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Stationarity 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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The results of this paper were first presented in Sections 2 and 3 of arXiv:2211.09604v1. We thank participants at seminars at Cambridge, Cyprus, Stanford and Oxford, for comments on earlier drafts of this work.
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This paper studies a class of multivariate threshold autoregressive models known as censored and kinked structural vector autoregressions (CKSVAR; SM21). These models feature endogenous regime switching induced by threshold-type nonlinearities, i.e.\ changes in coefficients and variances when one of the variables crosses a threshold, but differ importantly from a previous generation of vector threshold autoregressive models (VTAR; see e.g.\ TTG10) insofar as the autoregressive `regime' is determined endogenously, rather than being pre-determined. This notably allows the CKSVAR model to accommodate series that are subject to occasionally binding constraints, a leading example of which is provided by the zero lower bound (ZLB) on short-term nominal interest rates SM21,AMSV21,ILMZ20.
In this paper, we develop a set of sufficient conditions for the processes generated by the CKSVAR to be stationary, ergodic, and weakly dependent. These conditions are of interest, firstly, because the credibility of a structural model relies on its being able to generate plausible counterfactual trajectories and associated impulse responses, i.e.\ on its being adequate to replicate the most elementary time series properties of the data. If that data is apparently stationary, model configurations that instead give rise to explosive trajectories and responses should naturally be excluded from the parameter space (in a Bayesian setting, by assigning zero prior mass to these regions; cf.\ AMSV21, Sec.\ 3.3).\footnote{While we may want to allow some shocks to have permanent but bounded effects, as in a linear VAR with some unit roots, the nonlinearity in the CKSVAR prevents this from being straightforwardly treated as a mere boundary case of the stationary model; see DMW22 for a further discussion.} Secondly, frequentist inference on the parameters of these models relies on the series generated by the model, under the null hypothesis of interest, satisfying the requirements of laws of large numbers and central limit theorems for dependent data (Wool94Hdbk; PP97).
Because of the nonlinearity of the CKSVAR, the stationarity of the model defies the simple analytical characterisation that is available in a linear VAR. (In particular, it is insufficient to simply check the magnitudes of the autoregressive roots associated with some or all of the autoregressive `regimes' implied by the model: see (ref) below.) This is a problem routinely encountered in the literature on nonlinear time series models, and some of the approaches taken in that literature may be fruitfully applied here. Specifically, we rely on existing results from the theory of ergodic Markov processes (Tjo90AAP; MT09), casting the CKSVAR as an instance of the `regime switching' or VTAR models considered in that literature (see e.g.\ Tong90; Chan09; TTG10; HT13), in order to establish conditions sufficient for the series generated by a CKSVAR to be stationary, geometrically ergodic, and $\beta$-mixing (absolutely regular). Because the CKSVAR is continuous at the threshold (kink) and approximately homogeneous (of degree one), this can be characterised directly in terms of the stability of the deterministic part of the model, as per CT85AAP, yielding conditions for stationarity that are less conservative than those available for general VTAR models.
We develop a hierarchy of sufficient conditions for stability, only the most elementary of which, the joint spectral radius (JSR), has previously been applied to nonlinear time series models (e.g.\ Lieb2005; MS08JTSA; Saik08ET; KS2020ER). Though our criteria refer to quantities that cannot feasibly be computed exactly, they can be approximated numerically to a high degree of precision (arbitrarily well, given sufficient computation time, in some cases); we have developed the R package thresholdr to provide users of these models with a means of numerically verifying these criteria.\footnote{Available at: \url{https://github.com/samwycherley/thresholdr}}
The remainder of the paper is organised as follows. (ref) introduces the CKSVAR model and develops a canonical representation of the model, which is particularly amenable to our analysis. In (ref), we provide sufficient conditions for the CKSVAR to generate stationary, ergodic and weakly dependent time series. Those depend, in turn, on the stability of the deterministic part of the model, criteria for which are elaborated in (ref).
