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Cointegration with Occasionally Binding Constraints
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\footnotetext[1]{Department\ of Economics and Corpus Christi College; [email removed]}
\footnotetext[2]{Department of Economics and University College; [email removed].}
\footnotetext[3]{Department of Economics; [email removed]}
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We thank B. Beare, P. Bonomolo, H. P. Boswijk, A. Bykhovskaya, G. Cavaliere, R. Engle, J. Gao, D. Harris, X. Jiao, S. Johansen, I. Kasparis, D. Kristensen, M. Kulish, O. Lieberman, O. Linton, Y. Lu, T\@. Magdalinos, J. Morley, U. M{\"u}ller, L. Neri, B. Nielsen, A. Onatski, M. Plagborg-M{\o}ller, W.-K. Seo, A. Srakar, J. Stock, A. M. R. Taylor, and participants in seminars at Aarhus, BI Oslo, Cambridge, Cyprus, Melbourne, Monash, Oxford, St.\ Andrews, Stanford, Sydney, the VTSS workshop, the May 2023 Harvard conference in honour of Jim Stock and Mark Watson, and the 24th Zaragoza Workshop in Time Series Econometrics, for helpful comments on earlier drafts of this work. This research was supported by the European Research Council via Consolidator Grant 647152.
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Nonstationarity, in the form of highly persistent, randomly wandering time series, is ubiquitous in macroeconomics and finance. It presents both a challenge for inference (Stock1994; Watson1994) and an opportunity for the identification of dynamic causal effects (BlanchardQuah89). The canonical framework for modelling such series is as (common) stochastic trends generated by a linear vector autoregression (VAR) with unit roots, underpinned by the powerful Granger--Johansen representation theorem (Joh95, Ch.\ 4). However, this framework is inadequate when even one of the variables is subject to an occasionally binding constraint, such as the zero lower bound (ZLB) constraint on short-term nominal interest rates, which has recently gained particular prominence in macroeconomic policy analysis (e.g.\ Sum14BE; Will14; EMR19AEJM; Kocherlakota2019).
In an autoregressive model, an occasionally binding constraint naturally gives rise to nonlinearity in the form of multiple endogenously switching regimes, whose presence significantly complicates the links between unit roots and stochastic trends. While it has become increasingly common to introduce stochastic trends into (linear) empirical models of monetary policy by treating the `natural rate of interest' or `trend inflation' as latent random walks (LW03REStat; CS08AER; delNegroGiannoneGiannoniTambalotti2017; AndradeGaliLeBihanMatheron2019; BauerRudebusch2020; SchmittGroheUribe2022), little is understood about what would happen to the implied time series properties of the observable series, such as the actual inflation rate and the nominal rate of interest, if nonlinearities were introduced into such models via the ZLB constraint. There is thus a pressing need to extend our understanding of how stochastic trends may be modelled beyond the linear VAR framework, to more general settings capable of accommodating such nonlinearities.
This paper addresses this problem in the setting of the censored and kinked structural VAR (CKSVAR) model (SM21,AMSV21), which provides a flexible yet tractable framework within which to model time series that are subject to occasionally binding constraints, and more generally to threshold-type nonlinearities. In this model, which is otherwise like a linear structural VAR, one series is allowed to enter differently according to whether it lies above or below a threshold; e.g.\ in applications to monetary policy, this series may be taken to represent the stance of monetary policy in each period, which above zero coincides with the short-term policy rate, and below zero corresponds to the (unobserved) `shadow rate'. We provide a complete characterisation of how common linear and nonlinear stochastic trends may be generated in this model, via unit roots and appropriate generalisations of the usual rank conditions, providing the first extension to date of the Granger--Johansen representation theorem to a nonlinearly cointegrated setting. Our results, which describe the behaviour of both the short- and long-run components of the CKSVAR, are foundational for frequentist inference in this setting, in the presence of common stochastic trends (a treatment of which will be given elsewhere).
As in a linear VAR, unit roots are unavoidable if one wishes to apply the CKSVAR to series with stochastic trends. The usual criticism of simply estimating a model in differences -- that this obliterates the identifying long-run information carried by the cointegrating relations -- is here magnified by the threshold nonlinearity in the model, which dictates whether the affected variable (e.g.\ interest rates) should enter in levels or differences, and in turn prescribes appropriate forms for the other variables.\footnote{That these other variables (which enter the model linearly) cannot generally be replaced by their first differences can be seen from the vector error correction form of the CKSVAR given in (ref) below, from which it is evident that making this replacement (of $x_{t}$ by $\Delta x_{t}$) would amount to imposing the restriction that $\Pi^{x}=0$.} To put it another way, that nonlinearity prevents series from being simply `differenced to stationarity'. Moreover, we show that in our setting, unit roots are not a mere technical nuisance: rather, their presence may impart significant identifying power to the low frequency behaviour of the series. For instance, the possibility that cointegrating relations between series may change as one series crosses a threshold, something accommodated by the CKSVAR, may be utilised to test hypotheses about the relative effectiveness of unconventional monetary policy (see Examples (ref) and (ref) below; this is a problem that has been studied econometrically by e.g.\ GHP14JCMB; WX16JCMB; DGG20NBER; and ILMZ20).
In analysing the CKSVAR with unit roots, we make a major contribution to the literature on nonlinear cointegration. Here a long-standing open problem relates to how a (nonlinear) vector autoregression, which provides a unified description of the short- and long-run dynamics of a collection of time series, can generate nonlinear cointegration between those series -- where `nonlinear cointegration' is understood in the profound sense of those series having common nonlinear stochastic trends with possibly nonlinear cointegrating relations between those trends. As discussed in the recent review by Tjo20EctRev, despite the voluminous literature on the subject of `nonlinear cointegration', this problem has yet to be addressed at any reasonable level of generality.\footnote{While CGT17JBES make an important effort in this direction, their results are limited to a first-order bivariate VAR with two regimes, in which one of those regimes is delimited by a compact set, and so makes a negligible contribution to the long-run behaviour of the series generated by the model. Their results are thus markedly different from those obtained below.} Within the framework of the CKSVAR, we provide a resolution of this problem, showing that the model naturally gives rise to nonlinear cointegration, generating nonlinear common trend processes -- censored, regulated, and kinked Brownian motions (see (ref) below) -- that have not previously appeared in multivariate settings.
