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Common Trends and Long-Run Identification in Nonlinear Structural VARs
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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]}
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First version: April 2024. We thank B. Beare, J. Dolado, A.\ Escribano, S. Johansen, Y. Lu, B.\ Nielsen, B. Rossi, and seminar participants at UC3 Madrid, Oxford, Sydney, the 2024 BSE Summer Forum and the 26th Dynamic Econometrics Conference, for their helpful comments on earlier drafts of this paper.
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For more than four decades, the linear structural VAR (SVAR) model,
has played a central role in empirical macroeconomics. Structural VARs provide a satisfyingly coherent framework within which the problem of identifying causal relations, in the presence of simultaneity, may be approached by regarding the $p$ endogenous variables $z_{t}$ as generated by an underlying sequence of $p$ exogenous structural shocks $\varepsilon_{t}$. Viewed through the lens of these models, the identification problem reduces to one of identifying these structural shocks and their associated impulse responses: equivalently, of identifying the mapping $\Upsilon$ between $\varepsilon_{t}$ and $u_{t}$.
The early literature on structural VARs, following the seminal contribution of Sims80, developed contemporaneously with major advances in the modelling of common (stochastic) trends in macroeconomic time series. This latter work culminated in the development of the theory of the cointegrated VAR, and in particular the Granger--Johansen representation theorem (GJRT), which showed how a VAR could be configured so as to accommodate the presence of common stochastic trends, and the cointegrating relations that are dual to them (EG87Ecta; Joh91Ecta). This importantly demonstrated that the SVAR model could be reconciled with some of the evident properties of the (levels of) nonstationary macroeconomic time series, obviating the need to difference these series to stationarity prior to analysis (indeed, an implication of this work was that such pre-filtering was not only unnecessary, but could induce model misspecification). An important corollary to the GJRT was that by uncovering the mapping between the common trends and the endogenous variables, it made available a class of long-run identifying restrictions, relating to whether certain structural shocks were regarded as having permanent, or merely transient, effects (BlanchardQuah89; KPSW91AER).
Subsequent work has sought to extend the (S)VAR model so as to allow for various forms of nonlinearity: typically as modelled via some sort of regime switching, possibly with smooth transitions between these regimes (see e.g.\ Chan09; TTG10; HT13). More recently, interest in structural macroeconomic models that incorporate occasionally binding constraints, of which the zero lower bound on interest rates is a leading instance, has had its counterpart in the development of new class of regime-switching nonlinear VARs. These newer models are notably distinguished from the earlier nonlinear VAR literature by being able to accommodate endogenous regime switching, i.e.\ of allowing the current regime to depend on the current values of the endogenous variables, rather than being pre-determined, or being dependent on some exogenous switching process (SM21; AMSV21).
However, there remains a significant disconnect between the literature on nonlinear (S)VARs, and that on cointegration and common trends. In this respect, the situation is reminiscent of that of the literature on linear VARs in the early 1980s, with there being little theoretical understanding of how common trends may be accommodated within nonlinear VARs, such that the prior removal of common trends by pre-filtering remains a necessity. One area where significant efforts have been made to remedy this pertain to a certain class of `nonlinear VECM' models. These are VAR models that are specified directly in VECM form, with a linear cointegrating space, but in which the stationary components of the model (the equilibrium errors and the lagged differences) are permitted to enter nonlinearly (see e.g.\ EM02JTSA; BR04EctJ; Saik05JoE,Saik08ET; KR10JoE,KR13ET). This class of models has proved sufficiently tractable to facilitate a nonlinear extension of the GJRT, but the assumptions made in this literature are such as to leave a substantial portion of the space of nonlinear VAR models unexplored.
In this paper, we generalise (ref) in the direction of the additively time-separable SVAR
where $f_{0}:\mathbb{R}^{p}\ensuremath{\rightarrow}\mathbb{R}^{p}$ is continuous and has a continuous inverse (i.e.\ it is a homeomorphism), and each $f_{i}:\mathbb{R}^{p}\ensuremath{\rightarrow}\mathbb{R}^{p}$ is continuous.\footnote{For a detailed explanation of why the structural shocks enter (ref) in an apparently similar manner to (ref), see (ref) below.} We aim not only to extend the GJRT, but also to preserve exactly the kind of long-run identifying restrictions that are yielded by the GJRT in a linear setting, such that it remains possible to fruitfully discriminate between structural shocks on the basis of the permanence (or transience) of their effects. Via an associated VECM representation, this leads us to consider SVARs in which the image of the map \[ \pi(z)\coloneqq\sum_{i=1}^{k}f_{i}(z)-f_{0}(z) \] is an $r$-dimensional subspace of $\mathbb{R}^{p}$, a requirement that we term the common row space condition (CRSC); effectively, this entails a decomposition of the form $\pi(z)=\alpha\theta(z)$. This reverses a key assumption of the nonlinear VECM literature, where $\theta(z)=\beta^{\mathsf{T}}z$ is a fixed linear function, but $\alpha$ is allowed to vary (albeit only as a function of $\beta^{\mathsf{T}}z_{t-1}$ and $\Delta z_{t-1}$, or their lags). As we discuss, the principal motivation for the CRSC is that it ensures that only $q=p-r$ structural shocks may have permanent effects -- whereas if the CRSC is violated, all $p$ structural shocks may have permanent effects, even though $z_{t}$ has only $q<p$ common stochastic trends.
Our conditions on the nonlinear SVAR (ref) are otherwise rather general, and unlike the preceding literature allow for common stochastic trends to enter $z_{t}$ nonlinearly, thereby giving rise to nonlinear cointegrating relations between elements of $z_{t}$. (A rigorous discussion of `cointegration' in this nonlinear setting is deferred to a companion paper, in which the limiting distribution of the standardised process $n^{-1/2}z_{\smlfloor{n\lambda}}$ is derived; for a discussion of this in the context of the censored and kinked SVAR, see DMW22.) These conditions are presented in (ref), followed by a discussion of the structural shock identification problem that arises in this model. We further show that our conditions reduce to exactly those of the linear cointegrated VAR, when each $\{f_{i}\}_{i=0}^{k}$ is linear: and thus they do not appear to be particularly restrictive.