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. Define
and consider the model
which we may write more compactly as
where $\phi^{\pm}(\lambda)\coloneqq\phi_{0}^{\pm}-\sum_{i=1}^{k}\phi_{i}^{\pm}\lambda^{i}$ and $\Phi^{x}(\lambda)\coloneqq\Phi_{0}^{x}-\sum_{i=1}^{k}\Phi_{i}^{x}\lambda^{i}$, 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. If $b\neq0$, then by 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 may rewrite (ref) as
For the purposes of this paper, we may thus take $b=0$ without loss of generality. 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: see SM21, AMSV21 and ILMZ20.\footnote{The model formulated by AMSV21 falls within the scope of (ref), once the conditions necessary for their model to have a unique solution (for all values of $u_{t}$) are imposed: see their Proposition 1(i).} We shall follow the former in referring to (ref) as a censored and kinked structural VAR (CKSVAR) model. The following running example will be used to illustrate the concepts developed in this paper.
The CKSVAR encompasses both kinds of dynamic Tobit model as special cases (for applications of which, in both time series and panel settings, see e.g.\ DJ02FRB; DJH11; DSK12AE; LMS19; BMMV21JBF; and Byk21JBES).
As discussed by SM21 and AMSV21, the model (ref) is not guaranteed to have a unique solution for $(y_{t},x_{t})$, at least not for all possible values of $u_{t}$, unless certain conditions are placed on entries of the matrix \[ \Phi_{0}\coloneqq
=
\] of contemporaneous coefficients; or equivalently on the matrices
that respectively apply when $y_{t}$ is positive or negative. By SM21, (ref) has a unique solution -- the model is `coherent and complete' (see also GLM80Ecta) -- if the second condition of the following holds.
{{{{\scalefont{0.76}DGP}}}}
We say that a CKSVAR is 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 the following, whose proof appears in (ref).
The utility of the canonical representation lies in its rendering all the nonlinearity in the model as a more tractable function of the lags of $\tilde{y}_{t}$ alone. Because the time-series properties of a general CKSVAR are inherited from its canonical form, we may restrict attention to canonical CKSVAR models essentially without loss of generality. More precisely, for convenience we shall state our results below for canonical CKSVAR models, and then invert the mapping (ref)--(ref) to determine the implications of these results for general CKSVAR models. That is, we shall initially maintain the following.
{{{{\scalefont{0.76}DGP$^{\ast}$}}}}
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{{\ref*{exa:monetary}}}
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The nonlinearity of the CKSVAR makes the assessment of its stationarity a rather more complicated affair than it is for a linear (S)VAR. Though it may be natural to think of the model as having two autoregressive `regimes', corresponding respectively to positive and negative values of $\{y_{t}\}$, and associated with the autoregressive polynomials \[ \Phi^{\pm}(\lambda)\coloneqq
, \] the roots of $\Phi^{\pm}(\lambda)$ are not sufficient to characterise the stationarity of the model. What seems an intuitive criterion for stationarity, that all these roots should lie outside the unit circle, turns out to be neither necessary nor sufficient -- at best, it is merely necessary for another criterion for stability to be satisfied, as we develop below. Even in very special cases, such as a CKSVAR in which only lags of $y_{t}^{+}$ enter the model (as e.g.\ in AMSV21), it is possible to construct numerical examples of the following kind.
As the above example illustrates, a tractable analytic characterisation of stationarity is unlikely to be available in this setting. To make progress we draw on approaches developed in the literature on nonlinear time series models, specifically those pertaining to `regime switching' or `vector threshold autoregressive' (VTAR) models (see HT13), of which the CKSVAR is an instance. To make the connections with these models more apparent, let $\ensuremath{\mathbf{1}}^{+}(y)\coloneqq\ensuremath{\mathbf{1}}\{y\geq0\}$ and $\ensuremath{\mathbf{1}}^{-}(y)\coloneqq\ensuremath{\mathbf{1}}\{y<0\}$, and define\footnote{There is unavoidably some arbitrariness with respect to how these objects are defined when $y=0$, but since these only play a role in the model when multiplied by $y$, it does not matter what convention is adopted. Throughout the paper, we use the notation $\ensuremath{\mathbf{1}}^{\pm}(y)$ to ensure that all such definitions are mutually consistent.}
for $i\in\{1,\ldots,k\}$. Then for $z_{t}\coloneqq(y_{t},x_{t}^{\mathsf{T}})^{\mathsf{T}}$, the canonical CKSVAR
may be regarded as an autoregressive model with $2^{k}$ `regimes' corresponding to all the possible sign patterns of $(y_{t-1},\ldots,y_{t-k})$, with switches between those regimes occurring at each of the $k$ `thresholds' defined by $y_{t-i}=0$. While the existing literature provides results on the stationarity of a general class of regime-switching vector autoregressive models (see e.g.\ Lieb2005; Saik08ET; KS2020ER), because of that very generality we are able to improve on those results by exploiting the special structure of the CKSVAR model.