To clarify how our work relates to the existing literature on `nonlinear cointegration', and to explain why we have been able to make progress in an area that has previously seemed intractable, we briefly recall the two main strands of that literature. One strand (Tjo20EctRev, p. 657) starts from the vector error correction model (VECM) representation of a cointegrated VAR, and introduces nonlinearity into the error correction mechanism; a prototypical model is
in which the usually linear loadings $\alpha[\beta^{\mathsf{T}}z_{t-1}]=\alpha\xi_{t-1}$ on the equilibrium errors are replaced by a general nonlinear function. In the original `threshold cointegration' conception of this model, due to BF97IER, $g$ is piecewise linear, i.e.\ $g(\xi_{t})=\sum_{i=1}^{m}\alpha^{(i)}\ensuremath{\mathbf{1}}\{\xi_{t}\in\Xi_{i}\}\xi_{t}$, where each of the $\alpha^{(i)}$'s correspond to different `regimes'. The values of $\{\Gamma_{i}\}$ may also depend on lags of $\xi_{t}$ or $\Delta z_{t}$. (For regime-switching versions, including of the smoothed variety, see e.g.\ HS02JoE; Saik05JoE,Saik08ET; and Seo11ET; for versions in which $g$ is allowed to be a more general nonlinear function, but the $\{\Gamma_{i}\}$ matrices are fixed, see EM02JTSA; and KR10JoE,KR13ET). Notably, the nonlinearity in such models is wholly confined to the short-run dynamics: as in a linearly cointegrated VAR, there remains a globally defined cointegrating space spanned by the columns of $\beta$ (i.e.\ which is common to all `regimes'), and the limiting common trends remain a (vector) Brownian motion.
The other strand (Tjo20EctRev, pp. 658--666) takes as its starting point the triangular representation of a linearly cointegrated system, and introduces nonlinearity directly into the common trends, by specifying
Here $f(x_{t})$ replaces what would ordinarily be a linear function, with the consequence that the weak limit of $Y_{n}(\lambda)\coloneqq n^{-1/2}y_{\smlfloor{n\lambda}}$ will now be a nonlinear transformation of the limiting Brownian motions associated with $X_{n}(\lambda)\coloneqq n^{-1/2}x_{\smlfloor{n\lambda}}$. The errors $\varepsilon_{t}=(\varepsilon_{yt},\varepsilon_{xt})$ may be weakly dependent and cross-correlated, permitting $\{x_{t}\}$ to be endogenous. The function $f$ is typically estimated via some sort of regression, either parametrically (PP99ET,PP01Ecta; CW15JoE; LTG16AS) or nonparametrically (KMT07AS; WP09Ecta,WP16ET; Duffy17ET; DK21AS); there is also literature on specification testing in this setting (e.g.\ WP12AS; DGTY17JoE; WWZ18JoE; BRN20ET). Notable variants have used $f$ to model transitions between regimes with distinct linear cointegrating relations (SC04ET; GP06OBES), or allowed it to take the `functional coefficient' form $\beta(w_{t})x_{t}$ (Xia09JoE,PW23JoE).
In developing a VAR model that exhibits both nonlinearity in its short-run dynamics, as in (ref), and in the implied (long-run) common trends, as in (ref), this paper is the first to bridge the remarkably wide gulf that has existed between these two strands of the literature. The CKSVAR turns out to provide just enough flexibility to accommodate meaningful departures from linear cointegration, while retaining a sufficiently nice structure to be tractable. We show that depending on the rank conditions imposed on (submatrices of) the autoregressive polynomial evaluated at unity, the CKSVAR is capable of generating three distinct kinds of nonlinear cointegration, which we term: (i) regulated cointegration; (ii) kinked cointegration; and (iii) linear cointegration in a nonlinear VECM.
At a technical level, our contribution consists of identifying the alternative configurations of the model that give rise to cases (i)--(iii), which are essentially exhaustive of the possibilities here, and deriving analogues of the Granger--Johansen representation theorem in these three cases.\footnote{Our analysis is exhaustive with respect to the possibilities for generating series that are integrated of order one (in a suitably extended sense of the term; see (ref)) within the CKSVAR; higher orders of integration are not considered here.} In analysing case (i), our work relates to that of Cav05ET, LLS11Bern, GLY13JoE, and most closely to BD22, all of whom obtain convergence to regulated Brownian motions in univariate models. Case (ii), even in the univariate setting, does not appear to have been considered by any previous literature. While cases (i) and (ii) describe phenomena that are entirely new to the literature, case (iii) holds under a configuration of the model that falls within the very broad class of nonlinear VECMs considered by Saik08ET: but even here, our results regarding the ergodicity of the short-memory components of the model extend his, insofar as we are able to exploit certain properties of the CKSVAR that are not shared by all the models encompassed by his general framework.
We illustrate the economic significance of cases (i) and (ii) by demonstrating how each may arise in our running example of a stylised structural model of monetary policy in the presence of a zero lower bound, contingent on the values taken by certain model parameters. We further apply our results to determine the long-run properties of the structural model of ABH23mimeo, for which the presence of nonlinear transformations of potentially stochastically trending series (due to a `long-run Phillips curve') precludes the application of any pre-existing version of the Granger--Johansen representation theorem.
The remainder of the paper is organised as follows. (ref) introduces the CKSVAR model and the stylised structural macroeconomic models that we use as running examples. (ref) develops the heuristics of the model with unit roots, outlining the tripartite classification noted above. The main results of this paper, which extend the Granger--Johansen representation theorem to the CKSVAR model, are given in (ref). All proofs appear in the appendices.
