����������� ������ 54 (2002), 31–36

UDK 519.218

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M-ESTIMATES OF SETAR MODEL PARAMETERS

Dragan -Dorić

Abstract. For a stationary ergodic self-exiting threshold autoregressive model with single threshold parameter Chan (1993) obtained the consistency of the least-squares estimator and Qian (1998) proved it for the maximum likelihood estimator. The aim of this paper is to derive the similar results for the M-estimates of the same model under some regularity conditions.

1. Introduction

Nonlinear time series has drawn much attention in recent years and many classes of models have been proposed. One of the most popular is a class of threshold models characterized by piecewise linear processes separated according to the magnitude of threshold variable. When each linear regime follows an autoregressive process we have the well known threshold autoregressive (TAR) models. The major features of this class of models include limit cycles, jump resonance and interesting asymmetric features observed in economic and financial time series. Tong (1990) provides an excellent review of properties of that models. One of the TAR models is self-exciting threshold autoregressive (SETAR) model where threshold variable is one of the past observations. SETAR model can also exhibit nonlinear phenomena and Petruccelli (1992) shown that processes generated by this model may be viewed as an approximation to a more general class of nonlinear processes.

There are mainly two methods of parameter estimation which have been used in the literature of nonlinear time series analysis. The general theory of least squares estimates and maximum-likelihood estimators of nonlinear smooth autoregressive models is due to Klimko and Nelson (1978). Tjostheim (1986) extended these results to a general class of estimators. As in the threshold models the autoregression function is not differentiable at some points, these results are not applicable.

There are also results for estimators of nonlinear models with non-smooth, but Lipschitz continuous, autoregression function. Liebscher (2000) considered the model

X_{t+1} = g(X_{t},…, X_{t−p+1}; θ_{0}) + ε_{t+1}, t = p, p + 1,…

AMS Subject Classification: 62 M 10

Keywords and phrases: Autoregressive threshold model, M-estimates, strong consistency.

where {ε_{t}} is a sequence of i.i.d. random variables, θ_{0} ∈ Θ ⊂ R^{p} is the vector of the true parameters and g : R^{p} × Θ → R is a continuous function. He utilized a variational principle from stochastic optimization theory and proved strong convergence for some classes of robust estimators. Suppose that

Q_{n}(θ) = ∑_{t=p}^{n−1} ρ(X_{t+1} − g(X_{t}; θ)),

where X_{t} = (X_{t}, X_{t−1},…, X_{t−p+1})^{T}. Then M-estimator is determined by the function ρ: R → R. Liebscher (2000) assumed that ρ satisfies a nonuniform Lipschitz condition

|ρ(x) − ρ(y)| ≤ L_{ρ}(|x|^{α} + |y|^{α} + 1)|x − y|

for α > 0. It is possible to apply his result to continuous SETAR, but not for a general SETAR models.

Chan (1993) developed the strong consistency and limiting distribution of the conditional least-squares estimators in a SETAR(2; p, p) model. Qian (1998) obtained it for the maximum-likelihood estimates. In this paper we derive the similar results for the M-estimates. Section 2 presents the model and parameter estimates, while Section 3 describes assumptions and obtains the strong consistency of the M-estimates of the true parameters.

2. Model and parameter estimates

The process {X_{t}} is said to be a self-exciting threshold autoregressive process with threshold variable X_{t−d} if it is generated by SETAR(k; p_{1},…, p_{k}) model defined by

X_{t} = a_{i,0} + a_{i,1}X_{t−1} + ⋯ + a_{i,p_{i}}X_{t−p_{i}} + ε_{t}

for r_{i−1} < X_{t−d} ≤ r_{i}, i = 1, 2,…, k, where real numbers r_{i} satisfy

−∞ = r_{0} < r_{1} < ⋯ < r_{k} = +∞

and form a partition of the space of X_{t−d}. The innovation {ε_{t}} is an i.i.d. sequence of random variables independent of the past X_{t−1}, X_{t−2},…. The parameters r_{i} and the positive integer d are called the thresholds and the delay, respectively. The coefficients a_{i,j} and b_{i,j} are some real constants. Here, we only consider the case k = 2 and p_{1} = p_{2} = p,

X_{t} = \{ \begin{matrix}a_{1,0} + a_{1,1}X_{t−1} + ⋯ + a_{1,p}X_{t−p} + ε_{t}, & X_{t−d} ≤ r \\ a_{2,0} + a_{2,1}X_{t−1} + ⋯ + a_{2,p}X_{t−p} + ε_{t}, & X_{t−d} > r. \end{matrix}  (2.1)

Given the data X_{0}, X_{1},…, X_{n} generated from (2.1), the true parameter θ_{0},

