Cointegrated CKSVAR Analysis

• Cointegrated CKSVAR is a nonlinear censored and kinked SVAR model that employs threshold-based regime switching to capture distinct long-run cointegration relations.
• The model integrates regime-dependent coefficients and nonlinear error correction, facilitating structural decomposition and identification of common stochastic trends.
• Inference in Cointegrated CKSVAR leverages modified tests and joint spectral radius conditions to accurately capture stability and long-run dynamics.

Searching arXiv for recent and foundational papers on cointegrated CKSVAR and related nonlinear cointegration. Cointegrated CKSVAR denotes a cointegrated censored and kinked structural vector autoregression, a nonlinear SVAR in which one distinguished variable governs regime changes through thresholding or sign restrictions, while the system nevertheless admits a coherent long-run decomposition into common stochastic trends and cointegrating relations. In the canonical formulation, the state is $z_t=(y_t,x_t^\top)^\top$, with $y_t$ the variable subject to an occasionally binding constraint and $x_t\in\mathbb{R}^{p-1}$ the remaining variables. The central theoretical problem is to characterize when such a nonlinear system has $q$ common trends and $r=p-q$ cointegrating relations, how those relations depend on regime, and how long-run restrictions can still be formulated in structural form (Duffy et al., 2022, Duffy et al., 2024).

1. Structural form and canonical representation

A CKSVAR is built from the threshold decomposition

$$y_t^{+}=\max\{y_t,0\},\qquad y_t^{-}=\min\{y_t,0\},\qquad y_t=y_t^{+}+y_t^{-},$$

so the nonlinearity enters through the positive and negative parts of a single variable. In structural form, the CKSVAR($k$) is

$$\phi_0^{+} y_t^{+} + \phi_0^{-} y_t^{-} + \Phi_0^x x_t = c + \sum_{i=1}^k\left[\phi_i^{+} y_{t-i}^{+} + \phi_i^{-} y_{t-i}^{-} + \Phi_i^x x_{t-i}\right] + u_t,$$

or, equivalently, $$\phi^{+}(L)y_t^{+}+\phi^{-}(L)y_t^{-}+\Phi^x(L)x_t=c+u_t$$. The “censored” aspect refers to the truncation induced by $y_t^{+}$, while the “kinked” aspect refers to the regime-dependent slope change induced by $y_t$0 versus $y_t$1 (Duffy et al., 2022).

Under the coherence conditions imposed on the contemporaneous coefficients, the model can be transformed into a canonical representation in which the contemporaneous matrix is normalized to identity on $y_t$2. In the canonical form used for cointegration analysis,

$y_t$3

This two-regime, piecewise-linear SVAR is the baseline CKSVAR studied in the recent common-trend literature, including the formulation with $y_t$4 that linearizes the regime-dependent long-run relations in an augmented state space (Duffy et al., 30 Jul 2025).

2. Cointegration, common trends, and rank structure

Because threshold nonlinearity generally prevents ordinary difference stationarity from being the appropriate asymptotic notion, the CKSVAR literature introduces the weaker classifications $y_t$5 and $y_t$6. A process $y_t$7 is $y_t$8 if $y_t$9, and $x_t\in\mathbb{R}^{p-1}$0 if $x_t\in\mathbb{R}^{p-1}$1 for a nondegenerate limit process. This admits long-run limits such as regulated, censored, and kinked Brownian motions rather than only linear Brownian motion (Duffy et al., 2022).

Regime-specific long-run matrices are defined by

$x_t\in\mathbb{R}^{p-1}$2

The unit-root conditions require $x_t\in\mathbb{R}^{p-1}$3 to have $x_t\in\mathbb{R}^{p-1}$4 roots at $x_t\in\mathbb{R}^{p-1}$5 and all other roots outside the unit circle, with $x_t\in\mathbb{R}^{p-1}$6. In the important “case (ii)” configuration, both regimes have the same cointegration rank $x_t\in\mathbb{R}^{p-1}$7, the common-trend dimension is $x_t\in\mathbb{R}^{p-1}$8, and $x_t\in\mathbb{R}^{p-1}$9. Then

$q$0

with $q$1 allowed. The resulting cointegrating relations are nonlinear in levels: $q$2 or, equivalently, $q$3 in the augmented representation (Duffy et al., 30 Jul 2025).

The CKSVAR literature distinguishes three generic rank configurations.