We consider a VAR($k$) model in $p$ variables, in which one series, $y_{t}$, enters with coefficients that differ according to whether it is above or below a time-invariant threshold $b$, while the other $p-1$ series, collected in $x_{t}$, enter linearly. Defining
we specify that $(y_{t},x_{t})$ follow
or, more compactly,
where
for $\phi_{i}^{\pm}\in\mathbb{R}^{p\times1}$ and $\Phi_{i}^{x}\in\mathbb{R}^{p\times(p-1)}$, and $L$ denotes the lag operator. As in a linear SVAR, $\{u_{t}\}$ may be an i.i.d.\ sequence of mutually orthogonal structural shocks, but our results below also permit them to be cross-correlated or weakly dependent. Through an appropriate redefinition of $y_{t}$ and $c$, we may take $b=0$ without loss of generality, and will do so throughout the sequel.\footnote{Defining $y_{b,t}\coloneqq y_{t}-b$, $y_{b,t}^{+}\coloneqq\max\{y_{b,t},0\}$, $y_{b,t}^{-}\coloneqq\min\{y_{b,t},0\}$ and $c_{b}\coloneqq c-[\phi^{+}(1)+\phi^{-}(1)]b$, we can rewrite (ref) as \[ \phi^{+}(L)y_{b,t}^{+}+\phi^{-}(L)y_{b,t}^{-}+\Phi^{x}(L)x_{t}=c_{b}+u_{t}. \] } In this case, $y_{t}^{+}$ and $y_{t}^{-}$ respectively equal the positive and negative parts of $y_{t}$, and $y_{t}=y_{t}^{+}+y_{t}^{-}$. (Throughout the following, the notation `$a^{\pm}$' connotes $a^{+}$ and $a^{-}$ as objects associated respectively with $y_{t}^{+}$ and $y_{t}^{-}$, or their lags. If we want to instead denote the positive and negative parts of some $a\in\mathbb{R}$, we shall do so by writing $[a]_{+}\coloneqq\max\{a,0\}$ or $[a]_{-}\coloneqq\min\{a,0\}$.)
Models of the form of (ref) have previously been employed in the literature to account for the dynamic effects of censoring, occasionally binding constraints, and endogenous regime switching. SM21 proposed exactly this model, which he termed the censored and kinked structural VAR (CKSVAR) model, to describe the operation of monetary policy during periods when a zero lower bound may bind on the policy rate: in our notation, $y_{t}$ corresponds to his `shadow rate', expressing the central bank's desired policy stance, and $y_{t}^{+}$ to the actual policy rate. AMSV21 considered a model in which one variable is subject to an occasionally binding constraint, which although in its initial formulation is somewhat more general, reduces to an instance of the CKSVAR once the conditions necessary for the model to have a unique solution (for all values of $u_{t}$) have been imposed (see their Proposition 1(i)). This version of their model -- i.e.\ that in which the `private sector regression functions' are piecewise linear and continuous -- is thus accommodated by (ref).\footnote{See also ACHSV21RED, for a DSGE model with an occasionally binding constraint, in which agents' decision rules are approximated by functions with these properties.}
To put some economic flesh on the bones of the representation theory developed in Sections (ref) and (ref) below, we here introduce the running example of a stylised structural model of monetary policy in the presence of a zero lower bound (ZLB). This model provides a simple, economically interpretable framework in which we may illustrate the various forms of novel long-run behaviour permitted by the CKSVAR, by considering alternate parametrisations of the model. Moreover, as discussed in (ref) below, the model elucidates how that long-run behaviour may provide identifying information on the relative effectiveness of unconventional monetary policy (as compared with conventional rate-setting policy), i.e.\ on whether the zero lower bound really constrains the ability of a central bank to target inflation.
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While both SM21 and AMSV21 motivate and interpret (ref) as a structural model, empirically motivated reduced-form models of this kind have also appeared in the literature, particularly in the univariate ($p=1$) case of (ref), which encompasses the dynamic Tobit model (Maddala83; for applications, see e.g.\ DJ02FRB; DJH11; BMMV21JBF; and Byk21JBES).
We follow SM21 and AMSV21 in maintaining the following, which are necessary and sufficient to ensure that (ref) has a unique solution for $(y_{t},x_{t})$, for all possible values of $u_{t}$. Define \[ \Phi_{0}\coloneqq
=
, \] $\Phi_{0}^{+}\coloneqq[\phi_{0}^{+},\Phi_{0}^{x}]$ and $\Phi_{0}^{-}\coloneqq[\phi_{0}^{-},\Phi_{0}^{x}]$.
{{{{\scalefont{0.76}DGP}}}}
For a further discussion of these conditions, including why (ref)(ref) may be maintained without loss of generality when (ref)(ref) holds, see DMW23stat. As in that paper, we shall designate a CKSVAR as canonical if
While it is not always the case that the reduced form of (ref) corresponds directly to a canonical CKSVAR, by defining the canonical variables
where $\bar{\phi}_{0,yy}^{\pm}\coloneqq\phi_{0,yy}^{\pm}-\phi_{0,yx}^{\mathsf{T}}\Phi_{0,xx}^{-1}\phi_{0,xy}^{\pm}>0$ and $P^{-1}$ is invertible under (ref); and setting
where
we obtain a canonical CKSVAR for $(\tilde{y}_{t},\tilde{x}_{t})$. This is formalised by the following, which reproduces the first part of Proposition 2.1 in DMW23stat.
To distinguish between a general CKSVAR in which possibly $\Phi_{0}\neq\Ican[p]$, and its associated canonical form as given by (ref), we shall refer to the former as the `structural form' of the CKSVAR. Since the time series properties of a general CKSVAR are largely inherited from its derived canonical form, we shall often work with this more convenient representation of the system, and indicate this as follows.
{{{{\scalefont{0.76}DGP$^{\ast}$}}}}
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It is well known that a linear VAR can faithfully replicate the high persistence, random wandering and long-run co-movement that is characteristic of a great many macroeconomic time series, via the imposition of unit autoregressive roots and the familiar rank conditions (Joh95). The question thus arises as to whether, and how, such behaviour may also be generated within a CKSVAR, so that the model might still be applied to series for which a stationary CKSVAR would be inappropriate. As we shall see, it is possible not only to accommodate linear cointegration within the CKSVAR, but also to generate a variety of nonlinear forms of cointegration, owing to the richer class of common trend processes that the model supports. Moreover, as our examples below illustrate, such departures from linear cointegration may also aid in the identification of structural parameters.
In developing the CKSVAR with unit roots, we shall find it necessary to depart from the usual classification of processes according to their orders of integration, since the nonlinearity in the model generally prevents it from generating series that are difference stationary. This is an issue commonly encountered in regime-switching cointegration models: see e.g.\ the discussion in GP06OBES, where this motivates the definition of the `order of summability' of a time series, and the allied notion of `co-summability' as a generalisation of linear cointegration (BRG14JoE, pp.\ 335f.). While those concepts could well be applied to the CKSVAR, the following properties, which may be more easily verified, will suffice for our purposes.
