Cointegration

Definition

Cointegration describes a situation in which $n$ individually non-stationary $I(1)$ variables share $r < n$ stationary linear combinations — the cointegrating relations. The cointegrating vectors collect the linear weights that render those combinations stationary, and the loading matrix captures how fast each variable corrects toward the long-run equilibria.

Key Ideas

Cointegration is distinct from spurious regression: cointegrated series share a common stochastic trend, so their linear combination has finite variance even though each series individually wanders without bound.
The number of cointegrating relations $r$ (the cointegration rank) is testable via Johansen's (1991) likelihood ratio (LR) trace and max-eigenvalue tests.
The factorization $\Pi = \alpha\beta'$ separates long-run equilibria ( $\beta$ ) from short-run dynamics ( $\alpha$ ) but is not separately identified without further restrictions.
Cointegration and error correction are equivalent (Granger Representation Theorem): a cointegrated system always has a vector error correction model (VECM) representation and vice versa.

How It Works

Origins: Spurious Regression (Granger-Newbold 1974)

The cointegration concept emerged from the spurious regression problem. Granger and Newbold (1974) showed by simulation that regressing two independently generated random walks on each other by ordinary least squares (OLS) produces t-statistics that are systematically too large and $R^2$ that is systematically too high — with strongly autocorrelated residuals (Durbin-Watson $\approx 0$ ). Phillips (1986) provided the asymptotic theory: the t-statistic diverges to infinity as $T \to \infty$ and $\hat\beta$ does not converge to the true (zero) value. See Spurious Regression for full details.

The cointegration concept (Granger 1981) identifies the exception: if the regression residual $\varepsilon_t = y_t - \beta x_t$ is I(0), the regression is not spurious but instead captures a genuine long-run equilibrium. Cointegration is thus the condition under which levels regressions are valid.

Engle-Granger Two-Step Estimation

Engle and Granger (1987) developed the first systematic estimation method:

Step 1 (OLS for $\beta$ ): Estimate the cointegrating vector by OLS regression of $y_t$ on $x_t$ . Stock (1987) showed the OLS estimator $\hat\beta$ is superconsistent: it converges to $\beta$ at rate $T^{-1}$ rather than the usual $T^{-1/2}$ . The superconsistency means that pre-estimation error in $\hat\beta$ has no first-order effect on subsequent inference.
Step 2 (ML for $\alpha$ , $\Gamma$ ): Hold $\hat\beta$ fixed, substitute the estimated equilibrium error $\hat u_{t-1} = y_{t-1} - \hat\beta x_{t-1}$ into the error correction model (ECM), and estimate $(\alpha, \Gamma)$ by maximum likelihood. The second-stage estimators are consistent and asymptotically normal under standard conditions.

This two-step approach "opened the gates for a flood of applications" (Nobel Committee 2003) by giving applied economists a tractable way to incorporate long-run relationships. Johansen (1988, 1991) is the "second generation": rather than separating OLS and ML, he derives the maximum likelihood estimator (MLE) of the cointegrating space directly via reduced-rank regression, simultaneously obtaining β, α, and sequential LR tests for the rank $r$ . Johansen builds entirely on ML rather than partly on OLS, which gives efficiency gains and unified hypothesis testing.

Sims-Stock-Watson (1990): Two-Step Asymptotically Redundant

Sims, Stock, and Watson (1990) show via a canonical-regressor decomposition that when the Vector Autoregression (VAR) is estimated in levels by OLS, the limiting distribution of all coefficient estimators is identical to the distribution that would obtain if the cointegrating vector $\beta$ were known a priori. The superconsistency of the OLS estimator $\hat\beta$ (convergence at rate $T$ rather than $T^{1/2}$ ) means that plugging in $\hat\beta$ in place of the true $\beta$ has no first-order effect on any subsequent inference — it contributes nothing to the asymptotic distribution.

The practical implication is that the Engle-Granger two-step procedure is asymptotically redundant: there is no efficiency gain from estimating a cointegrating regression in Step 1 before estimating the error-correction model in Step 2, relative to simply estimating the levels VAR directly. The result rationalizes VAR-in-levels estimation as a complete alternative to the Johansen/Engle-Granger ECM approach for purposes of coefficient inference (though rank testing and structural long-run analysis still require explicit cointegration methods).

Three-Test Comparison (Dickey-Jansen-Thornton 1991)

Dickey-Jansen-Thornton (1991) survey three competing methods unified by reparameterizing the VAR( $p$ ) in error-correction form:

$\Delta Y_t = \Gamma_1 \Delta Y_{t-1} + \cdots + \Gamma_{p-1}\Delta Y_{t-p+1} - \psi Y_{t-p} + \varepsilon_t, \qquad \psi = I - A_1 - \cdots - A_p$

The rank of $\psi$ equals the number of cointegrating vectors $k$ ; the system therefore has $n - k$ common stochastic trends. The three approaches differ in how they estimate rank( $\psi$ ):

Engle-Granger (1987). Choose a normalization variable, regress it on the others by OLS, and apply an augmented Dickey-Fuller (ADF) test to the residuals. Critical values are non-standard. Main limitation: normalization-sensitive — different choices of dependent variable can yield different cointegrating vector estimates and different test outcomes.
Stock-Watson (1988). Factor-analyze $\Delta Y_t$ ; the $n-k$ directions of largest variance span the common stochastic trends, and their orthogonal complement contains the cointegrating relations. Avoids normalization sensitivity but requires choosing how many trends to extract.
Johansen (1988, 1991). Performs canonical correlation analysis of $\Delta Y_t$ and $Y_{t-p}$ after projecting out lagged-difference regressors. The factorization $\psi = \alpha\beta'$ yields ML estimates of the cointegrating vectors $\beta$ and loading matrix $\alpha$ , with sequential trace and max-eigenvalue LR tests for rank. Full ML gives efficiency gains over the OLS-based Engle-Granger approach.

Multiple comparison power loss. Sequentially testing rank 0, then rank 1, then rank 2, … inflates total type I error and shifts critical values far from standard Dickey-Fuller tables. Power falls as $n$ grows.

Economic interpretation caveat. The cointegrating vectors $\beta$ estimated from a reduced-form VAR are linear combinations of all structural equations — they cannot be interpreted as individual structural relations. A cointegrating vector that matches a theoretical identity (Fisher equation, quantity-theory identity) does so by coincidence of the reduced form, not because the system identifies structural parameters.