To state our results, we first recall the following (cf.\ Lieb2005, p.\ 671).
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 of such sequences, and a discussion of how this concept relates to other notions of ergodicity used in the literature, see Lieb2005.
While $\{z_{t}\}$ is not a Markov chain, $\b z_{t}\coloneqq(z_{t}^{\mathsf{T}},\ldots,z_{t-k+1}^{\mathsf{T}})^{\mathsf{T}}$ trivially is, as can be seen by putting (ref) in companion form. To establish that $\{\b z_{t}\}$ is $\mathcal{Q}$-geometrically ergodic, we shall make the following assumption on the innovation sequence $\{u_{t}\}$ in (ref), which allows for a certain form of conditional heteroskedasticity.
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{{{{\scalefont{0.76}ERR}}}}
We may now state our main result on the geometric ergodicity of the CKSVAR, which relates the geometric ergodicity of $\{\b z_{t}\}$ to the stability of deterministic system (or `skeleton' in the terminology of Tong90) associated to (ref), defined by
We say that such a system is stable if $\hat{z}_{t}\ensuremath{\rightarrow}0$ as $t\ensuremath{\rightarrow}\infty$, for every initialisation $\{\hat{z}_{t}\}_{t=-k+1}^{0}$. Our results for the CKSVAR follow as a corollary to a more general result for continuous and homogeneous (of degree one) autoregressive systems, given as (ref) in (ref). This in turn extends a fundamental result due to CT85AAP to the present setting, principally by relaxing their conditions on the innovations $\{u_{t}\}$.\footnote{Cf.\ Theorem 3.1 in CP99StSin, which also relaxes these conditions, but yields only the weaker conclusion of geometric ergodicity; whereas $\mathcal{Q}$-ergodicity additionally ensures that the stationary distribution has finite $m_{0}$th moment.} The proof of the following appears in (ref).
While the preceding reduces the problem to one of verifying the stability of the deterministic subsystem (ref), this still presents a formidable challenge. We therefore develop some sufficient conditions for stability -- and hence for ergodicity -- that can be evaluated numerically. Some of these concepts, particularly the joint spectral radius (JSR), have been previously applied to regime-switching and VTAR models, and to this extent there is an overlap between our results and those of Lieb2005 and KS2020ER. However, by exploiting the close connection between the ergodicity of a CKSVAR and the stability of its associated deterministic system, as per (ref), we can assess ergodicity using less conservative criteria than are available when working with a broader class of regime-switching autoregressive processes.
To formulate the criteria in abstract terms, let $\{w_{t}\}_{t\in\ensuremath{\mathbb{N}}_{0}}$ be a deterministic process, taking values in $\mathbb{R}^{d_{w}}$, that evolves according to a switched linear system: that is, a system in which there is a (possibly uncountable) set ${\cal I}\ni i$ of states, each of which is associated to a distinct autoregressive matrix $A[i]$. If $\{i_{t}\}\subset{\cal I}$ records the state of the system at each $t\in\ensuremath{\mathbb{N}}_{0}$, then $\{w_{t}\}$ evolves as
from some given $w_{0}\in\mathbb{R}^{d_{w}}$. Suppose further that the state in period $t$ is entirely determined by the value of $w_{t}$, via the mapping $\sigma:\mathbb{R}^{d_{w}}\ensuremath{\rightarrow}{\cal I}$. Then $i_{t}=\sigma(w_{t})$, and
Let ${\cal A}\coloneqq\{A[i]\}_{i\in{\cal I}}$ denote the set of autoregressive matrices, and $\mathscr{W}_{i}\coloneqq\sigma^{-1}(i)$, so that $\{\mathscr{W}_{i}\}_{i\in{\cal I}}$ partitions $\mathbb{R}^{d_{w}}$.
