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Stationarity and ergodicity of vector STAR models
Consider the vector smooth transition autoregressive model (see Hubrich and Terasvirta, 2013)
where $y_t$ and $\varepsilon_t$ are $n\times 1$ random vectors, $\mu_0$ and $\mu_1$ are $n\times 1$ intercept vectors, $\Phi_j$ and $\Psi_j$, $j=1,\ldots,p$, are $n\times n$ parameter matrices. We assume that the random vectors $\varepsilon_t$ are i.i.d., with zero mean and any positive definite covariance matrix and with a density bounded away from zero on compact subsets of $R^n$.
Furthermore, the continuous function of random variable $s_t$ and parameters $\gamma>0$ and $c$, $G(\gamma,c;s_t)$ takes values on $[0,1]$ and is called the transition function. Random variable $s_t$ is a function of $\{y_{t-j}, j=1,\ldots,p\}$. For example, it could be a logistic function
and $s_t=y_{t-j,i}$ for some $j\in\{1,\ldots,p\}$ and $i\in\{1,\ldots,n\}$.
The goal of this paper is to formulate conditions for the existence of a stationary solution to ((ref)). We don't aim to cover the most general case; instead, we provide an explicit treatment of the most popular models at the same time trying to keep exposition simple. We start with a simple model with two regimes and homoskedastic errors. Then we extend it to a more general situation with several regimes and regime-dependent error covariance matrix.
Conditions for stationarity exist for regime switching vector error correction models, see Bec and Rahbek (2004), Saikkonen (2005, 2008). We are not aware of corresponding conditions for vector STAR models.
The rest of the paper is organized as follows. Section 2 introduces the concept of the joint spectral radius of a set of matrices and shows how it can be used. Section 3 provides a numerical illustration. Section 4 contains our general result. Then we conclude.
For the model in Equation ((ref)) define matrices
The joint spectral radius (JSR) of a finite set of square matrices $\mathcal{A}$ is defined by
where $\mathcal{A}^j=\{A_1 A_2\ldots A_j:A_i\in\mathcal{A}, i=1,\ldots,j\}$ and $\rho(A)$ is the spectral radius of the matrix $A$, i.e.\ its largest absolute eigenvalue. See Jungers (2009) for the survey on the JSR.
Assumption R$'$ The joint spectral radius of matrices $B_1$ and $B_2$ is less than $1$, i.e.\ $\rho(\{B_1,B_2\})<1$.
Suppose that Assumption R$'$ is satisfied. Then there exists a solution to ((ref)), which is 1) strictly stationary 2) second-order stationary 3) $\beta$-mixing with geometrically decaying mixing numbers. This statement is a special case of Theorem (ref) below. Indeed, Equation ((ref)) is a particular case of Equation ((ref)) in Section (ref), setting in the latter $g=1$ and $\Omega_0=\Omega_1$.
There are a number of methods to approximate the JSR and verify Assumption {R$'$}, see Vankeerberghen, Hendrickx and Jungers (2014). For example, Gripenberg (1997) and Blondel and Nesterov (2005) describe algorithms to find an arbitrary small interval containing the JSR. The procedure of Blondel and Nesterov (2005) provides approximations to $\rho(\mathcal{A})$ of relative accuracy $1-\epsilon$ in time polynomial in $dim(B_1)^{(\ln card(\mathcal{A}))/\epsilon}$, where $dim(B_1)$ is the size of matrix $B_1$ and $card(\mathcal{A})$ is the number of matrices in set $\mathcal{A}$, which are small in a typical application in econometrics involving nonlinear dynamics. For the model in Equation ((ref)) computation is polynomial in $(np)^{(\ln 2)/\epsilon}$. In the next section we consider a model for the UK macroeconomic variables and approximate the JSR using a simple method which works out of the box. For that model, the computation is polynomial in $6^{(\ln 2)/\epsilon}$ and takes a few seconds.
