Ouliaris critical value for k regressors

#Johansen

Engle-Granger fixes one spread by regression. Johansen estimates the rank $r$ and the whole cointegrating matrix $\beta$ at once by Gaussian maximum likelihood on the error-correction system.

::: theorem [thm

] Concentrating out the lagged differences leaves residuals $R_{0t},R_{1t}$ with product moments $S_{ij}=T^{-1}\sum_t R_{it}R_{jt}^{\top}$. Gaussian maximum likelihood reduces to the ordered eigenvalues $\hat\lambda_1\ge\dots\ge\hat\lambda_n$ of $\lvert\lambda S_{11}-S_{10}S_{00}^{-1}S_{01}\rvert=0$, and

Under the rank-$r$ null $\mathrm{LR}_{\mathrm{tr}}(r)$ converges to a functional of an $(n-r)$-dimensional Brownian motion. :::

::: proof The $\hat\lambda_i$ are squared sample canonical correlations between $\Delta y_t$ and $y_{t-1}$ after the lags are removed. The $r$ largest are $O_p(1)$ and identify genuine cointegrating directions; the smallest $n-r$ are $O_p(T^{-1})$, so $-T\ln(1-\hat\lambda_i)=T\hat\lambda_i+o_p(1)$. On the non-cointegrated directions the $T\hat\lambda_i$ are eigenvalues of a matrix in the integrated regressors and the innovations, whose joint limit is the stated Brownian functional by the functional central limit theorem and the continuous-mapping theorem. :::

Johansen trace test for cointegrating rank
  input   series y, lag order, deterministic case
  1  concentrate out the lagged differences, leaving S00, S01, S11
  2  lambda_1 >= .. >= lambda_n <- eigenvalues of
     |lambda S11 - S10 S00^{-1} S01| = 0
  3  for r = 0, 1, .., n-1:
       LR_tr(r) <- -T sum_{i>r} ln(1 - lambda_i)
       if LR_tr(r) below its critical value, return rank r
  4  return full rank n

::: remark A constant or trend enters either inside the cointegrating space, where it sets the level or slope of $\beta^{\top}y_t$, or outside it, where it drives the common trends. Five nested cases are standard, from no deterministic term through a restricted constant, an unrestricted constant, a restricted trend, to an unrestricted trend, each with its own null limit for the rank statistics. Misreading a drift as an in-space mean inflates the apparent rank, so the test is run against the specification matching the data. :::

#Threshold cointegration

Linear error correction pulls back at a constant speed however far the system has strayed. When a band of inaction separates small departures from large ones, as a no-arbitrage region bounded by transaction costs does, the correction switches on only outside the band.

::: definition [def

] Let $z_t=\beta^{\top}y_t$ be the disequilibrium of a cointegrated system. The multivariate threshold error-correction model makes the dynamics regime-dependent in $z_{t-1}$,

the regime $j$ chosen by which interval cut by thresholds $\gamma_1<\gamma_2$ contains $z_{t-1}$. :::

::: remark With three regimes the outer two carry the correction while the middle one, $\gamma_1<z_{t-1}\le\gamma_2$, has $\alpha^{(2)}\approx0$, so inside the band the spread drifts as an uncorrected random walk and outside it is pulled back. The band is the region where the gain from correcting does not cover its cost, and a test of $\alpha^{(2)}=0$ against a correcting outer regime is a test for threshold cointegration [5]. The thresholds are fixed by minimising the residual sum of squares over a grid. :::

Cointegration completes the unit-root picture. Where the unit-root tests certify that each series is individually $I(1)$, the error-correction representation certifies that a combination of them is $I(0)$, splitting the system into $n-r$ common trends that wander and $r$ stationary spreads that revert. That decomposition of a nonstationary vector into permanent and transitory parts is the econometric backbone of mean-reversion strategies such as statistical arbitrage, and it sits one step beyond the autoregression whose reduced-rank coefficient matrix it reads.