The principal contribution of this paper is to provide an extension of the GJRT to the setting of a nonlinear VAR ((ref)), demonstrating how the model (ref) may be configured so as to generate common stochastic trends. This in turn permits an analysis of the stability of the steady state equilibria of the model, and a characterisation of the directions in which the shocks $u_{t}$ have only transient effects ((ref)), and thereby yields long-run identifying restrictions on $\Upsilon$ ((ref)). The GJRT also facilitates the calculation of the implied long-run multipliers associated with the shocks ((ref)). We subsequently return, in (ref), to a discussion of the role played by the CRSC in ensuring the availability of long-run identifying restrictions. Finally, (ref) elaborates on a class of models in which each $f_{i}$ is piecewise affine (or a smooth, convex combination of piecewise affine functions) showing how the abstract conditions introduced in (ref) may be specialised to this class of models. (ref) concludes. Proofs of all results appear in the appendices.
We consider an additively time-separable nonlinear VAR($k$) model for a $p$-dimensional time series $\{z_{t}\}_{t\in\mathbb{Z}}$, of the form
where $f_{i}:\mathbb{R}^{p}\ensuremath{\rightarrow}\mathbb{R}^{p}$ for $i\in\{0,1,\ldots,k\}$, and $\{u_{t}\}_{t\in\mathbb{Z}}$ is a sequence of `shocks' taking values in $\mathbb{R}^{p}$. (As a convenient location normalisation, we shall suppose throughout that $f_{i}(0)=0$ for all $i\in\{0,\ldots,k\}$.) By defining
for $j\in\{0,\ldots,k-1\}$, we can write the model in nonlinear VECM form as
where
(see also (ref) below).
In the linear VAR, where $\pi(z)=\Pi z$ is linear, the key assumption required for the model to generate common stochastic trends (i.e.\ cointegrated $I(1)$ processes) is that $\operatorname{rk}\Pi=r=p-q$, where $q$ is the number of unit roots in the autoregressive polynomial. Equivalently, this condition may be expressed in terms of the image of the map $z\ensuremath{\mapsto}\Pi z$, \[ \operatorname{im}\Pi=\{\Pi z\mid z\in\mathbb{R}^{p}\}=\operatorname{sp}\Pi, \] being an $r$-dimensional (linear) subspace of $\mathbb{R}^{p}$. In extending the GJRT to the setting of the nonlinear VAR (ref), the role of $\Pi$ (or more precisely, of $z\ensuremath{\mapsto}\Pi z$) will now be played by $\pi(z)$, and our high-level assumptions below will ensure that \[ \operatorname{im}\pi=\{\pi(z)\mid z\in\mathbb{R}^{p}\} \] remains an $r$-dimensional (linear) subspace of $\mathbb{R}^{p}$, exactly as in the linear case. We term this the common row space condition (CRSC). (Note that there is no corresponding restriction on $\ker\pi$; in particular, this will not be required to be a $q$-dimensional subspace of $\mathbb{R}^{p}$.) For a further discussion of this condition -- in particular, of the role that it plays in ensuring that the long-run identifying restrictions familiar from a linear cointegrated VAR are also available in the present setting -- see (ref) below.
To state our main assumptions on the model (ref), we first provide a more compact representation for (ref); its proof appears in (ref).
We next define a class ${\cal M}_{r}$ of nonlinear VAR models, that will be shown to be consistent with the presence of $q=p-r$ common stochastic trends in $\{z_{t}\}$, and $r$ `cointegrating relations' between those series. For some $\theta:\mathbb{R}^{p}\ensuremath{\rightarrow}\mathbb{R}^{r}$, define
and for $\alpha\in\mathbb{R}^{p\times r}$ with $\operatorname{rk}\alpha=r$,
Recall that a function $f:{\cal X\ensuremath{\rightarrow}{\cal Y}}$ is a homeomorphism if it is continuous, bijective, and has a continuous inverse. $f$ is bi-Lipschitz if there exists a $C_{0}<\infty$ such that $C_{0}^{-1}\smlnorm{x-x^{\prime}}\leq\smlnorm{f(x)-f(x^{\prime})}\leq C_{0}\smlnorm{x-x^{\prime}}$ for all $x,x^{\prime}\in{\cal X}$; we term any such $C_{0}$ a bi-Lipschitz constant for $f$. 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), where $\mathcal{A}^{t}\coloneqq\{\prod_{s=1}^{t}M_{s}\mid M_{s}\in\mathcal{A}\}$ is the set of $t$-fold products of matrices in ${\cal A}$, and $\rho(B)$ the spectral radius of $B$.
\needspace{3cm}
Our assumptions on the data generating process may now be stated.
{{{{\scalefont{0.76}CVAR}}}}
For the purposes of much of this paper, the sequence of innovations $\{u_{t}\}$ may be treated as a given non-random sequence in $\mathbb{R}^{p}$, since our results are either non-asymptotic, or take limits under the assumption that the shocks cease at some finite horizon $\tau$. However, it will occasionally be useful to indicate the implications of our results in cases where the following holds.
{{{{\scalefont{0.76}ERR}}}}
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To help cast further light on the defining conditions of class ${\cal M}_{r}$, we now show that when (ref) is specialised to a linear VAR, these conditions reduce to exactly the usual conditions for a linear VAR to generate common $I(1)$ components, as per the GJRT (Joh95, Ch.\ 4). This provides a clear indication that our conditions on the general, nonlinear VAR are not overly restrictive. By a linear VAR, we mean one in which $f_{i}(z)=\Phi_{i}z$ for some $\Phi_{i}\in\mathbb{R}^{p\times p}$, for $i\in\{0,\ldots,k\}$, so that (ref) becomes
Let $\Phi(\lambda)\coloneqq\Phi_{0}-\sum_{i=1}^{k}\Phi_{i}\lambda^{i}$ denote the associated autoregressive polynomial.
The proof of (ref) demonstrates that the factorisation required of $\pi(z)$ in (ref) has its direct counterpart in condition (ref) on the rank of $\Phi(1)$, while the high-level condition (ref), which here involves only the matrices
is equivalent to the familiar condition (ref) on the autoregressive roots.