In case of large matrices, the following simple but useful lemmas help to reduce dimensions of matrices and bound the JSR from below. To keep the exposition simple, we state them for sets consisting of two matrices, although they hold for any arbitrary (finite) number of matrices. Their proofs can be found in Protasov (1996), Blondel and Nesterov (2005) and Jungers (2009, Section 1.2.2).
In particular, $\max(\rho(B_1),\rho(B_2))\le\rho(\{B_1,B_2\})$, which gives a lower bound for the JSR which is easy to calculate. Also,
This condition is not sufficient for Assumption R$'$. Liebscher (2005), page 676, provides a simple example of a process, for which all eigenvalues are less than $1$ in absolute value, but the JSR is greater than $1$ and, moreover, the process is not ergodic.
Invariance under linear bijections is useful because transformations help to reduce the dimension of the problem. If the condition of the following lemma is satisfied, the set of matrices is called reducible and its JSR can be calculated from the JSR of smaller matrices.
A bivariate LSTAR model for joint movement of output growth ($y_{t1}$) and the interest rate spread ($y_{t2}$) in the UK is suggested by Anderson, Athanasopoulos, and Vahid (2007). In particular, the conditional mean of $y_t$, $\mu_t=\left(\mu_{t1},\mu_{t2}\right)'$, as a function of unknown parameters $b=\left(b_1,\ldots,b_{11}\right)'$ is
The logistic smooth transition autoregressive specification for output growth incorporates different regimes and smooth transitions between them.
Anderson, Athanasopoulos, and Vahid (2007) estimated the model by Maximum Likelihood (ML), while Kheifets (2018) propose tests of the following null hypothesis
where $\Sigma$ is a nonrandom positive definite covariance matrix and $\mathcal{I}_t$ is the information set generated by $y_{t-j}, j = 1,2,\ldots$.
Both papers rely on stationarity and ergodicity of the bivariate series. We assume that $y_t=\mu_t + \varepsilon_t$, where random vectors $\varepsilon_t$ are i.i.d.\ with zero mean and any positive definite covariance matrix with a density bounded away from zero on compact subset of $R^n$. Then, in our notation, $\gamma=b_6$, $c= b_7$ and $s_t=y_{t-2,1}$, and the system has stationary solution if Assumption R$'$ holds for the following matrices:
The sample used by Anderson, Athanasopoulos, and Vahid (2007) consists of $159$ quarterly time series observations, dating from 1960:3 to 1999:4, see Figure (ref). The data is available at \url{http://qed.econ.queensu.ca/jae/2007-v22.1/anderson-athanasopoulos-vahid/}. Output growth is calculated as $100$ $\times$ the difference of logarithms of seasonally adjusted real DGP, and the spread is the difference between the interest rates on 10 Year Government Bonds and 3 Month Treasury Bills.
Estimating by ML, we obtain the following coefficients $$\hat b = (0.35, 0.21, 0.15, 0.32, −0.52, 2.56, 0.68, 0.20, −0.14, 1.14, −0.26)′,$$ and standard errors $$ s.e. (\hat b) = (0.09, 0.09, 0.09, 0.11, 0.15, 0.29, 0.12, 0.09, 0.10, 0.10, 0.10) $$ (calculated using residual parametric bootstrap with $10000$ draws) and construct corresponding matrices $\hat B_1$ and $\hat B_2$. Imposing stationary initial values in the estimation is not feasible because for the considered LSTAR model (as well as most other nonlinear (V)AR models) the stationary distribution is not known. Thus, the ML estimation employed is necessarily conditional on fixed (and known) initial values.
The maximum eigenvalues of $\hat B_1$ and $\hat B_2$ are $0.9216$ and $0.8236$, both below $1$, therefore the necessary condition of Assumption R$'$ is satisfied. To obtain bounds on the joint spectral radius we use the JSR toolbox written in Matlab by Raphael Jungers, freely downloadable (with documentation and demos) from Matlab Central (\url{https://www.mathworks.com/matlabcentral/fileexchange/33202-the-jsr-toolbox}) The toolbox is described in Vankeerberghen, Hendrickx and Jungers (2014). In our case, the toolbox provides the upper and lower bounds which coincide up to the 4th digit and the value is $0.9216$, calculated in 9 iterations in 31 seconds. Therefore, Assumption R$'$ is satisfied.