Case	Rank pattern	Long-run implication
I (“regulated cointegration”)	$q$4	Common trends live in $q$5; regulated trend
II (“kinked cointegration”)	$q$6	Regime-dependent cointegration spaces; kinked trend
III (“linear cointegration in nonlinear VECM”)	$q$7	$q$8; trends only in $q$9

In Case I, the negative regime imposes one additional long-run restriction, and the shared trend becomes a regulated Brownian motion. In Case II, the rank is unchanged across regimes but the cointegrating space changes with the sign of $r=p-q$0, producing genuine nonlinear cointegration. In Case III, the constrained variable is stationary and the long-run nonstationarity is confined to $r=p-q$1, which yields a nonlinear VECM with linear cointegration (Duffy et al., 2022).

3. Representation theorems and long-run dynamics

The linear reduced-form foundation is supplied by the extended representation theorem for unit-root VARs. For

$r=p-q$2

if $r=p-q$3 is a unit root of multiplicity $r=p-q$4, then $r=p-q$5, and there exists an explicit matrix $r=p-q$6 such that $r=p-q$7. The nonstationary component is generated by the principal part of the Laurent expansion

$r=p-q$8

and annihilating that principal part isolates the stationary error-correction component. Closed forms are derived up to $r=p-q$9, with extension to higher orders argued by induction. For cointegrated CKSVAR, this provides the reduced-form template for permanent/transitory decomposition and for identifying common-trend directions through the pole structure of $$y_t^{+}=\max\{y_t,0\},\qquad y_t^{-}=\min\{y_t,0\},\qquad y_t=y_t^{+}+y_t^{-},$$0 (Faliva et al., 2021).

The nonlinear extension replaces the fixed linear cointegrating space with a regime-dependent or nonlinear one. In Case I of the CKSVAR theory, the normalized process satisfies

$$y_t^{+}=\max\{y_t,0\},\qquad y_t^{-}=\min\{y_t,0\},\qquad y_t=y_t^{+}+y_t^{-},$$1

so $$y_t^{+}=\max\{y_t,0\},\qquad y_t^{-}=\min\{y_t,0\},\qquad y_t=y_t^{+}+y_t^{-},$$2 is a regulated Brownian motion reflected at zero. In Case II,

$$y_t^{+}=\max\{y_t,0\},\qquad y_t^{-}=\min\{y_t,0\},\qquad y_t=y_t^{+}+y_t^{-},$$3

which is a kinked Brownian motion because the projection matrix changes when $$y_t^{+}=\max\{y_t,0\},\qquad y_t^{-}=\min\{y_t,0\},\qquad y_t=y_t^{+}+y_t^{-},$$4 changes sign. These results are described as the first extension to date of the Granger–Johansen representation theorem to a nonlinearly cointegrated setting (Duffy et al., 2022).

A more general framework is provided by additively time-separable nonlinear SVARs,

$$y_t^{+}=\max\{y_t,0\},\qquad y_t^{-}=\min\{y_t,0\},\qquad y_t=y_t^{+}+y_t^{-},$$5

with nonlinear VECM term $$y_t^{+}=\max\{y_t,0\},\qquad y_t^{-}=\min\{y_t,0\},\qquad y_t=y_t^{+}+y_t^{-},$$6. Under the common row space condition

$$y_t^{+}=\max\{y_t,0\},\qquad y_t^{-}=\min\{y_t,0\},\qquad y_t=y_t^{+}+y_t^{-},$$7

the nonlinear Granger–Johansen representation writes $$y_t^{+}=\max\{y_t,0\},\qquad y_t^{-}=\min\{y_t,0\},\qquad y_t=y_t^{+}+y_t^{-},$$8 as the inverse image, under a homeomorphism $$y_t^{+}=\max\{y_t,0\},\qquad y_t^{-}=\min\{y_t,0\},\qquad y_t=y_t^{+}+y_t^{-},$$9, of a $k$0-dimensional partial sum of shocks plus an exponentially stable equilibrium-error component. As a corollary, such models support the same kinds of long-run identifying restrictions as linearly cointegrated SVARs (Duffy et al., 2024).

4. Stationary components, error correction, and stability

The stationary side of cointegrated CKSVAR is governed by a nonlinear error-correction recursion rather than by a fixed linear companion matrix. In the linear-cointegration configuration with stationary constrained variable, the CKSVAR VECM admits the factorization

$k$1

and the stacked error-correction process

$k$2

satisfies

$k$3

If the associated deterministic subsystem is stable, then $k$4 is $k$5-geometrically ergodic, and in particular $k$6, $k$7, and $k$8 are geometrically ergodic (Duffy et al., 2023).