Money demand application (US quarterly, 1953.2–1988.4). Johansen finds 1 cointegrating (CI) vector for {real M1, real income $q$ , interest rate}; income elasticity $\approx 0.68$ – $0.85$ (unity not rejected); interest elasticity significantly negative. M2 and NM1M2 each yield 1 CI vector. Monetary base combined with $K$ yields 2 CI vectors with R3M but only 1 with R10Y — rank finding is sensitive to interest rate specification.

Extensions of Cointegration

Seasonal cointegration (Hylleberg, Engle, Granger, and Yoo 1990): when series are seasonally integrated ( $\Delta_4 x_t = x_t - x_{t-4} \sim I(0)$ but $x_t \not\sim I(0)$ ), cointegration is defined at the seasonal frequencies $\omega \in \{0, \pi/2, \pi\}$ separately. Unit roots at $\omega = \pi$ (biannual) and $\omega = \pi/2$ (quarterly) require different test regressions from the standard zero-frequency case.

Multicointegration (Granger and Lee 1990): when $x_t$ and $y_t$ are cointegrated with equilibrium error $e_t = y_t - \beta x_t$ , the cumulated disequilibrium $E_t = \sum_{s=1}^{t} e_s$ is I(1). If $E_t$ is cointegrated with $x_t$ or $y_t$ , the variables are multicointegrated — a richer long-run structure relevant to stock-flow models (e.g., sales and production: their difference is change in inventory, and the inventory level may be cointegrated with sales).

Threshold cointegration (Balke and Fomby 1997): error correction is nonlinear — adjustment toward equilibrium only occurs when the disequilibrium exceeds a threshold (transaction costs, menu costs). The ECM becomes $\Delta x_t = \alpha_1 e_{t-1} \mathbf{1}(|e_{t-1}| > \tau) + \ldots$ where $\tau$ is the threshold. Granger and Swanson (1996) provided the theoretical framework for nonlinear cointegration with transaction costs.

Granger Representation Theorem — Three Equivalent Forms (Engle-Granger 1987)

For an $N$ -dimensional I(1) vector $x_t$ with $r$ cointegrating vectors collected in $\alpha$ (so $\alpha'x_t \sim I(0)$ ), the following three representations are equivalent:

1. Error-correction (ECM):

$A(L)\Delta x_t = -\gamma\, z_{t-1} + d(L)\varepsilon_t, \qquad z_t = \alpha' x_t$

where $A(L)$ and $d(L)$ are finite-order matrix polynomials and $\gamma \neq 0$ (at least one variable corrects toward the equilibrium $z_t$ ).

2. Moving-average (Wold MA):

$x_t = C(1)\sum_{s=1}^{t}\varepsilon_s + C^*(L)\varepsilon_t$

where $C(1)$ has reduced rank $N - r$ (exactly $r$ zero eigenvalues — one for each cointegrating relation). The $N-r$ nonzero rows of $C(1)$ span the permanent component; the $r$ cointegrating vectors $\alpha$ are in the left null space of $C(1)$ .

3. Autoregressive:

Standard VAR representation with the ECM constraint $\Pi = \alpha\beta'$ imposed (where $\beta$ is the matrix of adjustment speeds and $\alpha$ is the cointegrating matrix; Engle-Granger use the notation reversed from Johansen's later convention). Specifically, the VAR has a unit root at $z=1$ of multiplicity $r$ , with all remaining roots outside the unit circle.

Key insight: The reduced rank of $C(1)$ is what distinguishes a cointegrated system from a plain multivariate random walk. If $C(1)$ had full rank $N$ , no stationary linear combination would exist. The rank deficiency by exactly $r$ is the algebraic fingerprint of $r$ cointegrating relations.

Seven Tests for Cointegration (Engle-Granger 1987)

All seven tests are applied to the OLS residuals $\hat z_t = y_t - \hat\beta' x_t$ from the cointegrating regression. Critical values are non-standard — obtained by Monte Carlo simulation (Tables I–III of the paper) — because $\hat z_t$ are generated residuals, not observed series. Standard Dickey-Fuller tables are invalid here.

Test	Statistic	Notes
CRDW (Cointegrating Regression Durbin-Watson)	Durbin-Watson on $\hat z_t$	Reject $H_0$ if DW $> c_\alpha$ ; simple but low power
DF	DF $t$ -statistic on $\Delta\hat z_t = \delta\hat z_{t-1} + e_t$	No lags; serially correlated residuals inflate size
ADF	ADF $t$ -statistic (recommended)	Add lagged $\Delta\hat z_{t-j}$ ; best power of all seven
RVAR	$F$ -type test from restricted bivariate VAR	Imposes $\hat\beta$ ; equivalent to DF for $N=2$
UVAR	Likelihood ratio test from unrestricted VAR	Does not impose $\hat\beta$ from step 1
ARVAR	Augmented version of RVAR	Adds lags; like ADF for RVAR
DVAR	VAR of differences + EC term	Compares restricted vs unrestricted

Recommendation: ADF on residuals is preferred. The CRDW is a quick check only. The VAR-based tests have lower power than ADF when $\hat\beta$ is well estimated.

Critical value example (from Table I, $N=2$ , $n=100$ , 5% level): CRDW $\approx 0.386$ ; DF $\approx -3.37$ ; ADF (2 lags) $\approx -3.17$ . These are substantially larger in absolute value than the corresponding unit-root critical values because the residuals are estimated rather than observed.

The VECM Representation

By the Granger Representation Theorem (Engle and Granger 1987), the cointegration hypothesis implies that the long-run impact matrix $\Pi$ in the VAR has rank $r$ . Writing the VAR in error correction form:

$\Delta x_t = \delta D_t + \sum_{i=1}^{p-1} \Gamma_i \Delta x_{t-i} + \Pi x_{t-1} + \varepsilon_t \tag{1}$

where $x_t \in \mathbb{R}^n$ , $D_t$ is a vector of deterministic components (constant, trend, dummies), and $\varepsilon_t \sim \mathcal{N}(0, \Sigma)$ .

Since $\mathrm{rank}(\Pi) = r$ , write:

$\Pi = \alpha\beta', \qquad \alpha, \beta \in \mathbb{R}^{n \times r} \tag{2}$

$\beta$ ( $n \times r$ ): cointegrating vectors — the $r$ stationary linear combinations $\beta'x_t$ (the long-run equilibria)
$\alpha$ ( $n \times r$ ): loading matrix (adjustment speeds) — the $r$ coefficients determining how fast each of the $n$ equations corrects toward equilibrium

Johansen (1991) Rank Determination

The cointegration rank $r$ is determined by Johansen's reduced rank regression. The test statistics are:

Trace statistic: $\lambda_{\mathrm{trace}}(r) = -T\sum_{i=r+1}^{n}\log(1-\hat\lambda_i)$
Max-eigenvalue statistic: $\lambda_{\max}(r) = -T\log(1-\hat\lambda_{r+1})$

where $\hat\lambda_i$ are eigenvalues from the generalized eigenvalue problem involving the concentrated likelihood. Critical values depend on the deterministic specification (constant inside/outside cointegrating space, trend).