In macroeconomics, a conventional starting point for empirical work is the structural (linear) VAR model, in which the observed series $\{z_{t}\}$ are regarded as being generated by an underlying $p$-dimensional i.i.d.\ sequence of structural shocks $\{\varepsilon_{t}\}$, whose elements are mutually orthogonal, and each of which have an economic interpretation (as e.g.\ an aggregate supply shock, a monetary policy shock, etc.).\footnote{In some treatments (e.g.\ LMS17JoE; GMR20REStud) the components of $\varepsilon_{t}$ are taken to be not merely uncorrelated, but mutually independent. Under non-Gaussianity, this is known to yield certain additional identifying restrictions. We emphasise that in the present work, only orthogonality between the components of $\varepsilon_{t}$ is maintained.} This perspective may also be adopted within the framework of (ref), reproduced here as
by supposing that there is a parametrisation $(c^{\varepsilon},\{f_{i}^{\varepsilon}\}_{i=0}^{k})$ of the model such that $\{z_{t}\}$ satisfies
We may suppose, as a convenient location and scale normalisation, that each of the elements of $\varepsilon_{t}$ have mean zero and unit variance, so that $\{\varepsilon_{t}\}$ is i.i.d.\ with $\ensuremath{\mathbb{E}}\varepsilon_{t}=0$ and $\ensuremath{\mathbb{E}}\varepsilon_{t}\varepsilon_{t}^{\mathsf{T}}=I_{p}$. For the purposes of this section, we shall also require $\varepsilon_{t}$ to be continuously distributed, with density $\varrho^{\varepsilon}$. (For simplicity of presentation, we also treat the order $k$ of the VAR as known, though only a known finite upper bound on the order of the VAR is required for the discussion that follows.)
The question then arises as to whether, or to what extent, the structural shocks \{$\varepsilon_{t}\}$ are identified by observation of $\{z_{t}\}$. To phrase this question more precisely, we first clarify how the model (ref) is parametrised. For $i\in\{0,\ldots,k\}$, let $\mathscr{F}_{i}$ denote a collection of functions $\mathbb{R}^{p}\ensuremath{\rightarrow}\mathbb{R}^{p}$ to which each $f_{i}$ respectively belongs. We also specialise (ref) to the case where $\{u_{t}\}$ is i.i.d., with a density $\varrho$ on $\mathbb{R}^{p}$ that belongs to some set $\mathscr{R}$, normalised to have $\ensuremath{\mathbb{E}} u_{t}=0$ and $\ensuremath{\mathbb{E}} u_{t}u_{t}^{\mathsf{T}}=I_{p}$. The parameter space for (ref) is then \[ \set P\coloneqq\{(\tilde{c},\{\tilde{f}_{i}\}_{i=0}^{k},\tilde{\varrho})\mid\tilde{c}\in\mathbb{R},\ \tilde{f}_{i}\in\mathscr{F}_{i}\text{ for }i\in\{0,\ldots,k\},\ \tilde{\varrho}\in\mathscr{R}\}. \] Under the regularity conditions given in (ref), each $(\tilde{c},\{\tilde{f}_{i}\}_{i=0}^{k},\tilde{\varrho})\in\set P$ completely specifies the density of $z_{t}$ conditional on $\vec{z}_{t-1}=(z_{t-1}^{\mathsf{T}},\ldots,z_{t-k}^{\mathsf{T}})^{\mathsf{T}}$. Following Matz09Ecta, we say that two points in $\set P$ are observationally equivalent if they imply exactly the same conditional density function, and thus the same likelihood for $\{z_{t}\}_{t=1}^{n}$, conditional on the initial values $\vec{z}_{0}$ (which as per (ref) we take to be arbitrary, and thus uninformative about the model parameters).
Let $\Upsilon\in\mathbb{R}^{p\times p}$ be an orthogonal matrix, and suppose that $\mathscr{R}$ is such that the density $\tilde{\varrho}$ of $\tilde{u_{t}}\coloneqq\Upsilon\varepsilon_{t}$ also lies in $\mathscr{R}$. Then it is evident from pre-multiplying (ref) through by $\Upsilon$, to obtain \[ \tilde{f}_{0}(z_{t})\coloneqq\Upsilonf_{0}^{\varepsilon}(z_{t})=\Upsilon c^{\varepsilon}+\sum_{i=1}^{k}\Upsilonf_{i}^{\varepsilon}(z_{t-i})+\Upsilon\varepsilon_{t}\eqqcolon\tilde{c}+\sum_{i=1}^{k}\tilde{f}_{i}(z_{t-i})+\tilde{u}_{t}, \] that $(\tilde{c},\{\tilde{f}_{i}\}_{i=0}^{k},\tilde{\varrho})$ must be observationally equivalent to $(c^{\varepsilon},\{f_{i}^{\varepsilon}\}_{i=0}^{k},\varrho^{\varepsilon})$. Thus, as is familiar from the linear structural VAR, the model parameters and the structural shocks are at best identified up to an unknown orthogonal transformation.