The preceding relates to the discussion of `cointegration' in MW13JoE, who consider a model with linear cointegration in which the common trends belong to a broad class of processes satisfying a weak convergence criterion (see their eq.\ (2)), which includes strict $I(1)$ and local-to-unity models as special cases, so that the vector of time series generated by the model is $I^{\ast}(1)$ as defined above. The bounded unit root processes of Cav05ET also provide an example of a series that converges weakly upon standardisation by $n^{-1/2}$, and so are $I^{\ast}(1)$, but not $I(1)$.
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{{\ref*{exa:dyntobit}}}
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Here we also need an enlarged notion of `cointegration' that is sufficiently general to encompass the possibilities of nonlinear cointegration accommodated by the CKSVAR model, particularly for the analysis of `case\ {{(ii)}}' below, such as is provided by the following (cf.\ GP06OBES, p. 817).
Our next step is to rewrite the CKSVAR, as in (ref) or (ref) above, in the form of a vector error-correction model (VECM). Define the autoregressive polynomials
\[ \Phi^{\pm}(\lambda)\coloneqq
, \]
and let $\Gamma_{i}^{\pm}\coloneqq-\sum_{j=i+1}^{k}\Phi_{j}^{\pm}\eqqcolon[\gamma_{i}^{\pm},\Gamma^{x}]$ for $i\in\{1,\ldots,k-1\}$, so that $\Gamma^{\pm}(\lambda)\coloneqq\Phi_{0}^{\pm}-\sum_{i=1}^{k-1}\Gamma_{i}^{\pm}\lambda^{i}$ is such that \[ \Phi^{\pm}(\lambda)=\Phi^{\pm}(1)\lambda+\Gamma^{\pm}(\lambda)(1-\lambda). \] Set $\pi^{\pm}\coloneqq-\phi^{\pm}(1)$ and $\Pi^{x}\coloneqq-\Phi^{x}(1)$. Then
where $\Delta\coloneqq1-L$ denotes the difference operator, and for clarity we note that $\Delta y_{t}^{+}=y_{t}^{+}-y_{t-1}^{+}$ (rather than being the positive part of $\Delta y_{t}$). In the case of a canonical CKSVAR, (ref) helpfully reduces to
While our main results apply to the general CKSVAR, the reader may find it helpful to work through the remainder of this section under the supposition that $(y_{t},x_{t})$ are generated by a canonical CKSVAR.
Just as in a linear (cointegrated) VAR, which corresponds to the special case of (ref) in which $\pi^{+}=\pi^{-}$ and $\gamma_{i}^{+}=\gamma_{i}^{-}$ for all $i\in\{1,\ldots,k-1\}$, the long-run dynamics will be governed by the matrix of coefficients on the lagged levels. More precisely, there are two such $p\times p$ matrices, $\Pi^{+}$ and $\Pi^{-}$, defined by
Although the canonical CKSVAR technically has $2^{k}$ distinct autoregressive `regimes' (corresponding to the possible sign patterns of $\b y_{t-1}\coloneqq(y_{t-1},\ldots,y_{t-k})^{\mathsf{T}}$; see also DMW23stat, Sec.\ 4.2), the behaviour of the CKSVAR with unit roots depends largely on the two regimes in which the elements of $\b y_{t-1}$ are all either positive or negative, which we shall loosely refer to as the `positive' and `negative' regimes, to which $\Pi^{+}$ and $\Pi^{-}$ correspond. This simplification occurs because whenever $y_{t}\sim I^{\ast}(1)$, it spends most of its time away from the origin, so that all elements of $\b y_{t-1}$ will have the same sign almost all of the time.
Our baseline assumptions on the CKSVAR with unit roots may now be stated.
{{{{\scalefont{0.76}CVAR}}}}
{{{{\scalefont{0.76}DET}}}}
{{{{\scalefont{0.76}ERR}}}}
In a linear VAR (i.e.\ where $\Phi^{+}(\lambda)=\Phi^{-}(\lambda)$) under (ref) the Granger--Johansen representation theorem (Joh95) implies that $z_{t}\coloneqq(y_{t},x_{t}^{\mathsf{T}})^{\mathsf{T}}\sim I^{\ast}(1)$ and that there exists a full column rank matrix $\beta\in\mathbb{R}^{p\times r}$ such that $\beta^{\mathsf{T}}z_{t}\sim I^{\ast}(0)$, where $r=r^{+}=r^{-}$. Thus $\{z_{t}\}$ is cointegrated in the sense of (ref), with cointegrating space $\operatorname{sp}\beta$. Here (ref) specialises to $c\in\operatorname{sp}\Phi(1)$, preventing the model from generating any common deterministic trends between the series; it has the same effect more generally when $\Pi^{+}\neq\Pi^{-}$. (We shall relax this condition, so as to allow for deterministic trends, in (ref) below.)
While linear cointegration may occur in a CKSVAR, other phenomena are possible, depending on the ranks of $\Pi^{+}$, $\Pi^{-}$ and $\Pi^{x}$. Within the framework of $I^{\ast}(0)$ and $I^{\ast}(1)$ processes, as delimited by (ref), there are three possibilities, each of which generate profoundly different trajectories for $\{y_{t}\}$. These are characterised in (ref); in each case $r$ (without a superscript) is defined such that it is possible to regard the system as having a cointegrating rank of at least $r$ in each `regime'. Since, $\Pi^{+}$ and $\Pi^{-}$ differ by only their first column, $r^{+}$ and $r^{-}$ may differ by at most one, so that the case where $r^{+}\neq r^{-}$ may be identified with case {{(i)}} in the table without loss of generality.\footnote{If we were to instead take $r^{+}=r^{-}+1$ in this case, the characteristic behaviour of the series in the positive and negative regimes, with $y_{t}^{+}\sim I^{\ast}(1)$ and $y_{t}^{-}\sim I^{\ast}(0)$, would now be reversed, but the key properties of the model would otherwise be unaltered.}
To develop some intuition for the properties of the model in these three cases, in advance of the representation theory developed in the next section, it is helpful to regard the long-run behaviour of the processes as being characterised by a space of common trends $\mathscr{M}$, defined as the set of non-stochastic (i.e.\ $u_{t}\coloneqq0$, $\forall t$) steady state solutions to (ref):
This space defines the domain of the limiting processes
In a linear VAR, $\mathscr{M}=\ker\Pi$ is a $q$-dimensional linear subspace of $\mathbb{R}^{p}$, the orthogonal complement of which is the cointegrating space; and $Z$ is a $p$-dimensional Brownian motion with rank $q$ covariance matrix -- i.e.\ it is a rank $q$ linear function of $U$ in (ref) above -- taking values in $\mathscr{M}$. Whereas in the CKSVAR, $\mathscr{M}$ no longer need be a linear subspace, but is instead a linear cone\footnote{Recall that a set $S\subset\mathbb{R}^{p}$ is termed a linear cone if $\lambda s\in S$ for every $s\in S$ and $\lambda\geq0$, i.e.\ if $S$ is closed under multiplication by non-negative scalars.} formed from the union of $\mathscr{M}^{+}$ and $\mathscr{M}^{-}$, the vectors orthogonal to which are the cointegrating vectors $\beta^{+}$ and $\beta^{-}$ on the half spaces $\mathscr{Z}^{+}$ and $\mathscr{Z}^{-}$ respectively (recall (ref) above). While $Z$ remains a function of $U$, that function need not be linear: indeed, in addition to (linear) Brownian motions (BMs), any of the following nonlinear processes may also appear among the limiting stochastic trends generated by the CKSVAR. (For further discussion of regulated BMs, see Har85book, Ch.\ 1.)