The crucial point is that stability cannot, in general, be inferred from the eigenvalues of the individual regime matrices alone. CKSVAR dynamics depend on admissible products of regime matrices induced by endogenous switching, so the relevant objects are the joint spectral radius, constrained joint spectral radius, and relaxed joint spectral radius. The relaxed joint spectral radius is designed to exploit the actual regime geometry and feasible transitions, and the paper states that the resulting conditions are less conservative than those typically available for general VTAR models (Duffy et al., 2023).

The nonlinear cointegration theory uses related stability requirements. In Case I and Case II, the transitory block is represented by regime-switching recursions whose joint spectral radius is required to be strictly less than one. This guarantees that the equilibrium-error terms, first differences, and auxiliary short-run states are $k$9, while the common-trend block remains $$\phi_0^{+} y_t^{+} + \phi_0^{-} y_t^{-} + \Phi_0^x x_t = c + \sum_{i=1}^k\left[\phi_i^{+} y_{t-i}^{+} + \phi_i^{-} y_{t-i}^{-} + \Phi_i^x x_{t-i}\right] + u_t,$$0 (Duffy et al., 2022).

5. Inference on rank and common trends

Inference in cointegrated CKSVAR proceeds along two distinct but complementary lines. The first concerns reduced-form rank determination under heteroskedasticity. In heteroskedastic VAR models with non-stationary unconditional volatility,

$$\phi_0^{+} y_t^{+} + \phi_0^{-} y_t^{-} + \Phi_0^x x_t = c + \sum_{i=1}^k\left[\phi_i^{+} y_{t-i}^{+} + \phi_i^{-} y_{t-i}^{-} + \Phi_i^x x_{t-i}\right] + u_t,$$1

standard Johansen procedures can be asymptotically distorted. Adaptive information criteria are built from a nonparametric estimator of $$\phi_0^{+} y_t^{+} + \phi_0^{-} y_t^{-} + \Phi_0^x x_t = c + \sum_{i=1}^k\left[\phi_i^{+} y_{t-i}^{+} + \phi_i^{-} y_{t-i}^{-} + \Phi_i^x x_{t-i}\right] + u_t,$$2, and the resulting ALS-BIC and ALS-HQC are weakly consistent for lag order and cointegration rank when $$\phi_0^{+} y_t^{+} + \phi_0^{-} y_t^{-} + \Phi_0^x x_t = c + \sum_{i=1}^k\left[\phi_i^{+} y_{t-i}^{+} + \phi_i^{-} y_{t-i}^{-} + \Phi_i^x x_{t-i}\right] + u_t,$$3 and $$\phi_0^{+} y_t^{+} + \phi_0^{-} y_t^{-} + \Phi_0^x x_t = c + \sum_{i=1}^k\left[\phi_i^{+} y_{t-i}^{+} + \phi_i^{-} y_{t-i}^{-} + \Phi_i^x x_{t-i}\right] + u_t,$$4. The same framework supports adaptive bootstrap pseudo-likelihood ratio tests, and the empirical application to the U.S. term structure reports that, with $$\phi_0^{+} y_t^{+} + \phi_0^{-} y_t^{-} + \Phi_0^x x_t = c + \sum_{i=1}^k\left[\phi_i^{+} y_{t-i}^{+} + \phi_i^{-} y_{t-i}^{-} + \Phi_i^x x_{t-i}\right] + u_t,$$5, BIC-based procedures select rank $$\phi_0^{+} y_t^{+} + \phi_0^{-} y_t^{-} + \Phi_0^x x_t = c + \sum_{i=1}^k\left[\phi_i^{+} y_{t-i}^{+} + \phi_i^{-} y_{t-i}^{-} + \Phi_i^x x_{t-i}\right] + u_t,$$6, while HQC and adaptive PLR procedures can select $$\phi_0^{+} y_t^{+} + \phi_0^{-} y_t^{-} + \Phi_0^x x_t = c + \sum_{i=1}^k\left[\phi_i^{+} y_{t-i}^{+} + \phi_i^{-} y_{t-i}^{-} + \Phi_i^x x_{t-i}\right] + u_t,$$7 or $$\phi_0^{+} y_t^{+} + \phi_0^{-} y_t^{-} + \Phi_0^x x_t = c + \sum_{i=1}^k\left[\phi_i^{+} y_{t-i}^{+} + \phi_i^{-} y_{t-i}^{-} + \Phi_i^x x_{t-i}\right] + u_t,$$8 depending on the penalty or significance level (Boswijk et al., 2022).