Remarkably, despite the much broader class of nonlinear transformations permitted by (ref), such orthogonal transformations delineate the only loci of non-identification in the nonlinear VAR. By (ref) in (ref), under certain conditions on the sets $\{\mathscr{F}_{i}\}_{i=0}^{k}$ and $\mathscr{R}$, and the parameters generating $\{z_{t}\}$, there exists a $\varrho\in\mathscr{R}$ such that the parameters $(c,\{f_{i}\}_{i=0}^{k},\varrho)$ of (ref) are observationally equivalent to those $(c^{\varepsilon},\{f_{i}^{\varepsilon}\}_{i=0}^{k},\varrho^{\varepsilon})$ of (ref), if and only if there is an orthogonal matrix $\Upsilon\in\mathbb{R}^{p\times p}$ such that
and thus the shocks $\{u_{t}\}$ in (ref) satisfy
(For a discussion of the conditions under which the preceding holds, see (ref): these are essentially weak smoothness conditions on the elements of $\{\mathscr{F}_{i}\}_{i=0}^{k}$ and $\mathscr{R}$, together with the requirement that $\{f_{i}^{\varepsilon}\}_{i=1}^{k}$be such as to ensure sufficient dependence of the r.h.s.\ of the model on lags of $z_{t}$.) Conversely, should (ref) fail to hold, then there will be at least some realisations of $\{z_{t}\}$ for which the likelihoods of $(c,\{f_{i}\}_{i=0}^{k},\varrho)$ and $(c^{\varepsilon},\{f_{i}^{\varepsilon}\}_{i=0}^{k},\varrho^{\varepsilon})$ will be distinct, and so the data is to this extent informative about these two alternative parametrisations of the model.\footnote{We do not claim, on the basis of (ref), that the parameters of the model are consistently estimable up to an orthogonal transformation. While it seems reasonable to suppose that consistent nonparametric estimation of the model would be possible (under regularity conditions) if $\{z_{t}\}$ is stationary and ergodic, the usual connection between identification and consistent estimation is attenuated when $\{z_{t}\}$ possesses some stochastic (or indeed, deterministic) trends, because of the non-recurrence of those trends in higher dimensions (see Bing01Hdbk; GP13JoE). Consistent estimation of the model parameters (up to $\Upsilon$) would in such cases likely require further restrictions on the functional form of $f_{i}$.}
Accordingly, when subsequently discussing the problem of structural shock identification, we shall suppose that the shocks $u_{t}$ appearing in (ref) are related to $\varepsilon_{t}$ in precisely the manner of (ref), yielding the structural nonlinear VAR model
where $\Upsilon\in\mathbb{R}^{p\times p}$ is an orthogonal matrix. In other words, on the strength of (ref), we shall regard observation of $\{z_{t}\}$ as being sufficient to identify the model parameters up to (and only up to) $\Upsilon$, and thence the structural shocks up to transformation by this same matrix. To identify the structural shocks, we thus seek additional restrictions sufficient to pin down (at least some of) the unknown elements of $\Upsilon$, such as will be provided by the long-run identifying restrictions developed in (ref) below.
The starting point for the analysis of the linear cointegrated VAR is the Granger--Johansen representation theorem (GJRT), which decomposes $z_{t}$ into the sum of: (i) an initial condition, (ii) a stochastic trend, and (iii) a weakly dependent process (which is stationary if $z_{t}$ is suitably initialised). This may be rendered in various ways: to facilitate the comparison with (ref) below (from which it follows in the linear case), we shall here write this for a linear VAR satisfying (ref){{{\scalefont{0.76}--3}}}, as
where: (i) $\abv{h}(\vec{z}_{0})$ captures the dependence on the initial values $\vec{z}_{0}=(z_{0}^{\mathsf{T}},\ldots,z_{-k+1}^{\mathsf{T}})^{\mathsf{T}}$; (ii) $\alpha_{\perp}^{\mathsf{T}}\sum_{s=1}^{t}u_{s}$ is a stochastic trend of dimension $q=\operatorname{rk}\alpha_{\perp}=p-r$; and (iii)
follows a VAR, where $\b{\alpha}$ and $\b{\beta}$ are as in (ref) above. The stability of this VAR follows from (ref), because membership of ${\cal M}_{r}$ implies, via (ref), that all the eigenvalues of $I_{p(k-1)+r}+\b{\beta}^{\mathsf{T}}\b{\alpha}$ must lie inside the unit circle. (It may also be noted from (ref) above that, in this case, $\Delta\b{\zeta}_{t}$ is itself a linear function of $\{\Delta z_{t-i}\}_{i=0}^{k-2}$.) $\b{\xi}_{t}$ may thus be rendered stationary through an appropriate choice of the initial conditions ($\vec{z}_{0}$ or, equivalently, $\b z_{0}$).
Before providing our extension of the GJRT to the general setting of (ref), we first note that the nonlinear counterpart of $\b{\xi}_{t}$ will continue to follow an autoregression of the form (ref), but now with time-varying coefficients, in particular with $\b{\beta}=\b{\beta}_{t}$ now depending on the level of $z_{t}$. While such processes cannot be stationary, we still have a well-defined notion of stability for such processes, which is sufficient to ensure that $\b{\xi}_{t}$ remains of strictly smaller stochastic order than the common trends.
A sufficient condition for exponential stability is that $\rho_{{\scriptstyle \mathrm{JSR}}}(\set A)<1$ (e.g.\ Jungers09, Cor. 1.1), which motivates the restriction on the JSR that appears in (ref). Notable consequences are that if $v_{t}=0$ for all $t\geq\tau$, then $w_{t}\ensuremath{\rightarrow}0$, while if $\{v_{t}\}$ and $w_{0}$ have uniformly bounded $2+\delta$ moments, then $\max_{1\leq t\leq n}\smlnorm{w_{t}}=o_{p}(n^{1/2})$. (For this last, see Lemma A.1 in DMW22.)
We now state our counterpart of the Granger--Johansen representation theorem for the model (ref), which constitutes the main result of this paper.
A first application of (ref) is to verify the stability of the (non-stochastic) steady state solutions to (ref). By considering (ref) when $z_{t-1}=\cdots=z_{t-k}=z$ for some $z\in\mathbb{R}^{p}$ and $u_{t}=0$, we see that for $z$ to be a steady state equilibrium, it must satisfy \[ 0=c+\pi(z)=\alpha[\mu+\theta(z)] \] or equivalently $\theta(z)=-\mu$, since $\operatorname{rk}\alpha=r$ and $\theta:\mathbb{R}^{p}\ensuremath{\rightarrow}\mathbb{R}^{r}$. Thus the set of steady state equilibria is given by the $q$-dimensional manifold
where the fact that $\mathscr{M}_{\mu}$ is indeed a $q$-dimensional manifold follows from the final equality, since $\chi:\mathbb{R}^{p}\ensuremath{\rightarrow}\mathbb{R}^{p}$ is a homeomorphism.