For $p=2$, various possible shapes of $\mathscr{M}$ are illustrated graphically in Figures (ref)--(ref), along with matching example trajectories for $(y_{t},x_{t})$. In all figures, $(y_{t},x_{t})$ are generated by a canonical CKSVAR with $k=1$, $c=0$, $u_{t}\ensuremath{\sim_{\ensuremath{\textnormal{i.i.d.}}}} N[0,I_{2}]$, and $y_{0}=x_{0}=0$; the specification of the model is completed by specifying $\pi^{+}$, $\pi^{-}$ and $\Pi^{x}$ in (ref), the values of which are given in each panel. The associated cointegrating vectors (on the half spaces $\mathscr{Z}^{+}$ and $\mathscr{Z}^{-}$) are the vectors orthogonal of $\mathscr{M}^{+}$ and $\mathscr{M}^{-}$, while the form of the limiting processes $Z$ is suggested by the shape of the set $\mathscr{M}$, where it concentrates. The main qualitative features of the three cases, as enumerated in (ref) above, are as follows.
The distinguishing characteristic of this case is that the common trends are restricted to the region where $y\geq0$, so that $Y$ will always be a Brownian motion regulated (from below) at zero, even though $y_{t}$ itself may take negative values. In the case that $q=1$, as e.g.\ when $p=2$ and $r=1$ as depicted in (ref), $X$ will also be a regulated process, i.e.\ we have cointegration where both processes share a common regulated stochastic trend. (When $q\geq2$, $X$ will also depend on $q-1$ additional linear BMs.) Accordingly, a model configured as in case {{(i)}} would be most appropriate when $y_{t}$ appears to wander randomly above a threshold, and makes only brief sojourns below that threshold.
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This case is perhaps more reminiscent of linear cointegration, which it accommodates as a special case. Here $\mathscr{M}^{+}$ and $\mathscr{M}^{-}$ trace out `half' subspaces of the same dimension $q$, but they need not be parallel, giving rise to a kink in $\mathscr{M}$ at the origin, with $\mathscr{M}$ itself being a linear cone. In general, $(Y,X)$ will follow a kinked Brownian motion driven by $p$ linear BMs, whose loadings, and the associated $r$ cointegrating relations, vary with the sign of $Y$. In the top panel of (ref), $y_{t}$ and $x_{t}$ are cointegrated, but with distinct cointegrating vectors $\beta^{+}=(1,-1)^{\mathsf{T}}$ or $\beta^{-}=(\tfrac{1}{2},-1)^{\mathsf{T}}$ applying on $\mathscr{Z}^{+}$ or $\mathscr{Z}^{-}$, i.e.\ when $y_{t}$ is positive or negative. We have a kind of `threshold cointegration', with the movement of $y_{t}$ across zero causing the model to switch between distinct cointegrating spaces (cf.\ SC04ET). This switch is reflected in the trajectories of $y_{t}$ and $x_{t}$: when $y_{t}\geq0$, the two series move together approximately one-for-one; whereas if $y_{t}\leq0$, $y_{t}$ changes by two units for every one-unit change in $x_{t}$, in the long run. Relative to case {{(i)}}, the trajectories of $\{y_{t}\}$ will now much more closely resemble those of a linear unit root process, and $\{y_{t}\}$ will accordingly tend to spend long stretches in both the positive and negative regions, with no tendency to revert to one or the other.
The bottom panel of (ref) presents an important special case where the cointegrating relationships are such that $\{x_{t}\}$ behaves like an $I^{\ast}(0)$ process when $y_{t}<0$, but cointegrates with $y_{t}$ when $y_{t}>0$; the limiting process $X$ will thus be a censored Brownian motion.
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This case, which is depicted in the (ref), entails that no trends are present in $\{y_{t}\}$, which is in fact a stationary process. The common trends are loaded entirely on $\{x_{t}\}$, and the cointegrating relationships between the elements of $x_{t}$ are unaffected by the sign of $y_{t}$, exactly as in literature on regime-switching nonlinear VECM models (see the text following (ref) above). For this reason, a model configured as in case {{(iii)}} is appropriate only when the univariate behaviour of $\{y_{t}\}$ appears stationary, in which case there will be frequent switches of `regime', particularly if the mean of $y_{t}$ is near zero.
We now proceed to develop analogues of the Granger--Johansen representation theorem for cases {{(i)}}--{{(iii)}}. These results are of interest in their own right, because by characterising the processes generated by the model, under alternative configurations of $\Pi^{\pm}$, they delimit the classes of observed time series that the model might be fruitfully applied to. Indeed, they indicate precisely the restrictions on $\Pi^{\pm}$ that might be appropriate for specific applications, according to whether $\{y_{t}\}$ is observed to either: wander randomly above a threshold, but spend only brief periods below it (case {{(i)}}); wander randomly on both sides of a threshold (case {{(ii)}}); or behave like a stationary process (case {{(iii)}}). Beyond such guidance, our results also lay the groundwork for the development of the asymptotics of likelihood-based estimators of the CKSVAR model in the presence of unit roots, by establishing the asymptotic behaviour of the (standardised) processes generated by the model.