We say (ref) is stable if $w_{t}\ensuremath{\rightarrow}0$ as $t\ensuremath{\rightarrow}\infty$, for every $w_{0}\in\mathbb{R}^{d_{w}}$ (what is more precisely termed `global asymptotic stability'; e.g.\ Ela05). The usual approach to establishing the stability of such a system is to construct a Lyapunov function. For our purposes, we may take this to be a function $V$ and an associated value $\gamma_{V}\in\mathbb{R}_{+}$ such that
As is well known, if we can find a $(V,\gamma_{V})$ with $\gamma_{V}<1$, the system is stable (Ela05): and thus the problem of establishing the stability of (ref) can be reformulated as one of showing that the stability degree \[ \gamma^{\ast}\coloneqq\inf\{\gamma_{V}\in\mathbb{R}_{+}\mid(V,\gamma_{V})\text{ satisfy (i)--(ii)}\} \] of the system is strictly less than unity. (While stability may be established using a Lyapunov function satisfying weaker conditions than (i)--(ii), for the systems considered here these conditions are not restrictive: see (ref).)
While the calculation of $\gamma^{\ast}$ is in general an undecidable problem, various upper bounds for it are known. The most elementary of these is provided by the joint spectral radius, which for a bounded collection of matrices ${\cal A}$ may be defined as (see e.g.\ Jungers09, Defn.\ 1.1) \[ \rho_{{\scriptstyle \mathrm{JSR}}}(\mathcal{A})\coloneqq\limsup_{t\ensuremath{\rightarrow}\infty}\sup_{B\in\mathcal{A}^{t}}\rho(B)^{1/t} \] where $\rho(B)$ denotes the spectral radius of $B$, and $\mathcal{A}^{t}\coloneqq\{\prod_{s=1}^{t}M_{s}\mid M_{s}\in\mathcal{A}\}$ is the collection of all possible $t$-fold products of matrices in ${\cal A}$. For each $\epsilon>0$, there exists a norm $\smlnorm{\cdot}_{\ast}$ such that
(see e.g.\ Jungers09, Prop.\ 1.4) whence $V(w)\coloneqq\smlnorm w_{\ast}$ trivially yields a valid Lyapunov function, and so $\gamma^{\ast}\leq\rho_{{\scriptstyle \mathrm{JSR}}}(\mathcal{A})$. Note that $\sup_{i\in{\cal I}}\rho(A[i])$ provides only a lower bound on $\rho_{{\scriptstyle \mathrm{JSR}}}(\mathcal{A})$, so control over the individual spectral radii is necessary, but not sufficient, for control over the JSR of ${\cal A}$.
The conservativeness of the JSR is readily apparent from the fact that it implies the stability of (ref) irrespective of the sequence $\{i_{t}\}$, so that a system adjudged to be stable by this criterion would remain so even if $\sigma$ in (ref) were replaced by an arbitrary mapping $\sigma^{\prime}:\mathbb{R}^{d_{w}}\ensuremath{\rightarrow}{\cal I}$. The JSR can thus be immediately improved upon by taking account of the constraints on the sequence $\{i_{t}\}$ of states permitted by the system. Suppose now that ${\cal I}$ is finite, and let ${\cal J}\subset{\cal I}\times{\cal I}$ denote the set of all pairs $(i,j)$ such that state $i$ may be followed by state $j$, i.e.\ $(i,j)\in{\cal J}$ if there exists a $w\in\mathscr{W}_{i}$ such that $A[\sigma(w)]w\in\mathscr{W}_{j}$; we say that $\{i_{t}\}$ is admissible by ${\cal J}$ if $(i_{t},i_{t+1})\in{\cal J}$ for all $t$. Then the constrained joint spectral radius (CJSR) may be defined as (cf.\ Dai12LAA, Defn.\ 1.2 and Thm. A; PEDJ16Auto, p.\ 243)
where now $\mathcal{A}^{t}({\cal J})\coloneqq\{\prod_{s=1}^{t}A[i_{s}]\mid\{i_{s}\}\text{ is admissible by }\mathcal{J}\}$. By PEDJ16Auto, for each $\epsilon>0$ there exists a family of norms $\{\smlnorm{\cdot}_{i}\}_{i\in\mathcal{I}}$ such that
whence $V(w)\coloneqq\sum_{i\in\mathcal{I}}\smlnorm w_{i}\ensuremath{\mathbf{1}}\{w\in\mathscr{W}_{i}\}$ is a Lyapunov function; it is evident that $\rho_{{\scriptstyle \mathrm{CJSR}}}(A[\cdot];{\cal J})\leq\rho_{{\scriptstyle \mathrm{JSR}}}(\mathcal{A})$, so that the CJSR yields an improved estimate of the stability degree of the system.