For the purposes of analysing the stability of $\mathscr{M}_{\mu}$, we shall suppose that the shocks $\{u_{t}\}$ are set to zero after some $\tau\in\ensuremath{\mathbb{N}}$: that is, $u_{s}=0$ for all $s\geq\tau+1$, and then consider how $z_{t}$ evolves from time $\tau$ forward. In other words, we shall fix the state of the model at time $\tau-1$ at
allow one final shock $u_{\tau}=u\in\mathbb{R}^{p}$ to occur at time $\tau$; and then evaluate the (non-stochastic) limit of $z_{t}=z_{t}(u;\mathfrak{z})$ as $t\ensuremath{\rightarrow}\infty$. We aim to show that $\mathscr{M}_{\mu}$ is strictly stable, in the sense of the following.
In other words, if $\mathscr{M}_{\mu}$ is strictly stable, then whatever the current state $\vec{z}_{\tau-1}=\mathfrak{z}$ of the model and the given limiting $z_{\infty}\in\mathscr{M}_{\mu}$, there is always a value for the $\tau$-dated shock $u_{\tau}$ that would lead $z_{t}$ to converge to $z_{\infty}$; every element in $\mathscr{M}_{\mu}$ may thus ultimately be `reached' from $\vec{z}_{\tau-1}$. (Note that this is not so trivial a matter as choosing $u_{\tau}=u$ such that $z_{\tau}=z_{\infty}$ immediately, since what is required is that $z_{t}$ converge to a steady state equilibrium at $z_{\infty}$.) Since $\{\b{\xi}_{t}\}$ is exponentially stable by (ref), it follows that $\b{\xi}_{t}\ensuremath{\rightarrow}0$ as $t\ensuremath{\rightarrow}\infty$ (see the discussion following (ref) above). Hence, noting that $\sum_{s=\tau}^{t}u_{s}=u_{\tau}=u$, we have by that result that \[ z_{t}(u;\mathfrak{z})=\chi^{-1}\left(
+S_{\chi}^{\mathsf{T}}\b{\xi}_{t}\right)\ensuremath{\rightarrow}\chi^{-1}\left(
\right)\eqqcolon z_{\infty}(u;\mathfrak{z})\in\mathscr{M}_{\mu}. \] Because the r.h.s.\ depends on $\mathfrak{z}$ only through $\abv{h}(\mathfrak{z})$, and as $u\in\mathbb{R}^{p}$ varies $\alpha_{\perp}^{\mathsf{T}}[\abv{h}(\mathfrak{z})+u]$ ranges freely over $\mathbb{R}^{q}$, we can induce $z_{\infty}(u;\mathfrak{z})$ to take any desired value in $\mathscr{M}_{\mu}$. We thus obtain the following.
As a direct consequence of the preceding, we obtain the following characterisation of the linear combinations of the shocks $u_{t}$ in (ref) that do not have permanent effects.
The significance of this result may be explained as follows. Suppose that, as in (ref) above, we have the nonlinear structural VAR
belonging to class ${\cal M}_{r}$. As noted in (ref), the parameters of this model are identified up to the unknown orthogonal matrix $\Upsilon\in\mathbb{R}^{p\times p}$, about which the data is entirely uninformative, and thus we seek additional restrictions that would pin down (at least some of) the elements of $\Upsilon$. To that end, suppose we partition the structural shocks as \[ \varepsilon_{t}=(\varepsilon_{(1),t}^{\mathsf{T}},\varepsilon_{(2),t}^{\mathsf{T}})^{\mathsf{T}} \] where $\varepsilon_{(1),t}$ takes values in $\mathbb{R}^{m}$, and collects the $m\in\{1,\ldots,r\}$ structural shocks that are regarded as having no permanent effect on $z_{t}$. Then (ref) implies that
which provides $qm$ `long-run identifying restrictions' on $\Upsilon$.\footnote{Note that (ref), or equivalently (ref), implies that at most $r$ elements of $\varepsilon_{t}$ may be such that the corresponding column of $\Upsilon$ lies in $\operatorname{sp}\alpha$, i.e.\ up to $r$ structural shocks may have purely transitory effects. As is familiar from the linear structural VAR, there is nothing here to preclude e.g.\ all shocks from having permanent effects; the number (and identities) of the $m\in\{0,\ldots,r\}$ structural shocks having purely transitory effects will thus depend on the identifying conditions that define those shocks.}
As discussed further in (ref) below, the availability of these long-run identifying restrictions is largely a consequence of the common row space condition, and provides one of the principal motivations for maintaining that condition in our model. Remarkably, despite the nonlinearity permitted by (ref), the identifying restrictions (ref) take exactly the same form as in a linear SVAR (see KL17book, Sec. 10.2).
Having obtained a characterisation of the attractor space $\mathscr{M}_{\mu}$ for $\{z_{t}\}$, we may also say something more about the permanent effect of a shock at time $t=\tau$. The limiting impulse responses or long-run multipliers can be computed by differentiating $z_{\infty}(u;\mathfrak{z})$ with respect to $u$, i.e. by computing the Jacobian
In view of (ref), postmultiplying this by $\Upsilon$ yields the long-run multipliers with respect to the structural shocks $\varepsilon_{t}$, at time $t=\tau$.
One concern we might have here, in the general nonlinear case, is the apparent dependence of these long-run multipliers on $\vec{z}_{t-1}=\mathfrak{z}\in\mathbb{R}^{kp}$, which potentially ranges over a very `large' ($kp$-dimensional) space. However, it will be noted that $z_{\infty}(u;\mathfrak{z})$ only depends on $\mathfrak{z}$ through $\alpha_{\perp}^{\mathsf{T}}\abv{h}(\mathfrak{z})$: and as shown in the proof of the next result, it is always possible to find a $z_{\mathfrak{z}}\in\mathscr{M}_{\mu}$ such that $\psi(z_{\mathfrak{z}})=\alpha_{\perp}^{\mathsf{T}}\abv{h}(\mathfrak{z})$. Therefore, for the purposes of computing the full set of possible long-run effects of shocks in the model, it suffices to consider cases in which the model is in a steady state equilibrium (i.e.\ where $z_{\tau-1}=\dots=z_{\tau-k}=z$ for some $z\in\mathscr{M}_{\mu}$) immediately prior to the incidence of the shock.