To facilitate the exposition, we shall initially suppose the data to be generated by a canonical CKSVAR, i.e.\ (ref) holds, and that there are no deterministic trends in cases {{(i)}} and {{(ii)}}, i.e.\ (ref) holds. Theorems (ref)--(ref) below are stated under these assumptions: the minor modifications required when (ref) is replaced by (ref) are given in the subsequent remarks, while a general treatment of deterministic trends follows in (ref). To avoid the need to specify how some of the quantities below should be defined when $k=1$, we shall treat this as a special case of the model with $k=2$, in which $\phi_{2}^{+}=\phi_{2}^{-}=0$ and $\Phi_{2}^{x}=0$; and thus henceforth $k\geq2$, unless otherwise stated. Because of the overlap between the arguments used to analyse the CKSVAR in each case, it will be occasionally necessary to redefine objects that have already appeared in the discussion of another case. While we have endeavoured to keep such notational conflicts to a minimum, and explicitly indicated wherever these arise, the reader is advised to treat each of the following three subsections, and the accompanying subsections of (ref) where the proofs of the theorems appear, as independent of each other.
Our proofs make repeated use of companion form representations of the VECM (ref), formulated slightly differently for each of cases {{(i)}}--{{(iii)}}. In this respect, the nearest analogue to our arguments, in the setting of a linear VAR, is provided by Han05EctJ.
Recalling (ref), we have the familiar factorisation \[ \Pi^{+}=\alpha^{+}\beta^{+\mathsf{T}}, \] where $\alpha^{+},\beta^{+}\in\mathbb{R}^{p\times r}$ have rank $r$. As we show below, $\beta^{+}$ spans the cointegrating space on $\mathscr{Z}^{+}=\mathbb{R}_{+}\times\mathbb{R}^{p-1}$ (recall (ref)), so that $(y_{t}^{+},x_{t})\sim I^{\ast}(1)$ but $\beta^{+\mathsf{T}}(y_{t}^{+},x_{t})\sim I^{\ast}(0)$. Moreover, since $y_{t}^{-}\sim I^{\ast}(0)$, it follows that $\beta^{+\mathsf{T}}(y_{t},x_{t})\sim I^{\ast}(0)$, so that the columns of $\beta^{+}$ are globally cointegrating vectors.
In a linear cointegrated VAR, no assumptions additional to (ref) and (ref) are needed, but the nonlinearity of the CKSVAR prevents our assumptions on the roots of $\Phi^{\pm}(\lambda)$ from being sufficient to ensure that the short-memory components (i.e.\ the equilibrium errors and the first differences) are indeed $I^{\ast}(0)$. For this reason, two further regularity conditions are required. To state these, let
where $\varphi^{\pm}\coloneqq[\phi_{2}^{\pm},\ldots,\phi_{k}^{\pm}]$, $D\coloneqq[I_{k-2},0_{(k-2)\times1}]$; $\b{\alpha}^{+},\b{\beta}^{+}\in\mathbb{R}^{kp\times[p(k-1)+r]}$ with
and $\b{\beta}_{1:p}^{+}$ denotes the first $p$ rows of $\b{\beta}^{+}$.\footnote{To avoid any ambiguity, the definition of $\b{\beta}^{+\mathsf{T}}$ in (ref) should be read `row-wise', with the `\raisebox{-2pt}{$\smash{\ddots}$}' signifying that successive rows of the matrix are formed by replicating the relevant block (i.e.\ that to the upper left of the `\raisebox{-2pt}{$\smash{\ddots}$}'), shifting it to the right by the width of the block; all other entries are zeros. Thus for $m\in\{1,\ldots,k-1\}$, rows $r+mp+1$ to $r+mp$ of $\b{\beta}^{+\mathsf{T}}$ are given by $[0_{p\times(m-1)p},I_{p},-I_{p},0_{p\times(k-m-1)}]$. The definitions given in e.g.\ (ref) below should be interpreted similarly.} We also define
for $\alpha_{\perp}^{+}$ and $\beta_{\perp}^{+}$ the orthocomplements of $\alpha^{+}$ and $\beta^{+}$. Let $\rho(M)$ denote the spectral radius of $M\in\mathbb{R}^{m\times m}$, and for $\mathcal{A}\subset\mathbb{R}^{m\times m}$ a bounded collection of matrices, let \[ \rho_{{\scriptstyle \mathrm{JSR}}}(\mathcal{A})\coloneqq\limsup_{t\ensuremath{\rightarrow}\infty}\sup_{B\in\mathcal{A}^{t}}\rho(B)^{1/t} \] denote its joint spectral radius (JSR; e.g.\ Jungers09, Defn.\ 1.1), for $\mathcal{A}^{t}\coloneqq\{\prod_{s=1}^{t}M_{s}\mid M_{s}\in\mathcal{A}\}$ the set of $t$-fold products of matrices in ${\cal A}$.\footnote{For a further discussion of the JSR, and references to the literature on methods for numerically approximating it, see DMW23stat.} Let $z_{t}\coloneqq(y_{t},x_{t}^{\mathsf{T}})^{\mathsf{T}}$.
{{{{\scalefont{0.76}CO{{(i)}}}}}}
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We turn next to the case in which the cointegrating rank $r$ is the same across both the positive and negative regimes, though the cointegrating space itself need not be. Here we also suppose that $\operatorname{rk}\Pi^{x}=r$, which as discussed in (ref) entails that $y_{t}\sim I^{\ast}(1)$. Under the foregoing, we must have $\pi^{\pm}\in\operatorname{sp}\Pi^{x}$, and so
where $\alpha\in\mathbb{R}^{p\times r}$, $\beta_{x}\in\mathbb{R}^{(p-1)\times r}$ and $\beta^{\pm}\in\mathbb{R}^{p\times r}$ have rank $r$, and $\theta^{\pm}\in\mathbb{R}^{p-1}$ is such that $\Pi^{x}\theta^{\pm}=\pi^{\pm}$. Let $\ensuremath{\mathbf{1}}^{+}(y)\coloneqq\ensuremath{\mathbf{1}}\{y\geq0\}$ and $\ensuremath{\mathbf{1}}^{-}(y)\coloneqq\ensuremath{\mathbf{1}}\{y<0\}$, and set $\beta(y)\coloneqq\beta^{+}\ensuremath{\mathbf{1}}^{+}(y)+\beta^{-}\ensuremath{\mathbf{1}}^{-}(y)$.\footnote{There is unavoidably some arbitrariness with respect to how such objects are defined when $y=0$, but since these only play a role in the model when multiplied by $y$, it does not matter which convention is adopted. Throughout the paper, we use the functions $\ensuremath{\mathbf{1}}^{\pm}(y)$ to ensure that all such definitions are mutually consistent.} Then we can define the equilibrium errors as \[ \xi_{t}\coloneqq\beta(y_{t})^{\mathsf{T}}z_{t}=\ensuremath{\mathbf{1}}^{+}(y_{t})\beta^{+\mathsf{T}}z_{t}+\ensuremath{\mathbf{1}}^{-}(y_{t})\beta^{-\mathsf{T}}z_{t}. \] Observe how (ref) implies that the `loadings' $\alpha$ of the equilibrium errors will be the same in both regimes, even though the cointegrating vectors that define those errors need not be. (Case {{(iii)}} entails the opposite, with fixed cointegrating vectors but loadings that depend on the sign of $y_{t}$: see (ref) below.)