Identification of $\beta$ and $\alpha$

The factorization $\Pi = \alpha\beta'$ is not unique: for any invertible $r \times r$ matrix $H$ , $(\alpha H^{-\prime})(\beta H)'= \alpha\beta' = \Pi$ . Identifying $\beta$ and $\alpha$ separately requires $r^2$ restrictions. Bauwens and Lubrano (1994a) use equation-by-equation linear restrictions on each cointegrating vector:

$R_i \beta_i = r_i, \qquad i = 1, \ldots, r \tag{3}$

where $R_i$ ( $s_i \times n$ ) and $r_i$ ( $s_i \times 1$ ) are known. A just-identified system imposes exactly $r^2$ restrictions in total (typically $r$ normalisations and $r(r-1)$ exclusion restrictions). An over-identified system imposes more, producing testable cross-equation restrictions on $\beta$ .

Bayesian Posterior of $\beta$ (Bauwens-Lubrano)

Let $Y$ ( $T \times n$ ) be the matrix of $\Delta x_t$ , $Z$ ( $T \times n$ ) the matrix of $x_{t-1}$ , and $X$ the matrix of the remaining regressors ( $D_t$ , $\Delta x_{t-i}$ ). Define:

$M_X = I_T - X(X'X)^{-1}X'$ $W_0 = Z'M_XZ, \qquad W_+ = Z'M_X\bigl[I_T - Y(Y'M_XY)^{-1}Y'\bigr]M_XZ$

Under a non-informative prior on $(\delta, \Gamma_i, \Sigma)$ and a prior $f_0(\beta)$ on the identified cointegrating vectors, the marginal posterior of $\beta$ is:

$f(\beta) \propto f_0(\beta)\; |\beta'W_0\beta|^{\,l_0}\; |\beta'W_+\beta|^{-l_+} \tag{4}$

where $l_0 = (T-k-r-n)/2$ and $l_+ = (T-k-r)/2$ , and $k$ is the number of columns of $X$ . This density has no closed form when $r > 1$ — integration over $\beta$ requires numerical methods (importance sampling or Gibbs Sampler).

Conditional Posterior of $\alpha \mid \beta$

Given $\beta$ , the model in (1) becomes a standard multivariate regression:

$\Delta x_t - \Gamma(L)\Delta x_{t-1} = \delta D_t + \alpha(\beta'x_{t-1}) + \varepsilon_t$

The conditional posterior of the stacked parameter block $(\delta', \mathrm{vec}(\alpha)')'$ is matrix-Student with expectation equal to the generalized least squares (GLS) estimator:

$\widehat{\alpha}'(\beta) = (\beta'W_0\beta)^{-1}\beta'W_* \tag{5}$

where $W_* = Z'M_XY$ . At each Gibbs step, $\alpha$ can be drawn jointly from this matrix-Student (or its moments can be accumulated for Rao-Blackwell estimation of $E(\alpha)$ ).

Bayesian ECM Forecasting: Informative Priors on Factor Loadings (Amisano-Serati 1999)

The Bayesian ECM (BECM) approach (LeSage 1990) estimates $\beta$ super-consistently via Johansen ML, then includes the disequilibrium terms $w_{t-1} = \hat\beta' y_{t-1}$ as regressors in a Bayesian VAR (BVAR) of differences. The $i$ -th equation becomes:

$\Delta y_{it} = \alpha_i' w_{t-1} + \sum_{l=1}^{k-1} \Gamma_{il}' \Delta y_{t-l} + \text{deterministics} + e_{it} \tag{BECM}$

The flat-prior problem. A natural prior assigns informative Minnesota-type priors to the $\Gamma$ lag coefficients (prior variance $\propto l^{-1.5}$ ) but diffuse priors to $\alpha$ (large $\lambda_{ij}$ ). This is inappropriate because $\hat\alpha$ converges at the standard $O_p(T^{-1/2})$ rate — not the super-consistent rate of $\hat\beta$ ( $O_p(T^{-1})$ ). Tight priors on $\Gamma$ combined with flat priors on $\alpha$ therefore over-weight the ECM correction relative to short-run dynamics, degrading short- and medium-horizon forecasts.

The fix (IP-BECM). Assign informative priors to $\alpha$ with hyperparameters $\lambda_{ij}$ of moderate size, tuned by the same grid-search procedure as the other hyperparameters. Empirically (Italian GDP, consumption, investment, 1970–1995), the IP-BECM model dominates all alternatives in 13/15 forecast horizon-equation combinations. The flat-prior FP-BECM only outperforms a standard Minnesota BVAR at long horizons (12+ steps), where the random-walk prior's ignorance of cointegration becomes the binding constraint.

Flat-Prior Pathology in BECM with Multiple Cointegrating Vectors (Felix-Nunes 2003)

Felix and Nunes (2003) provide the clearest empirical demonstration of what goes wrong when a diffuse prior is placed on error-correction factor loadings $\alpha$ in a BECM estimated with Johansen's multiple cointegrating vectors.

The mechanism. Johansen's trace test on a six-variable quarterly euro area system finds 5–6 cointegrating vectors, of which four are retained. A flat prior ( $\Omega \to \infty$ ) treats all four estimated equilibrium relations as equally informative and allows the model to fit arbitrary linear combinations of them. Because $\alpha$ estimates converge only at the slow $O_p(T^{-1/2})$ rate — unlike the super-consistent $O_p(T^{-1})$ rate of $\beta$ — a diffuse prior on $\alpha$ over-weights the ECM correction relative to the evidence actually contained in the data. With four loosely estimated equilibrium relations, this effectively overfits.

Empirical result (6 variables, 12 forecast horizons, avg root-mean-square error (RMSE) relative to RW = 1.000):

Model	Avg RMSE ratio
BVAR levels	0.731
BECM(EG)-IP (informative prior on α, single EG vector)	0.754
BECM(J)-IP (informative prior on α, 4 Johansen vectors)	0.805
BECM(EG)-FP (flat prior, single EG vector)	0.857
BECM(J)-FP (flat prior, 4 Johansen vectors)	1.278

The BECM(J)-FP is 28% worse than a random walk. The single Engle-Granger vector is less susceptible because it provides only one (better-estimated) equilibrium correction.