Because $\chi^{-1}$ is not necessarily differentiable everywhere, the long-run multipliers as defined in (ref) may not exist for every $z\in\mathscr{M}_{\mu}$. However, in most cases of interest, the subset $N\subset\mathscr{M}_{\mu}$ at which this occurs will be exceptionally small, corresponding e.g.\ in the case of piecewise affine models (see (ref) below) to points exactly on the boundary between two regimes. It is also possible that the Jacobian of $\chi^{-1}$ may fail to be invertible at exceptional points: though it must be invertible almost everywhere, since $\chi$ is invertible.
As introduced in (ref) above, a key characteristic of models in class ${\cal M}_{r}$ is the common row space condition (CRSC), that $\operatorname{im}\pi=\{\pi(z)\mid z\in\mathbb{R}^{p}\}$ is an $r$-dimensional linear subspace of $\mathbb{R}^{p}$; we noted there that no such restriction is imposed on $\ker\pi$. By contrast, the existing literature on nonlinear VECM models effectively reverses this condition by requiring that $\ker\pi$ be a $q$-dimensional subspace of $\mathbb{R}^{p}$, without necessarily restricting $\operatorname{im}\pi$ (see in particular KR10JoE). Loosely speaking, if we suppose that $\pi$ may be decomposed, for each $z\in\mathbb{R}^{p}$, as
where $\alpha,\beta:\mathbb{R}^{p}\ensuremath{\rightarrow}\mathbb{R}^{r}$, then the CRSC permits taking $\alpha(z)=\alpha$, if $\beta(z)$ is appropriately normalised, but leaves $\beta(z)$ otherwise unrestricted;\footnote{If $\beta(z)$ is instead normalised in some other way, such as e.g.\ $\beta(z)^{\mathsf{T}}=[I_{r},-A(z)]$, then $\alpha(z)$ may indeed vary with $z$. In this sense, it is not quite correct to regard the CRSC as requiring the loadings on the equilibrium errors to be constant, but rather that those loadings should lie in a fixed $r$-dimensional linear subspace; there may be nonlinear adjustment towards equilibrium, provided that it occurs along this subspace.} whereas in the nonlinear VECM literature, $\beta(z)=\beta$ is constant, with the result that there is a globally linear cointegrating space. The representation theory developed in the nonlinear VECM literature is thus strictly complementary to the present work. (Note that even the case where $\beta(z)=\beta$ in our framework is not encompassed by that literature, because we allow the dynamics of the implied VECM (ref) to depend on the level of $z_{t}$, whereas that literature requires that these be governed entirely by the stationary equilibrium errors $\beta^{\mathsf{T}}z_{t}$ and differences $\Delta z_{t}$.)
Recall that in the linear structural VAR model, common stochastic trends arise because some subset of the structural shocks (say, $q$ in number) have permanent effects on $z_{t}$, while the remaining $r=p-q$ shocks have only transitory effects. This provides not only an underpinning economic explanation for the patterns of long-run co-movement present in the data, but also a fruitful source of long-run identifying restrictions (following BlanchardQuah89, and KPSW91AER). The results developed in Sections (ref) and (ref) above show that these properties are preserved when we generalise the cointegrated linear SVAR to a nonlinear SVAR (ref) belonging to class ${\cal M}_{r}$. However, as the example developed in this section illustrates, these properties are highly sensitive to departures from the CRSC, even when the assumption of a globally linear cointegrating space is maintained (i.e.\ when $\beta(z)=\beta$ for all $z\in\mathbb{R}^{p}$ in (ref) above). In models where the CRSC fails, it will generally be the case that the direction in which an impulse $u_{t}=\delta$ has no permanent effect on $z_{t}$ will vary with the magnitude $\delta$. Thus, if we want to work with nonlinear SVAR models in which structural shocks may be distinguished on the basis of their long-run effects -- and in which, as a corollary, common stochastic trends arise because only a subset of these shocks have permanent effects -- then it would appear that the CRSC cannot be easily dispensed with.
To illustrate the consequences of a failure of the CRSC, we consider a nonlinear VAR(1), specified in VECM form as \[ \Delta z_{t}=a(\beta^{\mathsf{T}}z_{t-1})\beta^{\mathsf{T}}z_{t-1}+u_{t} \] where for $\Lambda:\mathbb{R}^{r}\ensuremath{\rightarrow}[0,1]$ a smooth function satisfying $\Lambda(0)=0$ and $\lim_{\smlnorm{\xi}\ensuremath{\rightarrow}\infty}\Lambda(\xi)=1$, and $\alpha,\tilde{\alpha}\in\mathbb{R}^{p\times r}$, each having rank $r$, \[ a(\xi)\coloneqq\Lambda(\xi)\tilde{\alpha}+[1-\Lambda(\xi)]\alpha, \] so that the model is one in which the kernel of $\pi(z)=a(\beta^{\mathsf{T}}z)\beta^{\mathsf{T}}$ is a fixed $q$-dimensional subspace (given by $\operatorname{sp}\beta$), but the image of $\pi$ is not a linear subspace. (If we further suppose that the eigenvalues of $I_{r}+\beta^{\mathsf{T}}\alpha$ lie strictly inside the unit circle, then the model satisfies conditions (A.2) and (A.3) of KR10JoE, who provide a GJRT for these models.) This model may be regarded as smoothly combining two regimes: an `inner' or `near equilibrium' regime in which
and an `outer' or `far from equilibrium' regime where $\tilde{\alpha}$ is replaced by $\alpha$. If the eigenvalues of $I_{r}+\beta^{\mathsf{T}}\tilde{\alpha}$ associated to the inner regime are also strictly inside the unit circle, then $\mathscr{M}_{0}=\beta_{\perp}$ is a strictly stable attractor in the sense of (ref) above.