The theorem below establishes that $\xi_{t}\sim I^{\ast}(0)$, and that $\beta^{+}$ and $\beta^{-}$ span the cointegrating spaces on $\mathscr{Z}^{+}$ and $\mathscr{Z}^{-}$ respectively. The limiting common trends are kinked Brownian motions given by a kind of projection of $U$ onto $\mathscr{M}$, defined in terms of
where $\theta(y)\coloneqq\ensuremath{\mathbf{1}}^{+}(y)\theta^{+}+\ensuremath{\mathbf{1}}^{-}(y)\theta^{-}$. Such objects as $P_{\beta_{\perp}}(y)$ take only two distinct values, depending on the sign of $y$, and we routinely use the notation $P_{\beta_{\perp}}(+1)$ and $P_{\beta_{\perp}}(-1)$ to indicate these. Similarly to case {{(i)}}, beyond our assumptions on the ranks of $\Pi^{\pm}$ and $\Pi^{x}$, two further regularity conditions are needed to ensure that the system is well behaved. To state these, let $\b{\alpha},\b{\beta}(y)\in\mathbb{R}^{[k(p+1)-1]\times[r+(k-1)(p+1)]}$ with
where $\Gamma_{i}\coloneqq[\gamma_{i}^{+},\gamma_{i}^{-},\Gamma_{i}^{x}]$ for $i\in\{1,\ldots,k-1\}$, and
so that $S(y_{t})z_{t}=(y_{t}^{+},y_{t}^{-},x_{t}^{\mathsf{T}})^{\mathsf{T}}$, where $z_{t}=(y_{t},x_{t}^{\mathsf{T}})^{\mathsf{T}}$.
{{{{\scalefont{0.76}CO{{(ii)}}}}}}
Finally, we consider the other case with a regime-invariant cointegrating rank $r$, but in which $\operatorname{rk}\Pi^{x}=r-1$. Then we have the factorisation
where $\alpha_{x}\in\mathbb{R}^{p\times(r-1)}$ and $\beta_{x}\in\mathbb{R}^{(p-1)\times(r-1)}$ have rank $r-1$, and $\alpha^{\pm},\beta\in\mathbb{R}^{p\times r}$ have rank $r$. (Note that the dimensions of $\beta_{x}$ here differ from those in case {{(ii)}}.) The model is thus one in which the positive and negative regimes share a common cointegrating space, which contains $e_{p,1}$. Thus we would expect $y_{t}$ to behave like a stationary process: and indeed, the model falls within the very general framework of Saik08ET, whose results could be applied to establish the stationarity and ergodicity of the equilibrium errors \[ \xi_{t}\coloneqq\beta^{\mathsf{T}}z_{t}=
, \] and hence of $y_{t}$. Our technical contribution here is to exploit the structure of the CKSVAR so as to permit his conditions, which refer to the JSR of a collection of autoregressive matrices, to be relaxed to merely requiring the stability of a certain deterministic subsystem (for which control over the JSR is a sufficient but not necessary condition).\footnote{It should be emphasised that we do not claim to be relaxing the conditions of Saik08ET for the general class of regime-switching error correction models considered in that paper. Rather, we are able to exploit the fact that our model (the CKSVAR configured as per (ref)) is a special case of that framework to obtain weaker sufficient conditions for the ergodicity of $\beta^{\mathsf{T}}z_{t}$, in our setting.}
Our only regularity condition on the system, in this case, is a stability condition of this kind. To present the system to which this applies, define
for $i\in\{1,\ldots,k\}$, $\b y_{t-1}\coloneqq(y_{t-1},\ldots,y_{t-k})^{\mathsf{T}}$, and recognise that the factorisation (ref) applies more generally to \[ \Pi(\b y_{t-1})\coloneqq\sum_{i=1}^{k}\Phi(y_{t-i})-I_{p}=
\beta^{\mathsf{T}}\eqqcolon\alpha(\b y_{t-1})\beta^{\mathsf{T}}. \] Further, define $\b{\alpha}(\b y_{t-1}),\b{\beta}\in\mathbb{R}^{kp\times[r+(k-1)p]}$ as
for
all of which depend only on the signs of the elements of $\b y_{t-1}$. Collect the short memory components of the model as \[ \chi_{t}\coloneqq
=
\] for $\beta_{x,\perp}\in\mathbb{R}^{p-1}$ having $\operatorname{rk}\beta_{x,\perp}=p-r$, such that $\beta_{x,\perp}^{\mathsf{T}}\beta_{x}=0$. Letting $\b{\chi}_{t}\coloneqq(\chi_{t}^{\mathsf{T}},\ldots,\chi_{t-k+1}^{\mathsf{T}})^{\mathsf{T}}$, we show in the proof of (ref) below that \[ \chi_{t}=Bc+M[I_{p(k-1)+r}+\b{\beta}^{\mathsf{T}}\b{\alpha}(\b G\b{\chi}_{t-1})]\b H\b{\chi}_{t-1}+Bu_{t} \] for $B\in\mathbb{R}^{p\times p}$ invertible, $M\in\mathbb{R}^{p\times[p(k-1)+r]}$, $\b G\in\mathbb{R}^{k\times pk}$ and $\b H\in\mathbb{R}^{[p(k-1)+r]\times kp}$. $\{\b{\chi}_{t}\}$ thus evolves according to a regime-switching VAR, a Markov process that will be stationary and geometrically ergodic under the conditions given below.
The first of these conditions relates to the innovations $\{u_{t}\}$: for technical reasons, requiring that $u_{t}$ have a (conditional) Lebesgue density that is bounded away from zero, an assumption that is common in the literature on ergodic Markov processes, greatly facilitates our analysis.