The fix. Assign a finite informative prior variance Ω to α, tuned by the same grid search used for the Minnesota hyperparameters. This shrinks the factor loading estimates toward zero and limits the damage from the four loosely estimated Johansen vectors. BECM(EG)-IP is the second-best model overall and the best for real GDP at forecast horizons beyond 12 quarters. The result corroborates the theoretical argument in Amisano-Serati (1999) that informative priors on α are essential when α is not super-consistently estimated. See Minnesota Prior for the full Felix-Nunes hyperparameterization.

ECM vs. BVAR Forecasting: LeSage (1990) Evidence

LeSage (1990) provides the largest-scale empirical comparison of ECM and BVAR forecasting to that point: 50 Ohio industries, monthly labor market data (manhours N, nominal wages W, prices P), rolling 25-month evaluation over 1983–1985 with 12-step-ahead horizons. Five models compete: ECM, unrestricted VAR, MVAR (Minnesota prior), BVAR (block-recursive prior), and BECM (Minnesota prior on lags, diffuse prior on EC term).

Cointegrated industries (7 of 50). The ECM dominates at forecasting horizons 3–12 months, producing ~20% less mean absolute percentage error (MAPE) at month 12 than any VAR or BVAR. The improvement is monotonically increasing with horizon — near-term ECM forecasts are similar to or slightly worse than VAR, while long-run forecasts are markedly better. Bayesian shrinkage (MVAR, BVAR) degrades performance for cointegrated series because the random-walk prior is misspecified: it ignores the long-run equilibrium. The BECM performs nearly as well as the ECM at horizons 5–12 months — contradicting Engle and Yoo's (1987) prediction that the misspecified prior would dominate and produce inferior forecasts.

Possibly cointegrated (5 industries, mixed ADF/DF results). The BVAR (block-recursive) wins at all 12 horizons — suggesting the variables are not truly cointegrated despite borderline test statistics. LeSage proposes using forecast comparisons as a practical tie-breaker when cointegration tests give conflicting results.

Non-cointegrated (38 industries). The MVAR (Minnesota prior) produces the best short-horizon forecasts (months 1–7); the BECM produces the best long-horizon forecasts (months 8–12), with ECM second. A regression of ADF t-statistics on forecast errors shows no systematic relationship, ruling out the low-power explanation: the error-correction variable itself is responsible for the long-horizon improvement even without confirmed cointegration.

Mechanism. In the BECM, the EC variable enters with a diffuse prior while the autoregressive lags are shrunk toward a random walk. Tighter shrinkage on the VAR terms increases the relative importance of the EC term, giving the practitioner direct control over the short-run vs. long-run balance. A sufficiently loose prior recovers the ECM as a limiting case.

Bayesian Vector Error Correction Model (BVEC) Forecasting: Single Known Cointegrating Vector (Stark 1998)

When the cointegrating vector is known or imposed a priori, Bayesian estimation can proceed with a diffuse prior on the error correction loading while retaining informative (Minnesota-type) shrinkage on the short-run dynamics. Stark (1998) demonstrates this approach for a 7-variable U.S. macroeconomic system.

Model specification. Stark's BVEC(5) system comprises {ΔRGDP, Δ²PGDP, ΔRFF, ΔRPIM, ΔU, ΔRM2, ΔRTB}, estimated 1960Q4–1997Q4. A single EC term $\text{RFF}_{t-1} - \text{RTB}_{t-1}$ (the federal funds rate minus the 10-year Treasury bond rate) enters all equations, imposing the cointegrating vector $(1,-1)$ between the two interest rates as a known I(0) relation. The loading $\delta_i$ on the EC term receives a diffuse prior in each equation, following LeSage (1990) and Joutz-Maddala-Trost (1995). Estimation uses Theil's mixed estimator, which coincides with the Bayesian posterior mean under normality.

EC significance and forecast relevance. In a rolling regression experiment (mid-1975–1997Q4), the spread enters significantly in the ΔRGDP equation ( $p = 0.003$ ), ΔRFF ( $p = 0.068$ ), ΔU ( $p = 0.000$ ), and ΔRTB ( $p = 0.063$ ). Removing the EC term raises the two-year unemployment RMSE from 0.805% to 1.113% — a 38% deterioration — with mixed effects on inflation and GDP.

Fisher relation: imposing cointegration can worsen forecasts. When a second EC term $\text{RFF}_{t-1} - \Delta\text{PGDP}_{t-1}$ is added (imposing real rate stationarity as a Fisher relation), the two-year inflation RMSE rises from 1.52% to 2.36%. This illustrates a general principle: if the economic theory underlying a cointegrating restriction is empirically fragile, imposing it creates more specification error than it eliminates. The Fisher effect is weak enough in the U.S. data that the constraint actively damages inflation forecast accuracy.

Gonzalo-Granger Common Stochastic Trends (1995)

In a cointegrated system with $m$ I(1) variables and cointegrating rank $r$ , there are $m - r$ common stochastic trends — the permanent components shared by all variables. Gonzalo and Granger (1995) provide an explicit identification and estimation method.

Let $\alpha_\perp$ ( $m \times (m-r)$ ) be the orthogonal complement of the loading matrix $\alpha$ , satisfying $\alpha'\alpha_\perp = 0$ . The common trends are defined as:

$W_t = \alpha_\perp' x_t \tag{GT}$

The identifying restriction is that the equilibrium errors $z_t = \beta' x_t$ do not Granger-cause $W_t$ at very low frequencies — i.e., the transitory component has no persistent effect on the permanent component. Under this restriction, $(\alpha_\perp, \beta_\perp)$ span the permanent and transitory components in a way that is simultaneously rotation-free and economically interpretable.

The full system decomposition is:

$x_t = \underbrace{A_1 W_t}_{\text{permanent}} + \underbrace{A_2 z_t}_{\text{transitory}}, \qquad A_1 = \beta_\perp(\alpha_\perp'\beta_\perp)^{-1}, \quad A_2 = \alpha(\beta'\alpha)^{-1}$

This is the multivariate analogue of the Beveridge-Nelson decomposition for a single series: just as the Beveridge-Nelson decomposition splits $x_t$ into a random-walk (permanent) and a stationary (transitory) part, the Gonzalo-Granger decomposition splits the $m$ -dimensional system into $m-r$ random-walk trends $W_t$ and $r$ stationary equilibria $z_t$ .