Now suppose we were to repeat the analysis of (ref) for this model: i.e.\ fixing the state $z_{\tau-1}=\mathfrak{z}\in\mathbb{R}^{p}$ of the model in time $\tau$, having the model impacted by a final shock $u_{\tau}=u$, and then computing $\lim_{t\ensuremath{\rightarrow}\infty}z_{t}(u;\mathfrak{z})=z_{\infty}(u;\mathfrak{z})$ in the absence of any further shocks (so $u_{t}=0$ for all $t\geq\tau+1$). We may then ask: what values of $u$ would leave $z_{\infty}(u;\mathfrak{z})$ unchanged? That is, we would like to determine the set \[ \{u\in\mathbb{R}^{p}\mid z_{\infty}(u;\mathfrak{z})=z_{\infty}(0;\mathfrak{z})\}. \] Let us further suppose that $\beta^{\mathsf{T}}z_{\tau-1}=\beta^{\mathsf{T}}\mathfrak{z}=0$, so that $z_{\tau-1}\in\mathscr{M}_{0}$, and the model is in equilibrium prior to the incidence of $u_{\tau}$. Locally to $\mathscr{M}_{0}$, we would expect shocks in the direction of $\operatorname{sp}\tilde{\alpha}$ to have no permanent effects; whereas further from $\mathscr{M}_{0}$, shocks in directions lying progressively closer to $\operatorname{sp}\alpha$ should have no permanent effects.
(ref) illustrates this is indeed the case for a bivariate ($p=2$) model with
and $\Lambda(\xi)=2\smlabs{\Phi(\xi)-0.5}$, where $\Phi$ denotes the Gaussian cdf. (Without loss of generality, we take $\mathfrak{z}=0$.) For values of $\smlnorm u$ between $0$ and $10$, the figure reports the (unique) direction $\delta_{\smlnorm u}$ such that $u_{\tau}=\smlnorm u\delta_{\smlnorm u}$ has no permanent effect on $z_{t}$; for the sake of comparability with $\tilde{\alpha}$ and $\alpha$, both of which have unit first element, this is reported in terms of the ratio $\delta_{\smlnorm u,2}/\delta_{\smlnorm u,1}$. We see here that $\delta_{\smlnorm u,2}/\delta_{\smlnorm u,1}$ takes the value $0.5=\tilde{\alpha}_{2}$ when $\smlnorm u=0$, and tends towards $0.25=\alpha_{1}$ as $\smlnorm u$ grows (equalling $0.26$ when $\smlnorm u=20$). As a consequence \[ \ensuremath{\bigcap}_{u\in\mathbb{R}^{p}}\operatorname{sp}\delta_{\smlnorm u}=\emptyset, \] and as such there is no direction along which the shocks $u_{t}$ will only have transient effects. There is thus no possibility of discriminating between the underlying structural shocks $\varepsilon_{t}$ according to whether not they have permanent effects: each must have some such effect, to an extent that varies with the magnitude of the shock.
Finally, we introduce a class of regime-switching models in which the conditions required for our results may be verified relatively straightforwardly. Suppose that for each $i\in\{0,\ldots,k\}$,
where $\{\set Z^{(\ell)}\}_{\ell=1}^{L}$ is a collection of convex sets that partition $\mathbb{R}^{p}$, $\{\bar{\phi}_{i}^{(\ell)}\}_{\ell=1}^{L}\subset\mathbb{R}^{p}$ and $\{\Phi_{i}^{(\ell)}\}_{\ell=1}^{L}\subset\mathbb{R}^{p\times p}$.\footnote{If instead $f_{i}(z)=\sum_{\ell=1}^{L}\ensuremath{\mathbf{1}}\{z\in\set Z_{i}^{(\ell)}\}(\bar{\phi}_{i}^{(\ell)}+\Phi_{i}^{(\ell)}z)$, with a partition $\{\set Z_{i}^{(\ell)}\}_{\ell=1}^{L_{i}}$ that depends on $i$, the model can nonetheless be written in terms of $\{f_{i}\}_{i=0}^{k}$ of the form (ref), i.e.\ with a lag-independent partition $\{\set Z^{(\ell)}\}_{\ell=1}^{L}$, by forming each $\set Z^{(\ell)}$ from a refinement of the sets in $\{\set Z_{i}^{(\ell)}\}_{\ell=1}^{L_{i}}$, as $i$ ranges over $\{0,\ldots,k\}$.} When these parameters are such that $f_{i}$ is continuous for each $i\in\{0,\ldots,k\}$, we shall say that each $f_{i}$ is a piecewise affine function, and refer to the model as a piecewise affine VAR. (We do not consider cases in which $f_{i}$ may be discontinuous; so continuity is always implied when a function or VAR is described as being piecewise affine.)
