Abstract
Large sample statistical analysis of threshold autoregressive models is usually based on the assumption that the underlying driving noise is uncorrelated. In this paper, we consider a model, driven by Gaussian noise with geometric correlation tail and derive a complete characterization of the asymptotic distribution for the Bayes estimator of the threshold parameter.
Similar content being viewed by others
References
Caner, M., Hansen, B. E. (2001). Threshold autoregression with a unit root. Econometrica, 69(6), 1555–1596.
Cappé, O., Moulines, E., Rydén, T. (2005). Inference in hidden Markov models. New York: Springer Series in Statistics, Springer.
Chan, K. S. (1993). Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model. The Annals of Statistics, 21(1), 520–533.
Chan, N. H., Kutoyants, Y. A. (2010). Recent developments of threshold estimation for nonlinear time series. Journal of the Japan Statistical Society, 40(2), 277–303.
Chan, N. H., Kutoyants, Y. A. (2012). On parameter estimation of threshold autoregressive models. Statistical Inference for Stochastic Processes, 15(1), 81–104.
Dedecker, J., Doukhan, P. (2003). A new covariance inequality and applications. Stochastic Processes and their Applications, 106(1), 63–80.
Doukhan, P., Louhichi, S. (1999). A new weak dependence condition and applications to moment inequalities. Stochastic Processes and their Applications, 84(2), 313–342.
Hansen, B. E. (2011). Threshold autoregression in economics. Statistics and its Interface, 4(2), 123–127.
Ibragimov, I. A., Has’minskiĭ, R. Z. (1981). Statistical estimation. Asymptotic theory. In Applications of Mathematics (Vol. 16). New York: Springer.
Kutoyants, Y. A. (2012). On identification of the threshold diffusion processes. Annals of the Institute of Statistical Mathematics, 64(2), 383–413.
Ling, S., Tong, H. (2005). Testing for a linear MA model against threshold MA models. The Annals of Statistics, 33(6), 2529–2552.
Ling, S., Tong, H., Li, D. (2007). Ergodicity and invertibility of threshold moving-average models. Bernoulli, 13(1), 161–168.
Liptser, R. S., Shiryaev, A. N. (2001). Statistics of random processes. II Applications. In Applications of Mathematics (New York) (Vol. 6, expanded ed.). Berlin: Springer.
Liu, W., Ling, S., Shao, Q. M. (2011). On non-stationary threshold autoregressive models. Bernoulli, 17(3), 969–986.
Meyer, R. M. (1973). A Poisson-type limit theorem for mixing sequences of dependent “rare” events. Annals of Probability, 1, 480–483.
Meyn, S., Tweedie, R. L. (2009). Markov chains and stochastic stability (2nd ed.). Cambridge: Cambridge University Press.
Pham, D. T., Chan, K. S., Tong, H. (1991). Strong consistency of the least squares estimator for a nonergodic threshold autoregressive model. Statistica Sinica, 1(2), 361–369.
Tong, H. (1983). Threshold models in nonlinear time series analysis. In Lecture Notes in Statistics (Vol. 21). New York: Springer.
Tong, H. (2011). Threshold models in time series analysis—30 years on. Statistics and its Interface, 4(2), 107–118.
Tsay, R. S. (1989). Testing and modeling threshold autoregressive processes. Journal of the American Statistical Association, 84(405), 231–240.
Author information
Authors and Affiliations
Corresponding author
Additional information
P. Chigansky is supported by ISF grant 314/09.
Appendix: Ergodic lemmas used in the proof
Appendix: Ergodic lemmas used in the proof
The proofs in Sect. 2 use the ergodic properties of the processes, summarized in the following lemmas. Our standing assumption is (10).
Lemma 6
For all integers \(j\ge 0\) and \(p\ge 1\),
and
with a positive constant \(r_1<1\) and constants \(R_1\) and \(R_2\), independent of \(\theta _0\).
Proof
For \(j\ge 1\),
Further, by Jensen’s inequality
and hence
Similarly, with \(\rho := |\rho ^+|\vee |\rho ^-|\)
and
which gives (45). \(\square \)
Lemma 7
The Markov chain \((X_j, \xi _j)\) has the unique invariant measure under \(\mathbb{P }_{\theta _0}\), with uniformly bounded probability density \(p(x,y;\theta _0)\) satisfying
where \(\widetilde{\mathbb{P }}_{\theta _0}\) is the corresponding stationary probability on \((\Omega ,\mathcal{F })\).
Moreover, the chain is geometrically ergodic, i.e., there exist positive constants \(C\) and \(r<1\), such that for a measurable function \(|h|\le 1\) and \(m\ge k\)
and consequently, for an \(\mathcal{F }_{m,\infty }\)-measurable random variable \(|H|\le 1\)
Finally, \((X_j,\xi _j)\) is geometrically mixing, i.e., for measurable functions \(|g|\le 1\), \(|h|\le 1\)
In particular, (48) and (49) hold with the stationary expectation \(\widetilde{\mathbb{E }}_{\theta _0}\).