Economic interpretation. In a bivariate system of income $y_t$ and consumption $c_t$ with one cointegrating vector (the consumption-income ratio), there is one common trend (say, permanent income) and one transitory component (cyclical deviations from the long-run ratio). The Gonzalo-Granger decomposition identifies which linear combination of $(y_t, c_t)$ constitutes the common trend without imposing arbitrary normalization.

Granger (1997) caution. The decomposition provides the common stochastic trends even when the series are not strictly I(1) but merely "persistent" — the ECM-based construction is robust to near-unit-root and other persistent processes that fail to reject the I(1) null but are not pure unit root processes.

Practical Cautions on Johansen Estimation (Granger 1997)

Despite being the dominant frequentist approach to cointegration, the Johansen (1991) MLE has several practical limitations that applied economists sometimes overlook:

Lag length sensitivity. The cointegrating rank estimate depends on the number of lags in the underlying VAR. Different lag choices can alter both the estimated rank $r$ and the associated cointegrating vectors, with no objective way to choose when information criteria disagree.
Super-consistency asymptotics. The rate- $T^{-1}$ convergence of $\hat\beta$ ("super-consistency") is an asymptotic result that may not dominate in typical macro sample sizes (T = 50–200 observations). In finite samples the practical advantages over OLS may be limited.
Over-parameterization. A VAR with many lags can accommodate a large number of parameters; many estimated coefficients will be uninformative and the model far from parsimonious. Standard practice of "interpret each coefficient" is misleading when the system is lightly identified.
Identification. Without additional restrictions, the $r$ cointegrating vectors are only identified up to an invertible $r \times r$ rotation — the columns of $\hat\beta$ are not individually interpretable. Imposing economic theory to identify the relations "may give interesting results but historically econometric models that have been forced to obey some economic theory have usually performed less well than unconstrained models" (Granger 1997).
Long-run information accumulates slowly. Even with super-consistency, the useful information about $\beta$ grows only at rate $T^{-1}$ ; in data-limited settings (short time series, frequent structural breaks) posterior uncertainty about the long-run equilibrium remains wide.

Optimal Inference and the LAMN/LGF Dichotomy (Phillips 1991)

Phillips (1991) establishes the theoretical foundation for why full-system ML dominates OLS for estimating cointegrating vectors. The key is a distinction between two asymptotic frameworks arising from how the cointegrating structure is parameterized.

Triangular ECM representation. Phillips works with the system:

$y_{1t} = By_{2t} + u_{1t} \tag{P1}$ $\Delta y_{2t} = u_{2t} \tag{P2}$ $\Delta y_t = -EAy_{t-1} + v_t \tag{P3}$

where $B$ ( $n_1 \times n_2$ ) is the cointegrating vector appearing linearly in (P1). This triangular form separates the $I(1)$ components explicitly and is the key structural feature.

LAMN vs. LGF. The likelihood of the triangular ECM — which eliminates unit roots from the parameterization — is locally asymptotically mixed normal (LAMN): the normalized score converges to a mixture of normals (mixing on the random Fisher information matrix $G = \int_0^1 S_2 S_2' dt$ ), enabling chi-squared test statistics and Cramér-Rao efficiency bounds. By contrast, an unrestricted VAR in levels implicitly estimates unit roots alongside all other parameters and delivers a locally Gaussian functional (LGF) likelihood — producing nonstandard, nuisance-parameter-dependent distributions and no optimal inference theory.

Theorem 1 (iid errors): mixed-normal MLE. Under the triangular ECM, the full-system MLE satisfies:

$T(\hat{B} - B) \Rightarrow \left(\int_0^1 dS_{1\cdot2}\,S_2'\right)\left(\int_0^1 S_2 S_2'\right)^{-1}$

which is conditionally $N(0,\,\Omega_{11\cdot2}\otimes G)$ given $G=\int_0^1 S_2 S_2'\,dt$ — a mixed normal (MN) distribution. The MLE is symmetrically distributed, median unbiased, and asymptotically efficient. Wald, LR, and Lagrange multiplier (LM) tests all converge to $\chi^2_q$ under this parameterization.

OLS and simultaneous-equations bias. OLS satisfies:

$T(\hat{B}^* - B) \sim \left(A\int_0^1 dS\,S_2' + \Sigma_{12}\right)\left(\int_0^1 S_2 S_2'\right)^{-1}$

The additional $\Sigma_{12}$ term (cross-equation innovation covariance) induces a simultaneous-equations bias whenever $u_{1t}$ and $u_{2t}$ are correlated. OLS estimators of the cointegrating vector are not median unbiased and are asymptotically inefficient.

Single-equation ECM. The single-equation partial MLE is efficient iff $\Sigma_{12} = 0$ (strict exogeneity of $y_{2t}$ ). In the typical macro setting where all variables are jointly determined, this condition fails and single-equation ML is biased.

Transient dynamics need not be estimated. A key practical implication: only a consistent long-run covariance estimate $\hat\Omega = 2\pi\hat{f}(0)$ is required — the transient dynamics ( $A$ , $E$ , etc.) need not be jointly estimated by ML. This justifies the Phillips-Hansen (1990) fully modified OLS (FM-OLS) approach.

General linear processes (Theorem 1'). The mixed-normal limit holds for general linear process errors with long-run covariance $\Omega = 2\pi f(0)$ — the spectral density at frequency zero replaces the short-run $\Omega$ .

Granger Causality Rank Conditions (Toda-Phillips 1993)

Toda and Phillips (1993) showed that standard Wald $\chi^2$ asymptotics for Granger noncausality tests in levels VARs fail generically when $y_t \sim I(1)$ . The critical quantity is the rank of the subblock of the cointegrating matrix corresponding to the candidate causal variable. Partition $y_t = (y_{1t}', y_{2t}', y_{3t}')'$ where $y_{3t}$ ( $n_3 \times 1$ ) is the potential cause and let $\beta_3$ denote the last $n_3$ rows of $\beta$ :

rank $(\beta_3) = n_3$ ("sufficient cointegration with respect to $y_3$ "): in the levels VAR (OLS), the Wald test $F_{WD} \xrightarrow{d} \chi^2_{n_1 n_3 k}$ (Theorem 1). In the Johansen ECM the same holds (Theorem 3, condition 1).
rank $(\beta_3) < n_3$ with no cointegration: $F_{WD}$ converges to a nonstandard but nuisance-parameter-free limit whose critical values can be tabulated (Theorem 2).
rank $(\beta_3) < n_3$ with cointegration: the limit distribution depends on nuisance parameters (innovation correlations) and cannot be recovered from OLS levels VAR alone — because OLS estimators of $\beta$ suffer from simultaneous-equations bias (Phillips 1991). No valid test is available without ML estimation.
In the Johansen ECM, the second sufficient condition for $\chi^2$ validity is rank $(\Gamma_1) = n_1$ where $\Gamma_1$ = first $n_1$ rows of the loading matrix $\alpha$ (Theorem 3, condition 2).