Notice that the last column of both matrices consists of zeros. That means that the set $\{B_1,B_2\}$ is reducible to $\{B_{1,1},B_{2,1}\}$ (the other two matrices are $1$ by $1$ zeros):
so $\rho\{B_1,B_2\}=\rho\{B_{1,1},B_{2,1}\}$ by Lemma 3 and 4. The JSR toolbox performs this reduction.
We now state our general result for a model with $g+1$ regimes and heteroskedasticic errors,
where $y_t$ and $\varepsilon_t$ are $n\times 1$ random vectors, $\mu_0$ and $\mu_{i}$ are $n\times 1$ intercept vectors, $\Phi_j$ and $\Psi_{i,j}$, $j=1,\ldots,p$, are $n\times n$ parameter matrices, and $\Omega_i$ is positive definite and $i=1,\ldots,g$. We assume that random vectors $\varepsilon_t$ are i.i.d.$(0,I_n)$ with a density bounded away from zero on compact subsets of $R^n$. If regime switches are only in conditional means, i.e. variances coincide in all regimes, $\Omega_0=\Omega_i$ for all $i=1,\ldots,g$, then the last term is simplified to $\Omega_0^{1/2}\varepsilon_t$. The transition function is common for all components of vector $y_t$, and it is a continuous function of random variable $s_t$ and parameters $\gamma$ and $c$ and takes values on $[0,1]$. Moreover, $G_1 + ... + G_g \le 1$ and $s_t$ is a function of $\{y_{t-j}, j=1,\ldots,p\}$ as before.
For $i=1,\ldots,g$ define matrices
Assumption R The joint spectral radius of the matrices defined above is less than $1$, i.e.\ $\rho(\{B_1,\ldots,B_g, B_{g+1}\})<1$.
We will use the Markov chain theory on which Mayen and Tweedie (1993) is a standard reference. In particular, for geometric ergodicity of a Markov chain see Meyn and Tweedie (1993), Chapter 15.
In this paper we describe conditions for stationarity and ergodicity of vector STAR models. The sufficient condition is that the joint spectral radius of certain matrices is below $1$. This condition can be checked using recently introduced toolboxes from computational mathematics.
For linear models, for example for the (causal) univariate autoregressive model of order $1$, it is known that stationarity holds if the autoregressive coefficient is in $(-1,1)$, otherwise the time series is nonstationary. For the considered vector LSTAR model, the stationarity holds if the joint spectral radius is below $1$. However, it is only a sufficient condition for stationarity and ergodicity. Thus, if the joint spectral radius equals one or is larger than one, we can only conclude that it is not possible to verify stationarity and ergodicity by using the employed criterion and in principle it is possible that stationarity and ergodicity still holds. The conclusion is similar to that discussed above if the value of the joint spectral radius based on estimates is smaller than one but deemed to be so close to one that, due to estimation errors, it can be one or even larger than one with high probability.
An interesting extension of the model considered here would be to add regressors $x_t$, as in e.g. Hubrich and Terasvirta (2013)
where $x_t$ are $k\times 1$ random vectors, $\Gamma$ and $\Xi$ are $n\times k$ parameter matrices. Suppose that $x_t$ is an exogenous random vector. Then no results on stationarity can hold true if $x_t$ is nonstationary, and even if $x_t$ is assumed stationary and ergodic Theorem 1 in Saikkonen (2008) is not applicable because without further assumptions model (8) cannot be cast into the required Markov chain form.
Igor Kheifets gratefully acknowledges financial support from the Spanish Ministerio de Ciencia, Innovacion y Universidades under Grant ECO2017-86009-P and thanks the faculty and staff of the New Economic School for their hospitality during his visits to Moscow. Pentti Saikkonen thanks the Academy of Finland (grant number 1308628) for financial support.