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{{{{\scalefont{0.76}ERR$^{\prime}$}}}}
Our main condition on the model parameters is the following.
{{{{\scalefont{0.76}CO{{(iii)}}}}}}
Finally, to state our main result for case {{(iii)}}, we recall the following (cf.\ Lieb2005, p. 671; MS08JTSA, pp. 460f.).
If $\{w_{t}\}$ is $\mathcal{Q}$-geometrically ergodic, it will be stationary if given a stationary initialisation, i.e.\ if $w_{0}$ also has distribution $\pi$; moreover, it will have geometrically decaying $\beta$-mixing coefficients. For these and further properties, and a discussion of how this concept relates to other notions of ergodicity used in the literature, see Lieb2005.
To simplify the exposition of our results for cases {{(i)}} and {{(ii)}} above, we have so far maintained (ref), which prevents the model from generating any common deterministic trends. Since this is likely to be restrictive in applications -- there being many macroeconomic series that exhibit both stochastic and deterministic trends -- it is important to clarify that this simplifying assumption may be almost entirely dispensed with.
The presence of deterministic trends complicates the analysis of cases {{(i)}} and {{(ii)}}, because these deterministic trends will generally dominate any common stochastic trends, such that the limit theory developed in Theorems (ref) and (ref) no longer applies directly to the standardised process $Z_{n}(\lambda)\coloneqq n^{-1/2}z_{\smlfloor{n\lambda}}$. Nonetheless, as explained below, these results continue to provide an accurate description of the asymptotics of this process upon linear detrending. (Since we did not maintain (ref) in the context of case {{(iii)}}, we have nothing to say about that case in this section.)
We therefore now contemplate a model in which (ref) is relaxed, i.e.\ in which it is no longer required that $c\in\operatorname{sp}\Pi^{+}\ensuremath{\cap}\Pi^{-}$. In a linear cointegrated VAR, an unrestricted intercept $c\in\mathbb{R}^{p}$ permits the model to impart deterministic trends to all elements of $z_{t}=(y_{t},x_{t}^{\mathsf{T}})^{\mathsf{T}}$, but in such a way that these deterministic trends are eliminated by the cointegrating relations (simultaneously with the common stochastic trends; see Joh95, Sec. 5.7). This remains true in the CKSVAR, with the caveat that if a deterministic trend is imparted to $y_{t}$, then this dominant drift component will asymptotically push $y_{t}$ so far into either the positive or negative region (depending on the sign of the drift) as to render the threshold nonlinearity at zero irrelevant -- and so the series would come to be adequately described by a linear VAR.
Since we would only apply the CKSVAR when $\{y_{t}\}$ spent an appreciable portion of the sample on both sides of zero, it is appropriate to work within an asymptotic framework in which both regions continue to be visited with nonvanishing probability as $T\ensuremath{\rightarrow}\infty$.\footnote{Here we maintain that the location of the threshold delimiting the regimes is normalised to zero; of course the argument continues to apply if that threshold were instead fixed at some other finite level.} The cleanest way to ensure this is to restrict $c$ such that a deterministic trend may be imparted to $x_{t}$, but not to $y_{t}$. In the context of cases {{(i)}} and {{(ii)}}, essentially the same condition is required, but is denoted slightly differently so as to be consistent with the notation used in the analysis of these two cases. (Recall the definitions of $P_{\beta_{\perp}^{+}}$ and $P_{\beta_{\perp}}(+1)$ appearing in (ref) and (ref) above.)
{{{{\scalefont{0.76}DET$^{\prime}$}}}}
To state our results in terms of the structural form of the CKSVAR, we shall say that `(ref){{{\scalefont{0.76}$^\prime$}}} holds' whenever (ref) holds with (ref)(ref) replaced by (ref), and mutatis mutandis for (ref){{{\scalefont{0.76}$^\prime$}}} (see part (i) of Remarks (ref) and (ref) above). To deal with the possible presence of a deterministic trend in $x_{t}$, in case {{(i)}} we define the linearly detrended processes \[
=z_{t}^{d}\coloneqq z_{t}-(P_{\beta_{\perp}^{+}}c)t \] noting that $y_{t}^{d}=y_{t}$ under (ref); in case {{(ii)}}, $z_{t}^{d}$ is defined similarly with $P_{\beta_{\perp}^{+}}$ replaced by $P_{\beta_{\perp}}(+1)$. Define $X_{n}^{d}(\lambda)\coloneqq n^{-1/2}x_{\smlfloor{n\lambda}}^{d}$ to be the standardised process corresponding to $x_{t}^{d}$. The only modification that then needs to be made to the conclusions of Theorems (ref) and (ref) above is that $X_{n}$ should be replaced by $X_{n}^{d}$.
The CKSVAR provides a flexible yet tractable framework in which to structurally model vector time series subject to an occasionally binding constraint, such as the zero lower bound on interest rates, and more general threshold nonlinearities. Nonetheless, even that seemingly limited amount of nonlinearity radically changes the properties of the model relative to a linear VAR. When unit autoregressive roots are introduced into the model, it is able to accommodate varieties of long-run behaviour that cannot be generated within a linear VAR, such as nonlinear common stochastic trends (censored, regulated and kinked Brownian motions) and cointegrating relationships that may be regime-dependent. This is not merely a theoretical curiosity but rather something that, as our examples illustrate, allows the long-run properties of the model to carry useful identifying information on structural parameters, as might pertain e.g.\ to the relative effectiveness of unconventional monetary policy.
Our results provide a complete characterisation of the forms of nonlinear cointegration (between processes $\ast$-integrated of order one) generated by the CKSVAR. In deriving these, we have given the first treatment of how nonlinear cointegration, in the profound sense of nonlinear common stochastic trends and nonlinear cointegrating relations, may be systematically generated within a (nonlinear) VAR, and thus the first extension of the Granger--Johansen representation theorem to a nonlinearly cointegrated setting. The special structure of the CKSVAR makes this problem peculiarly tractable, while being flexible enough to generate interesting departures from linear cointegration. Our results indicate how progress may now be made in the analysis of more general nonlinear VARs with unit roots, while our representation theory provides the foundations for inference on cointegrating relations in the CKSVAR. Our findings with respect to these problems will be reported elsewhere; some initial results regarding inference in this setting, in the univariate case, are given in BD22 and DJ25.
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