The practical implication: Granger causality tests must be conducted in a Johansen ECM framework after testing the cointegrating rank $r$ and the rank of $\beta_3$ . See Granger Causality for the full asymptotic theory and sequential testing procedure.

Wald Statistics under Rational Expectations Restrictions (Warne 1997)

Warne (1997) extends the Sims-Stock-Watson (1990) result to nonlinear restrictions on VAR coefficients, with particular focus on nonlinear cross-equation (NCE) restrictions implied by rational expectations (RE) models. The key insight is that RE restrictions typically constrain the row space of the long-run impact matrix $A(1) = I - \sum_{i=1}^p A_i$ , and this is precisely why standard χ² asymptotics fail.

Setup. Write the VAR in the canonical regressor decomposition of Sims-Watson (1994): $y_t = B_c z_t + \varepsilon_t$ , where $z_{1t}$ collects mean-zero stationary regressors, $z_{3t}$ collects the $n - r - k_4$ non-stationary components, and $z_{4t}$ is the deterministic trend. The coefficient blocks $\hat\delta_1, \hat\delta_2$ converge at rate $T^{1/2}$ , but $\hat\delta_3$ converges at rate $T^{3/2}$ — different rates yield different contributions to the limiting Wald distribution.

Proposition 1. Under the null $F(\beta) = 0$ (smooth nonlinear restrictions), the Wald statistic converges to: $W \Rightarrow (P^*(\delta)[V^{-1}\otimes I]\phi)'(P^*(\delta)[V^{-1}\otimes \Omega]P^*(\delta)')^{-1}(P^*(\delta)[V^{-1}\otimes I]\phi)$ where $\phi = [\phi_1', \phi_2', \phi_3', \phi_4']'$ has components drawn from Brownian functionals, and $P^*(\delta) = \lim_{T\to\infty} P_T^*(\delta)$ is the limiting Jacobian. This distribution is non-χ² unless the restrictions do not constrain the $\delta_3$ coefficients.

NCE restrictions from RE models. A linear RE model generates restrictions of the form $\sum_{i=0}^s N_i E[y_{t+i}|\mathscr{A}_t] = \lambda$ , where $N_i$ are known matrices and $\lambda$ is a known vector. Under the null these imply nonlinear cross-equation restrictions on $\beta$ . The Jacobian $P_1(\delta)$ — the block corresponding to the non-stationary regressors — fails to have full row rank whenever $r < n$ , because the NCE restrictions constrain the row space of $A(1)$ and thus act on the cointegrating space. This is the structural reason the Wald statistic has a nonstandard limit.

Lower bound for r. Conveniently, the NCE restrictions provide a lower bound for the cointegration rank: the rows of $N_0^\perp$ (the complement induced by the model matrices) must themselves be cointegrating vectors, so $\mathrm{rank}(N_0^\perp) \leq r$ . This bound selects the cointegrating space needed for the simulation, making the approach practical.

Expectations hypothesis application. For the term spread between one- and three-month U.S. bond yields (Campbell-Shiller 1991 data, January 1952–February 1987), the cointegration rank lower bound is $r \geq 1$ , $\mu = 0$ (no trend in the spread), giving $D$ and $P(\delta)$ explicitly. The Wald statistic $W = 59.48$ (q = 8 restrictions) is decisively rejected at the 99th percentile under all three distributions: χ² (32.00), simulated nonstandard asymptotic (33.86), and empirical bootstrap (37.58). Campbell-Shiller (1991) reached the same conclusion.

Small-sample caution. Monte Carlo evidence indicates the simulated asymptotic critical value at nominal 5% corresponds to an empirical rejection rate of approximately 10% when $T = 422 - p$ . Practitioners should use low nominal significance levels (1–2%) when applying this approach.

Deterministic Specifications in Johansen Rank Tests (Lütkepohl 1999)

The limiting null distribution of the Johansen LR trace statistic $LR(r_0) = -T\sum_{j=r_0+1}^K \log(1-\hat\lambda_j)$ depends not only on $K-r_0$ but also on which deterministic terms are assumed present in the data-generating process. Lütkepohl (1999, Table 1) classifies five cases by assumptions on the intercept $\mu_0$ and trend slope $\mu_1$ in $y_t = \mu_0 + \mu_1 t + x_t$ :

Case	Assumption	Interpretation
1	$\mu_0 = \mu_1 = 0$	No deterministic terms; rarely applicable
2	$\mu_0$ free, $\mu_1 = 0$	Intercept only, no trend
3	$\mu_0$ free, $\mu_1 = 0$ , intercept restricted to cointegrating space	No linear trend in levels; intercept absorbed into $\beta$
4	$\mu_0$ free, $\mu_1 \neq 0$ , $\beta'\mu_1 = 0$	Linear trend in levels but not in cointegrating relations
5	$\mu_0$ , $\mu_1$ both free	Linear trend allowed in cointegrating relations

In applied work Case 4 (trend in levels, no trend in cointegrating relations) is most common. For each case the critical values of the trace and max-eigenvalue statistic $LR_{\max}(r_0) = -T\log(1-\hat\lambda_{r_0+1})$ differ and are tabulated separately.

Saikkonen-Lütkepohl prior trend adjustment. An alternative for Cases 2–4 is to first estimate the mean/trend parameters by a feasible GLS procedure, subtract the fitted deterministic component $\hat\mu_0 + \hat\mu_1 t$ from $y_t$ , and then apply the rank test to the adjusted series. Saikkonen and Lütkepohl (1997, 1998, 1999) show this "trend-first" procedure attains higher local power than the standard Johansen test under the respective null. In simulations (Lütkepohl-Saikkonen 1999), none of the three standard test variants uniformly dominates, but prior trend adjustment is generally preferred when the trend specification is known.

Prior Specification on the Cointegrating Space (Warne 2006)

The cointegrating matrix $\beta$ is only identified up to an invertible $r \times r$ rotation — so the object with natural prior meaning is not $\beta$ itself but the cointegrating space $\mathrm{col}(\beta)$ , which lives on the Grassmann manifold $\mathcal{G}(n,r)$ (the set of all $r$ -dimensional subspaces of $\mathbb{R}^n$ ). Warne (2006) parametrizes the space by a companion matrix $\Psi \in \mathbb{R}^{(n-r)\times r}$ via $B \equiv \beta(c'\beta)^{-1} = \binom{c}{\Psi}$ , where $c$ is a fixed $r \times r$ normalization matrix.