θ_{0} = (A_{1}^{T} , A_{2}^{T} , r, d)^{T}  (2.2)

where

A_{i} = (a_{i,0}, a_{i,1},…, a_{i,p})^{T} i = 1, 2  (2.3)

should be estimated. We assume that θ_{0} is an interior point of the parameter space R^{2p+2} × \overline{R} × \{1, 2,…, p\} where \overline{R} = R ∪ \{−∞, ∞\}. There exists a compact subset

M-estimates of SETAR model parameters 33

K of R^{2p+2} such that θ_{0} is an interior point of Θ = K × \overline{R} × \{1, 2,…, p\}. We define the estimator θ̂_{n} as a global minimizer of a criterion Q_{n}(θ) on a compact subset Θ

θ̂_{n} ∈ argmin_{θ∈Θ} Q_{n}(θ),  (2.4)

where Q_{n}(θ) = ∑_{i=p}^{n} ρ_{k}(e_{k}(θ), θ) and e_{k}(θ) = X_{k} − E_{θ} (X_{k} | F_{k−1}). Least-squares estimators represent a special case of θ̂_{n} for ρ_{k}(x) = ρ(x) = x^{2}. The minimization can be done in two steps. Assume that θ = (B_{1}^{T} , B_{2}^{T} , s, c)^{T} is a general parameter in Θ and let A = (A_{1}^{T} , A_{2}^{T})^{T} and B = (B_{1}^{T} , B_{2}^{T})^{T}.

1. For fixed s ∈ \overline{R} and c ∈ \{1, 2,…, p\} let B_{n}(s, c) be defined by

B_{n}(s, c) ∈ argmin_{B∈K} Q_{nsc}(B),

where Q_{nsc}(B) = Q_{n}(B, s, c) = Q_{n}(θ).

2. Let r̂_{n} i d̂_{n} be the values of s and c for which

Q_{n}(B_{n}(r̂_{n}, d̂_{n}), r̂_{n}, d̂_{n}) = min_{s∈\overline{R},c∈\{1,2,... ,p\}} Q_{n}(B_{n}(s, c), s, c)

and let Â_{n} = B_{n}(r̂_{n}, d̂_{n}).

For any θ ∈ Θ, by definition of B_{n}, r̂_{n} and d̂_{n}, we have
Q_{n}(θ) = Q_{n}(B, s, c) ≥ Q_{n}(B_{n}(s, c), s, c)
≥ Q_{n}(B_{n}(r̂_{n}, d̂_{n}), r̂_{n}, d̂_{n}) = Q_{n}(Â_{n}^{T} , r̂_{n}, d̂_{n}) = Q_{n}(θ̂_{n})

and, hence, θ̂_{n} = (Â_{n}^{T} , r̂_{n}, d̂_{n})^{T}.

3. Assumptions and strong consistency

Assume that ρ_{k} = ρ(e_{k}). The model (2.1) can be represented as

X_{t} = h(X_{t}, θ_{0}) + ε_{t}, t ≥ 1

where, for x = (x_{1},…, x_{p}) ∈ R^{p}, θ ∈ Θ,

h(x, θ) = (a_{1,0} + ∑_{i=1}^{p} a_{1,i}x_{i})I(x_{d} ≤ r) + (a_{2,0} + ∑_{i=1}^{p} a_{2,i}x_{i})I(x_{d} > r).

Let h(x, θ) = h_{sc}(x, B) = h(x, B, s, c),

ḣ_{sc}(x) = \frac{∂}{∂B} h_{sc}(x, B)  (3.1)
z = (1, x^{T})^{T} .  (3.2)

Then,

ḣ_{sc}(x, B) = (z^{T} I(x_{c} ≤ s), z^{T} I(x_{c} > s))^{T}  (3.3)

and

h(x, θ) = B^{T} ḣ_{sc}(x)  (3.4)

for s ∈ \overline{R}, x ∈ R^{p} and c ∈ \{1,…, p\}. Equations (3.1), (3.2) and (3.3) imply that

|ḣ_{sc}(x)| ≤ \sqrt{1 + ∥x∥^{2}}.  (3.5)

From the last inequality and (3.4) we have

|h(x, θ)| ≤ ∥B∥\sqrt{1 + ∥x∥^{2}}.  (3.6)

Furthermore, for any s_{1} ∈ \overline{R}, s_{2} ∈ \overline{R} i c ∈ \{1, 2,…, p\}

|ḣ_{s_{1}c}(x) − ḣ_{s_{2}c}(x)| ≤ \sqrt{2(1 + ∥x∥^{2})} · I(min\{s_{1}, s_{2}\} < x_{c} ≤ max\{s_{1}, s_{2}\})

≤ \sqrt{2(1 + ∥x∥^{2})} · I(|x_{c} − s_{2}| ≤ |s_{1} − s_{2}|) (3.7)

To derive the consistency of the estimates we shall need the following assumptions on the class of function ρ.