The form of the Lyapunov function implicitly provided by the CJSR in turn suggests how this construction may be further improved upon. For $V(w)=\sum_{i\in\mathcal{I}}\smlnorm w_{i}\ensuremath{\mathbf{1}}\{w\in\mathscr{W}_{i}\}$ to be a Lyapunov function, it is sufficient that: (i) such an inequality as (ref) hold only for $w\in\mathscr{W}_{i}$ such that $A[i]w\in\mathscr{W}_{j}$, rather than for all $w\in\mathbb{R}^{d_{w}}$; and (ii) each $\smlnorm{\cdot}_{i}$ satisfy $\smlnorm w_{i}>c_{i}\smlnorm w$ on $w\in\mathscr{W}_{i}$, for some $c_{i}>0$, i.e.\ $\smlnorm{\cdot}_{i}$ need not itself be a norm. Relaxing (ref) in this manner, and replacing each norm $\smlnorm{\cdot}_{i}$ by some mapping $\smldblangle{\cdot}_{i}:\mathbb{R}^{d_{w}}\ensuremath{\rightarrow}\mathbb{R}$ from a class of functions ${\cal C}$, leads us to define the relaxed joint spectral radius (RJSR) for class ${\cal C}$ as\footnote{When ${\cal C}$ is the set of all norms on $\mathbb{R}^{d_{w}}$, and the `$\forall w\in\mathscr{W}_{i}$, etc.'\ qualifiers are replaced by `$\forall w\in\mathbb{R}^{d_{w}}$', (ref) provides a valid characterisation of the CJSR: see PEDJ16Auto.}\needspace{3\baselineskip}
Clearly the preceding provides an upper bound for $\gamma^{\ast}$. Whether it provides a lower bound for $\rho_{{\scriptstyle \mathrm{CJSR}}}(A[\cdot];{\cal J})$ depends on the choice of $\mathcal{C}$; this is the case if e.g.\ $\mathcal{C}$ is taken to be the set of norms on $\mathbb{R}^{d_{w}}$, or some class capable of approximating these norms arbitrarily well (such as homogeneous polynomials).
In practice, the main difficulty with all of these objects is computational, with the exact computation of the (C)JSR known to be an NP--hard problem. Nonetheless, significant progress has been made in calculating approximate upper bounds for both of these quantities, with an (in principle) arbitrarily high degree of accuracy (PJ08LAA,LPJ19ITAC,LPJ20SJCO), using semi-definite (SDP) and sum of squares (SOS) programming. With respect to approximating the RJSR, there is naturally the risk of choosing ${\cal C}$ to be too broad a class of functions that the approximation of $\rho_{{\scriptstyle \mathrm{RJSR}},{\cal C}}$ becomes infeasible. As explained in (ref), building on the approach of FTCMM02Auto, we therefore take as our starting point the SDP and SOS programs used to approximate the (C)JSR, relaxing the analogues of (ref) in the direction of (ref), for the case where the $\{\mathscr{W}_{i}\}_{i\in{\cal I}}$ are convex cones (as is appropriate for a CKSVAR). This entails relating ${\cal C}$ to either a certain class of quadratic functions (for the SDP program) or homogeneous polynomials of a given (even) order (for the SOS program), and has the additional benefit that our approximate calculations of $\rho_{{\scriptstyle \mathrm{RJSR}},{\cal C}}$ are guaranteed to provide a lower bound on the estimate of the $\rho_{{\scriptstyle \mathrm{CJSR}}}$ yielded by these programs, and thus a less conservative estimator $\gamma^{\ast}$ in practice. All of these quantities are calculated by our R package, thresholdr.