Flat-prior pathology. A flat prior on $\Psi$ does not induce a uniform distribution over $\mathcal{G}(n,r)$ ; instead it overweights the region near $(c'\beta)^{-1} \to \infty$ (i.e., near normalization breakdown), as shown by Strachan and van Dijk (2003). This produces distorted posterior inference on the cointegrating relations.

Villani/Warne prior (uniform on Grassmann manifold). The matrix- $t$ prior

$\Psi \sim t_{(n-r)\times r}(0,\; c_\perp'c_\perp,\; c'c,\; 0)$

where $c_\perp$ is the orthogonal complement of $c$ , is algebraically equivalent to a uniform distribution over $\mathcal{G}(n,r)$ and therefore free of normalization artifacts. Warne (2006) combines this with a conjugate prior on the remaining parameters:

$\Omega \sim \mathrm{IW}(A, q)$
$\mathrm{vec}(\alpha)|\beta,\Omega,r \sim \mathcal{N}(0,\; (\beta'\beta)^{-1}\otimes\lambda_\alpha^2\Omega)$
$\Gamma_i|\Omega \sim \mathcal{N}(0,\; \Sigma_{\Gamma_i}\otimes\Omega)$ with $\Sigma_{\Gamma_i} = (\lambda_b^2/i^{2\lambda_l})I_n$ — Minnesota-style geometric shrinkage, penalising higher lags more strongly
$\Phi$ flat (deterministic terms)

The $\Gamma|\Omega$ prior is Warne's key addition relative to Villani (2005b): it introduces informative short-run dynamics analogous to a Litterman/Minnesota prior within the cointegrated VECM.

Posterior Mode of the Cointegrating Space (Warne 2006)

After integrating out $(\Phi, \Gamma, \alpha, \Omega)$ analytically, the marginal posterior of $\Psi$ is proportional to a matrix- $t$ density whose mode solves the generalized eigenvalue problem

$\bigl|\lambda\, S_{11} - S_{10}\, S_{00}^{-1}\, S_{01}\bigr| = 0$

where $S_{ij}$ are Johansen-style concentrated moment matrices augmented by prior terms and rescaled by $(T+p+q+r+m+1)^{-1}$ (Warne 2006, Proposition 4). The $r$ eigenvectors corresponding to the largest $r$ eigenvalues give the posterior modal cointegrating space. This is algebraically identical to the Johansen (1991) MLE but computed on modified moment matrices that incorporate prior information — the frequentist and Bayesian modal estimators coincide in form.

Rank and Lag-Order Posterior via Marginal Likelihood (Warne 2006)

Because the cointegration rank $r$ and lag order $k$ are discrete unknowns, Warne computes the joint posterior $p(r,k|Y)$ from the marginal likelihood $p(Y|r,k)$ using a Chib (1995) identity applied to the VECM:

$\log p(Y|r) = \underbrace{\log p(Y|\tilde\alpha,\tilde\Psi,r)}_{\text{analytic}} + \underbrace{\log p(\tilde\alpha,\tilde\Psi|r)}_{\text{analytic}} - \underbrace{\log p(\tilde\Psi|\tilde\alpha,Y,r)}_{\text{matrix-}t,\text{ analytic}} - \underbrace{\log p(\tilde\alpha|Y,r)}_{\text{Rao-Blackwell}}$

evaluated at posterior estimates $(\tilde\alpha, \tilde\Psi)$ . The last term is estimated without additional MCMC by averaging the analytic Gaussian conditional over Gibbs draws: $\hat{p}(\tilde\alpha|Y,r) = G^{-1}\sum_{i=1}^G p(\tilde\alpha|\Psi^{(i)},Y,r)$ .

Key simplification (Corollary 1). Under the informative $\Gamma|\Omega$ prior, the marginal likelihood $p(Y|k)$ at rank $r=n$ (full rank, unrestricted VAR) is available in closed form — no MCMC required. This allows the entire $(r,k)$ grid to be evaluated efficiently: Bayesian lag-order selection at full rank uses the closed form; MCMC is only needed for $r < n$ . See Lag-Order Selection Criteria.

Bartlett's paradox. If all parameters — including $(\alpha,\Omega)$ — are given improper flat priors, the marginal likelihood $p(Y|r)$ integrates to infinity and rank posteriors are undefined. Villani's proper prior on $(\alpha,\Omega)$ is the minimal fix that renders the comparison well-defined.

VECM for Multi-Population Mortality (Zhou et al. 2013)

Cointegration arises naturally in multi-population mortality modeling. Under the Lee-Carter structure (Lee and Carter 1992), the log central death rate for population $i$ at age $x$ and time $t$ is $\ln(m_{x,t}^{(i)}) = \alpha_x^{(i)} + \beta_x\kappa_t^{(i)}$ , where $\kappa_t^{(i)}$ is a non-stationary mortality index. The non-divergence hypothesis — that mortality levels in related populations cannot drift apart indefinitely — translates exactly into the requirement that $(\kappa_t^{(1)}, \kappa_t^{(2)})$ are cointegrated with cointegrating vector $(1, -1)$ :

$\kappa_t^{(1)} - \kappa_t^{(2)} \xrightarrow{t\to\infty} \text{constant}$

This motivates the bivariate VECM for the $\kappa$ factors:

$\Delta\kappa_t^{(1)} = \rho^{(1)}(\kappa_{t-1}^{(1)} - \kappa_{t-1}^{(2)}) + \phi_0' + \phi_1\Delta\kappa_{t-1}^{(1)} + \phi_2\Delta\kappa_{t-1}^{(2)} + \epsilon_t^{(1)}$ $\Delta\kappa_t^{(2)} = \rho^{(2)}(\kappa_{t-1}^{(1)} - \kappa_{t-1}^{(2)}) + \theta_0' + \theta_1\Delta\kappa_{t-1}^{(1)} + \theta_2\Delta\kappa_{t-1}^{(2)} + \epsilon_t^{(2)}$

The adjustment coefficients $\rho^{(1)} < 0$ and $\rho^{(2)} < 0$ (both estimated negative in the UK data: $-0.208$ and $-0.105$ ) confirm that both populations are pulled back toward equilibrium, but at different speeds.