Piecewise affine VARs, of which the CKSVAR of SM21 is a recent instance, provide a flexible but tractable means of introducing nonlinearity into a vector autoregressive model. Defining \[ \ensuremath{\mathbf{1}}^{(\ell)}(z)\coloneqq\ensuremath{\mathbf{1}}\{z\in\set Z^{(\ell)}\}, \] we see that \[ f_{0}(z_{t})-\sum_{i=1}^{k}f_{i}(z_{t-i})=\sum_{\ell=1}^{L}\ensuremath{\mathbf{1}}^{(\ell)}(z_{t})(\bar{\phi}_{i}^{(\ell)}+\Phi_{i}^{(\ell)}z_{t})-\sum_{i=1}^{k}\sum_{\ell=1}^{L}\ensuremath{\mathbf{1}}^{(\ell)}(z_{t-i})(\bar{\phi}_{i}^{(\ell)}+\Phi_{i}^{(\ell)}z_{t-i}). \] This is a kind of endogenous regime-switching VAR, in which the sets $\{\set Z^{(\ell)}\}_{\ell=1}^{L}$ demarcate $L$ distinct `regimes'. However, unlike in typical models of this kind, there is no requirement that the same `regime' apply e.g.\ to $z_{t}$ and $z_{t-1}$ -- each may lie in a different member of $\{\set Z^{(\ell)}\}_{\ell=1}^{L}$ -- and so it might be more correct to say that there are a total of $L^{k}$ autoregressive regimes, once all the possible patterns of membership of $\{z_{t-i}\}_{i=0}^{k}$ in the sets $\{\set Z^{(\ell)}\}_{\ell=1}^{L}$ are allowed for. Nonetheless, it turns out that a fruitful approach to analysing the long-run behaviour of these systems focuses attention on those $L$ `linear submodels' that arise when each of $\{z_{t-i}\}_{i=0}^{k}$ lie in the same $\set Z^{(\ell)}$, to each $\ell\in\{1,\ldots,L\}$ of which we may associate the autoregressive polynomial
The conditions for membership of ${\cal M}_{r}$, for piecewise affine VARs, will parallel (ref){{{\scalefont{0.76}--3}}}. Before stating these, we first give a result that reduces (ref) to a condition on the JSR of a collection of only $L$ matrices (rather than a collection of uncountably many matrices). To state it, observe that
In this context, we note that (ref) effectively requires that
where $\alpha,\beta^{(\ell)}\in\mathbb{R}^{p\times r}$ with $\operatorname{rk}\alpha=\operatorname{rk}\beta^{(\ell)}=r$. Further, let $\Gamma_{j}^{(\ell)}\coloneqq-\sum_{j=i+1}^{k}\Phi_{j}^{(\ell)}$, $\b{\Gamma}^{(\ell)}\coloneqq[\Gamma_{i}^{(\ell)}]_{i=1}^{k-1}$, and
Since the $p(k-1)+r$ eigenvalues of $I_{p(k-1)+r}+\b{\beta}^{(\ell)\mathsf{T}}\b{\alpha}$ correspond to the inverses of the non-unit roots of $\Phi^{(\ell)}(\lambda)$, a necessary, though not sufficient condition for \[ \rho_{{\scriptstyle \mathrm{JSR}}}(\{I_{p(k-1)+r}+\b{\beta}^{(\ell)\mathsf{T}}\b{\alpha}\}_{\ell=1}^{L})<1 \] is that the eigenvalues of $I_{p(k-1)+r}+\b{\beta}^{(\ell)\mathsf{T}}\b{\alpha}$ should lie strictly inside the unit circle, for all $\ell\in\{1,\ldots,L\}$. Thus, if $\operatorname{rk}\Phi^{(\ell)}(1)=q=p-r$, as per (ref), then $\Phi^{(\ell)}(\lambda)$ will have $q$ roots at unity, and all others strictly outside the unit circle (this follows e.g.\ from arguments given in the proof of (ref)). In other words, for a piecewise affine model to belong to class ${\cal M}_{r}(\alpha,\abv b,\abv{\rho})$ (for some $\abv{\rho}<1$), it is essentially necessary that each of its $L$ linear submodels satisfy the usual conditions for a linear VAR to give rise to $q$ common trends and $r$ cointegrating relations (i.e.\ conditions (ref) and (ref) above).
It remains to consider (ref), i.e.\ the requirement that $f_{0}$ be a homeomorphism. For the purposes of verifying this condition, two important special cases of the piecewise affine VAR are:
In these models, the results of GLM80Ecta provide necessary and sufficient conditions for $f_{0}$ to be invertible, which can be expressed in terms of the determinants of $\{\Phi_{0}^{(\ell)}\}_{\ell=1}^{L}$. We thus have the following
Since it may be verified that a linear VAR satisfying (ref){{{\scalefont{0.76}--3}}} is itself a piecewise linear model satisfying (ref){{{\scalefont{0.76}--3}}} above, (ref) may be construed as a special case of the preceding (though for expository reasons, a separate proof of that result is given in (ref)).
The model (ref) may also be extended to allow for smooth transitions between the $L$ regimes. In the literature on smooth transition (vector) autoregressive models, the conventional approach (e.g.\ HT13) is to replace the indicator functions $\ensuremath{\mathbf{1}}\{z\in\set Z^{(\ell)}\}$ by smooth maps $\Lambda^{(\ell)}(z)$, so that now \[ f_{i}^{\mathrm{ST}}(z)=\sum_{\ell=1}^{L}\Lambda^{(\ell)}(z)(\bar{\phi}_{i}^{(\ell)}+\Phi_{i}^{(\ell)}z), \] where $\Lambda^{(\ell)}(z)\in[0,1]$ and $\sum_{\ell=1}^{L}\Lambda^{(\ell)}(z)=1$ for all $z\in\mathbb{R}^{p}$, so that $f_{i}^{\mathrm{ST}}(z)$ is a always a smooth, convex combination of the affine functions $\{z\ensuremath{\mapsto}\bar{\phi}_{i}^{(\ell)}+\Phi_{i}^{(\ell)}z\}_{\ell=1}^{L}$. While models of this kind may be accommodated within our framework (under certain regularity conditions), the fact that the gradient of $f_{i}^{\mathrm{ST}}$ is not a convex combination of those underlying affine regimes makes it difficult to reduce (ref) to a bound on the JSR of a finite collection of matrices, in the manner of (ref). As an alternative specification that allows for smooth transitions between regimes, while also keeping (ref) tractable, consider
where $f_{i}$ is a (continuous) piecewise affine function as in (ref) above, and $K$ is a smooth kernel with mean zero. Then we have the following.
The requirement in (ref) that $f_{0}$ be linear is needed principally to facilitate the verification of (ref), because of the role played by $f_{K,0}^{-1}$ there. We expect that it should be possible to extend the preceding to allow $f_{0}$ to be a general piecewise affine function, albeit possibly at the cost of additional regularity conditions. (Indeed, it may be shown that the smooth counterpart $f_{0,K}$ of $f_{0}$ is a homeomorphism if every $\Phi\in\operatorname{co}\{\Phi_{0}^{(\ell)}\}_{\ell=1}^{L}$ is invertible, thereby satisfying (ref).)
This paper has extended the Granger--Johansen representation theorem to a flexible class of additively time-separable, nonlinear SVAR models. This shows that such models may be applied directly to time series in which (common) stochastic trends are present, without the need for pre-filtering, thus avoiding the potential for misspecification that this entails. As an important corollary to our results, we show that these models are capable of supporting the same kinds of long-run identifying restrictions as are available in linear cointegrated SVARs. A companion paper to the present work provides limit theory for $n^{-1/2}z_{\smlfloor{n\lambda}}$, and a further discussion of nonlinear cointegration, under the assumptions maintained here.