Proof
The transition kernel of the process \((X_j,\xi _j)\) has a positive density with respect to the Lebesgue measure:
and hence in the terminology of Meyn and Tweedie (2009), it is \(\psi \)-irreducible and aperiodic. Further, a ball \(B_R\) of radius \(R>0\) around the origin is a small set with respect to, e.g., the measure
and \(V(x,y)=|x|+|y|\) satisfies the drift condition
for sufficiently large \(R\). By Theorem 15.0.1 in Meyn and Tweedie (2009), it follows that there exists a unique invariant probability measure \(\pi \) and for any measurable \(h(x,y)\le V(x,y)\),
with positive constants \(C\) and \(r<1\), i.e., (47) holds. Since \(\widetilde{\mathbb{E }}_{\theta _0} H = \widetilde{\mathbb{E }}_{\theta _0} h(X_m,\xi _m)\) and \(\mathbb{E }_{\theta _0} (H|\mathcal{F }_k) = \mathbb{E }_{\theta _0} (h(X_m,\xi _m)|\mathcal{F }_k)\) with \(h(x,y):= \mathbb{E }_{\theta _0}(H|X_m=x,\xi _m=y)\), the claim (48) follows from (45) and (47). Since the transition kernel \(P\) has a bounded continuously differentiable density with respect to the Lebesgue measure, so does the invariant measure \(\pi \) and (46) follows. The mixing inequality (49) follows from Theorem 16.1.5 in Meyn and Tweedie (2009). \(\square \)
The theory, used in the proof of the previous lemma, does not directly apply to the Markov chain \((X_j,\xi _j,\widehat{\Xi }_j)\) [see (16) for the definition of \(\widehat{\Xi }_j\)], since it is generated by a \((3+d)\)-dimensional recursion, driven by two dimensional noise. This typically excludes \(\psi \)-irreducibility. Fortunately, for our purposes the following weaker properties are sufficient:
Lemma 8
The Markov process \((X_j,\xi _j,\widehat{\Xi }_j)\) has the unique invariant measure. Let \(\widetilde{\mathbb{P }}_{\theta _0}\) denote the corresponding stationary probability (by uniqueness, the stationary probabilities \(\widetilde{\mathbb{P }}_{\theta _0}\), introduced in Lemmas 7 and 8, coincide). Then for a measurable function \(h(x,y,z)\), satisfying \(|h(x,y,z)|<1\) and the Lipschitz condition
with a positive constant \(L\),
for some positive constants \(C\) and \(q<1\) and all integers \(m\ge \ell \ge 0\).
Proof
Under the stationary measure \(\widetilde{\mathbb{P }}_{\theta _0}\) from Lemma 7, we can extend the definition of \((X_j,\xi _j)\) to the negative integers and define
where \(b:=\frac{a}{1+\gamma }\) and \(c:=\frac{a\gamma }{1+\gamma }\). The distribution of \((X_0,\xi _0,\widehat{\Xi }_0)\) is invariant. To establish uniqueness, let \(\mu \) and \(\mu ^{\prime }\) be two invariant measures and note that by Lemma 7 their \((X,\xi )\) marginals coincide. Hence,
where \(\nu \) is the invariant measure of the process \((X_j,\xi _j)\) and \(\mu (x,y;\mathrm{d}z)\) and \(\mu ^{\prime }(x,y;\mathrm{d}z)\) are corresponding regular conditional probabilities. Let \((X_j,\xi _j,\widehat{\Xi }_j)\) and \((X_j,\xi _j,\widehat{\Xi }^{\prime }_j)\) be the solutions of the recursions (2), (3) and (7) with \(u:=u_k\), \(k=0,{\ldots },d\) subject to the initial conditions \((X_0,\xi _0,\widehat{\Xi }_0)\) and \((X_0,\xi _0,\widehat{\Xi }^{\prime }_0)\), where \((X_0,\xi _0)\) is sampled from \(\nu \) and \(\widehat{\Xi }_0\) and \(\widehat{\Xi }^{\prime }_0\) are sampled from \(\mu (X_0,\xi _0;\mathrm{d}z)\) and \(\mu ^{\prime }(X_0,\xi _0;\mathrm{d}z)\). Note that \(\widehat{\Xi }^{\prime }_j-\widehat{\Xi }_j= b^j(\widehat{\Xi }^{\prime }_0-\widehat{\Xi }_0)\) and hence for any uniformly continuous function \(g\)
Since uniformly continuous functions form a measure defining class, the uniqueness follows.
To derive the bound (50), note that for \(\ell \le m\)
Using the bound (45), we get
By the triangle inequality
Note that \(h(X_m,\xi _m, J_2)\) is measurable with respect to \(\mathcal{F }_{\frac{1}{2} (m+\ell ),\infty }\) and by (48)
By the Lipschitz property of \(h\) and (52), we have
Similar bound holds for the last term in (53) and the claim follows with \(q:=\sqrt{|b|\vee r}\). \(\square \)
About this article
Cite this article
Chigansky, P., Kutoyants, Y.A. Estimation in threshold autoregressive models with correlated innovations. Ann Inst Stat Math 65, 959–992 (2013). https://doi.org/10.1007/s10463-013-0402-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-013-0402-4