Key advantages over the RWAR (random walk + autoregressive) baseline (Cairns et al. 2011):

Symmetric: neither population is assumed dominant. In the UK example the smaller population (insured lives) leads the larger (England & Wales [E&W] males) at lags 3 and 13 — the RWAR assumption is empirically wrong.
Automatic non-divergence: the error-correction mechanism enforces the cointegrating relation without an explicit parameter constraint (unlike the VAR alternative).
Better fit: Bayesian information criterion (BIC) = 174.6 (VECM) vs. 176.4 (VAR) vs. 179.2 (RWAR); residuals flat vs. systematic downward trends in RWAR and VAR.

See Lee-Carter Model and Longevity Basis Risk for the broader context.

GARCH Heteroscedasticity and Cointegration Testing (BDV 1998)

Standard Johansen (1991) rank tests assume that the VECM innovations $\varepsilon_t$ are homoscedastic. In practice, financial data exhibit volatility clustering (GARCH effects), which inflates the residual variance estimate and can reduce the apparent spread between eigenvalues in the reduced-rank regression — biasing tests toward non-rejection of the null of no cointegration.

VECM-BEKK model. Bauwens, Deprins, and Vandeuren (BDV, 1998) estimate a bivariate VECM for the long rate $R_t$ and short rate $r_t$ of five countries with BEKK-GARCH innovations (see GARCH and BEKK-GARCH):

$\Delta \begin{pmatrix} R_t \\ r_t \end{pmatrix} = \mu + \sum_{i=1}^{k-1} \Gamma_i \Delta \begin{pmatrix} R_{t-i} \\ r_{t-i} \end{pmatrix} + \alpha\beta' \begin{pmatrix} R_{t-1} \\ r_{t-1} \end{pmatrix} + \varepsilon_t, \qquad \varepsilon_t | \mathcal{F}_{t-1} \sim \mathcal{N}(0, H_t)$

where $H_t$ follows a BEKK(1,1) process. Estimation is by quasi-maximum likelihood (QML): the full Gaussian log-likelihood $\ell(\theta) = -\frac{1}{2}\sum_t [\log|H_t| + \varepsilon_t'H_t^{-1}\varepsilon_t]$ is maximized jointly over all VECM and GARCH parameters.

Present value motivation. Campbell and Shiller (1987) show that if long rates are rational expectations of future short rates, the spread $R_t - \gamma r_t$ must be stationary, implying $\beta = (1, -\gamma)'$ . Under the pure expectations hypothesis, $\gamma = 1$ . BDV accept $\gamma = 1$ for France, UK, and USA but not Belgium and Germany.

Main empirical findings (BDV 1998):

GARCH sharpens cointegration evidence. Johansen tests on raw residuals reject cointegration for UK and Germany; fitting the joint VECM-BEKK shifts the evidence toward cointegration rank 1 for both countries.
Asymmetric adjustment. In all five countries, $|\hat\alpha_{r}| > |\hat\alpha_{R}|$ : the short rate $r_t$ corrects faster toward the long-run equilibrium than the long rate $R_t$ , consistent with central banks targeting short-term rates.
Multivariate IGARCH (Integrated GARCH). The covariance-stationarity eigenvalue $\rho(A_1\otimes A_1 + G_1 \otimes G_1) > 1$ in all countries — the unconditional covariance matrix does not exist.
Off-diagonal volatility spillovers. The diagonal BEKK restriction is rejected for all countries except the USA, implying shocks to one rate feed into conditional volatility of the other.

The broader methodological lesson is that joint estimation of long-run (cointegrating) and short-run volatility (GARCH) dynamics is preferable to two-step approaches that ignore the interaction. See GARCH and BEKK-GARCH for the BEKK model details and covariance stationarity condition.

Steady State and Cointegration (Villani 2008)

The steady state VECM of Villani (2008) augments (1) with informative priors on the mean growth rate $\gamma = E(\Delta x_t)$ and on the mean of the cointegrating relations $\mu_0 = E(\beta'x_t)$ :

$\Gamma(L)(\Delta x_t - \gamma) = \alpha(\beta'x_{t-1} - \mu_0 - \mu_1 t) + \varepsilon_t$

with the constraint $\beta'\gamma = \mu_1$ . See Steady State VAR for details.

Why It Matters

Long-run equilibrium relationships (PPP, consumption-wealth ratio, money demand) are cointegration hypotheses — cointegration testing is the empirical standard for verifying them. Purchasing Power Parity (PPP) example: Engel-Kim (1999) fail to find cointegration between U.S. prices, U.K. prices, and the nominal exchange rate with vector $(1,-1)$ over 106 years of monthly data — the real exchange rate has a permanent non-stationary component. See Purchasing Power Parity.
Ignoring cointegration and estimating a VAR in differences discards long-run information; estimating in levels is valid but inefficient.
The VECM provides separate inference on the long-run parameters ( $\beta$ ) and short-run dynamics ( $\Gamma_i, \alpha$ ), enabling hypothesis tests about economic relationships.
Variance analogue — co-persistence (Bollerslev-Engle 1993): the same algebraic structure extends to second moments. IGARCH series may share a common persistent variance factor, so a portfolio (co-persistent vector) has a stationary conditional variance even though each individual series is IGARCH. Co-persistence = fewer persistent variance factors than assets; the co-persistent vector eliminates the explosive eigenvectors of $A_1+B_1$ . See GARCH and BEKK-GARCH.

Open Questions

Determination of $r$ is sensitive to the choice of deterministic specification and sample length; pretesting uncertainty propagates into subsequent inference. The Warne (2006) framework quantifies this sensitivity via $p(r,k|Y)$ rather than hiding it behind a point estimate.
When $r$ is treated as unknown, Bayesian model averaging over $r \in \{0,\ldots,n\}$ is possible (Koop-Potter-Strachan 2005; Warne 2006) but computationally demanding for large $n$ .
High-dimensional cointegrated systems ( $n > 10$ ) present challenges for both frequentist testing and Bayesian posterior computation.
The opportunity cost semi-elasticity in money demand applications can be data-uninformative even in large samples (Warne 2006 euro area M3 finding), illustrating that posterior uncertainty may remain wide regardless of sample size.
I(1) pre-classification is an imperfect foundation for cointegration analysis: stochastic unit root processes (Granger-Swanson 1994), near-unit-root AR processes, and I(0) with broken trends all fail to reject the I(1) null yet behave differently in long-run relationships. The Gonzalo-Granger (1995) common-trends construction is robust to such misclassification, but the Johansen rank tests are not.