The second concerns direct inference on the number of common trends in a nonlinear CKSVAR. For the two-regime piecewise-linear SVAR with known sign-based nonlinear cointegration, the standard Breitung multivariate variance ratio test is not valid because linear combinations of $$\phi_0^{+} y_t^{+} + \phi_0^{-} y_t^{-} + \Phi_0^x x_t = c + \sum_{i=1}^k\left[\phi_i^{+} y_{t-i}^{+} + \phi_i^{-} y_{t-i}^{-} + \Phi_i^x x_{t-i}\right] + u_t,$$9 fail to isolate all nonlinear cointegrating directions. The modified procedure instead works with the augmented state

$$\phi^{+}(L)y_t^{+}+\phi^{-}(L)y_t^{-}+\Phi^x(L)x_t=c+u_t$$0

forms

$$\phi^{+}(L)y_t^{+}+\phi^{-}(L)y_t^{-}+\Phi^x(L)x_t=c+u_t$$1

and tests $$\phi^{+}(L)y_t^{+}+\phi^{-}(L)y_t^{-}+\Phi^x(L)x_t=c+u_t$$2 with

$$\phi^{+}(L)y_t^{+}+\phi^{-}(L)y_t^{-}+\Phi^x(L)x_t=c+u_t$$3

The paper proves that the modified statistic yields correct inferences regarding the number of common trends, whereas the unmodified test tends to infer a higher number of common trends than are actually present when cointegrating relations are nonlinear (Duffy et al., 30 Jul 2025).

The asymptotic theory for that test requires a new law-of-large-numbers-type result for stable but nonstationary regime-switching autoregressions. The core device is a dual linear process approximation, under which averages of functions of the nonlinear equilibrium-error process converge to regime-occupation-weighted mixtures of the stationary plus- and minus-regime limits (Duffy et al., 30 Jul 2025).

6. Long-run identification, scope, and limitations

Relative to linear Johansen theory, several features are preserved. Cointegration is still characterized by a decomposition into trend directions and equilibrium-error directions; $$\phi^{+}(L)y_t^{+}+\phi^{-}(L)y_t^{-}+\Phi^x(L)x_t=c+u_t$$4 remains the number of common stochastic trends and $$\phi^{+}(L)y_t^{+}+\phi^{-}(L)y_t^{-}+\Phi^x(L)x_t=c+u_t$$5 the rank of cointegration; and structural shocks can still be partitioned into permanent and transitory disturbances by long-run restrictions. In the nonlinear SVAR framework, if shocks are written as $$\phi^{+}(L)y_t^{+}+\phi^{-}(L)y_t^{-}+\Phi^x(L)x_t=c+u_t$$6, then a shock has no permanent effect if and only if it lies in $$\phi^{+}(L)y_t^{+}+\phi^{-}(L)y_t^{-}+\Phi^x(L)x_t=c+u_t$$7, equivalently

$$\phi^{+}(L)y_t^{+}+\phi^{-}(L)y_t^{-}+\Phi^x(L)x_t=c+u_t$$8

This is the nonlinear analogue of the usual long-run zero restrictions from cointegrated SVAR analysis (Duffy et al., 2024).

At the same time, the CKSVAR literature imposes important scope conditions. The nonlinear cointegration results are derived for a known piecewise-linear sign-based form, with two regimes and threshold normalized to zero; the modified common-trend test assumes that this nonlinear form is known; and the principal asymptotic theory often uses the restriction that no deterministic trends appear in the variables. The stationarity papers are explicit that their criteria are sufficient conditions tied to the deterministic subsystem and to refinements of the joint spectral radius, rather than simple closed-form checks. The linear Laurent-expansion theory is fully explicit only up to $$\phi^{+}(L)y_t^{+}+\phi^{-}(L)y_t^{-}+\Phi^x(L)x_t=c+u_t$$9, although higher-order extension is argued to be within reach by induction (Duffy et al., 30 Jul 2025, Duffy et al., 2023, Faliva et al., 2021).

The resulting concept of cointegrated CKSVAR is therefore technically narrower than the phrase might suggest, but within that scope it is precise. It covers piecewise-linear structural systems with occasionally binding constraints, regime-dependent short-run adjustment, and either linear or nonlinear long-run equilibria; it supplies Granger–Johansen-type representation theorems for those systems; and it restores the possibility of common-trend analysis and long-run structural identification in settings where differencing to stationarity would discard the central nonlinearity of interest (Duffy et al., 2022).