In the present setting, the role of (ref) is played by (ref), the deterministic system associated to a canonical CKSVAR, rendered here in companion form (and suppressing the `hat' decorations) as
where $\b y_{t-1}=(y_{t-1},\ldots,y_{t-k})^{\mathsf{T}}$, and \[ \b F(\b y_{t-1})\coloneqq
. \] Here there are (at most) $\smlabs{{\cal I}}=2^{k}$ distinct states. If we associate with each $i\in{\cal I}$ a distinct $s(i)\in\{0,+1\}^{k}$, we can take \[ \mathscr{Z}_{i}\coloneqq\{\b y_{t-1}\in\mathbb{R}^{k}\mid\ensuremath{\mathbf{1}}^{+}(y_{t-j})=s_{j}(i),\ \forall j\in\{1,\ldots,k\}\}\times\mathbb{R}^{p-1}, \] $\sigma(\b z_{t-1})=i$ if $\b z_{t-1}\in\mathscr{Z}_{i}$, and $\b F[i]$ to be the autoregressive matrix that applies in this case.
Letting ${\cal F}\coloneqq\{\b F[i]\}_{i\in{\cal I}}$, the JSR of ${\cal F}$ provides a first estimate of stability degree $\gamma^{\ast}$ of the system. However, since the sign pattern of the first $k-1$ elements of $\b y_{t-1}$ must propagate forwards to $\b y_{t}$, any given state $i\in{\cal I}$ can only be succeeded by two other states (which differ only according to the sign of $y_{t}$). Thus we can generally obtain a tighter estimate via the CJSR, which reduces the set of possible transitions from the $\smlabs{{\cal I}}^{2}=2^{2k}$ implicitly permitted by the calculation of the JSR, to $2\smlabs{{\cal I}}=2^{k+1}$. An even lower estimate of $\gamma^{\ast}$ -- which may prove decisive for establishing the stability of (ref) -- can be obtained via the RJSR, at the cost of greater computation time. (ref) thus has a noteworthy advantage over approaches that utilise bounds on the (C)JSR to directly establish the ergodicity of the corresponding stochastic system, because the evolution of the deterministic system is much more tightly constrained, particularly when the innovations $\{u_{t}\}$ have full support, as assumed in (ref).
For especially tractable instances of the CKSVAR, additional sufficient conditions for the stability of (ref) may also be available. For example, a result of DJH11 implies that for the censored dynamic Tobit ((ref) above), a sufficient condition for stability is that the `censored' autoregressive polynomial $1-\sum_{i=1}^{k}[\phi_{i}^{+}]_{+}\lambda^{i}$ should have all its roots outside the unit circle. (Interestingly, as their result indicates, control over the the roots of the various autoregressive regimes is not even necessary to ensure the stability of a CKSVAR.)
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{{\ref*{exa:monetary}}}
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To illustrate the practical utility of the stability criteria developed above, we compute upper bounds for the JSR, CJSR and RJSR (using our R package, thresholdr) for some of our running examples. These upper bounds are denoted with a `bar' (as $\bar{\rho}_{{\scriptstyle \mathrm{JSR}}}$, etc.)\ to distinguish them from the actual quantities, which cannot be computed exactly.
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{{\ref*{exa:monetary}}}
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{{\ref*{exa:univariate}}}
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{{\ref*{exa:explosive}}}
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The CKSVAR provides a flexible yet tractable framework in which to structurally model vector time series that are subject to an occasionally binding constraint, and more general threshold nonlinearities. Nonetheless, even that seemingly limited amount of nonlinearity radically changes the properties of the model. The usual criteria for stationarity and ergodicity, in terms of the roots of the autoregressive polynomial(s), no longer apply, being neither necessary nor sufficient for the CKSVAR to generate a stationary time series. We must instead employ other criteria, such as the joint spectral radius or the related quantities developed above, to establish stationarity via control over the deterministic subsystem of the CKSVAR.
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