Condition 1. For x ∈ R, y ∈ R and some α > 0, β ≥ 1, γ > 0 function ρ satisfies

|ρ(x) − ρ(y)| ≤ α (|x|^{β} + |y|^{β} + 1) · |x − y|^{γ}.

Condition 2. δEρ(ε_{1} + a) > Eρ(ε_{1}) for a ≠ 0.

Moreover, suppose the existence of some moments of the process {X_{t}}.

Condition 3. E |X_{t}|^{2β} < ∞ and E |X_{t}|^{2γ} < ∞. First, we need to prove the following statement.

Lemma 1. Suppose that {X_{t}} in model (2.1) is stationary and ergodic and the Conditions 1 and 3 hold. Then, for any θ ∈ Θ

E sup_{θ^{∗}∈U_{θ}(ν)} |ρ(e_{1}(θ^{∗})) − ρ(e_{1}(θ))| → 0 (ν → 0),

where U_{θ}(ν) is a ν neighborhood for θ given by

U_{θ}(ν) = {θ^{∗} ∈ Θ : θ^{∗} = (B^{∗^{T}} , s^{∗}, c)^{T} , |B^{∗} − B| < ν, |s^{∗} − s| < ν, ν > 0} .

Proof. Let ε_{1}(θ) = X_{1} − h(X_{0}, θ) and δ(x, θ^{∗}) = h(x, θ) − h(x, θ^{∗}). Then,

δ(x, θ^{∗}) = h_{s,c}(x, B) − h_{s,c}(x, B^{∗}).

From (3.4) and (3.5) we conclude

|h_{sc}(x, B) − h_{sc}(x, B^{∗})| ≤ ∥B − B^{∗}∥\sqrt{1 + ∥x∥^{2}}.

By (3.3) and (3.7) we have

|h_{sc}(x, B^{∗}) − h_{s^{∗}c}(x, B^{∗})| ≤ ∥B^{∗}∥\sqrt{2(1 + ∥x∥^{2})}I(min\{s, s^{∗}\} < s_{c} ≤ max\{s, s^{∗}\})

≤ ∥B^{∗}∥\sqrt{2(1 + ∥x∥^{2})}I(|x_{c} − s| ≤ |s^{∗} − s|). (3.8)

M-estimates of SETAR model parameters 35

Thus for θ^{∗} ∈ U_{θ}(ν) and s ∈ R, x ∈ R^{p}

|δ(x, θ^{∗})| ≤ |h_{sc}(x, B) − h_{s^{∗}c}(x, B)| + |h_{s^{∗}c}(x, B) − h_{s^{∗}c}(x, B^{∗})|
≤ (\sqrt{2}∥B∥I(|x_{c} − s| ≤ |s^{∗} − s|) + ∥B − B^{∗}∥) \sqrt{1 + ∥x∥^{2}}
≤ (\sqrt{2}∥B∥I(|x_{c} − s| ≤ ν) + ν) \sqrt{1 + ∥x∥^{2}}.  (3.9)

It follows from Conditions 1. and inequality

(a + b)^{β} ≤ 2^{β−1}(a^{β} + b^{β}), a > 0, b > 0, β ≥ 1

that for α > 0, β ≥ 1 i γ > 0

Δρ(θ, θ^{∗}) = |ρ(e_{1}(θ^{∗})) − ρ(e_{1}(θ))|
= |ρ(X_{1} − h(X_{0}, θ^{∗})) − ρ(X_{1} − h(X_{0}, θ))|

≤ α (|X_{1} − h(X_{0}, θ^{∗})|^{β} + |X_{1} − h(X_{0}, θ)|^{β} + 1) |h(X_{0}, θ) − h(X_{0}, θ^{∗}|^{γ}

= α (|ε_{1} + δ(X_{0}, θ_{0}, θ^{∗})|^{β} + |ε_{1} + δ(X_{0}, θ_{0}, θ)|^{β} + 1) |δ(X_{0}, θ, θ^{∗})|^{γ}

≤ α(2^{β}|ε_{1}|^{β} + 2^{β−1}|δ(X_{0}, θ_{0}, θ^{∗})|^{β} + 2^{β−1}|δ(X_{0}, θ_{0}, θ)|^{β} + 1)|δ(X_{0}, θ, θ^{∗})|^{γ}

≤ α (2^{β}|ε_{1}|^{β} + 2^{β−1}∥A − B^{∗}∥^{β}
· (1 + ∥X_{0}∥^{2})^{β/2} + 2^{β−1}∥A − B∥^{β}(1 + ∥X_{0}∥^{2})^{β/2} + 1)
· |δ(X_{0}, θ, θ^{∗})|^{γ}
= α (2^{β}|ε_{1}|^{β} + 1) |δ(X_{0}, θ, θ^{∗})|^{γ}+
+ α2^{β−1} (∥A − B^{∗}∥^{β} + ∥A − B∥^{β}) (1 + |X_{0}|^{2})^{β/2}|δ(X_{0}, θ, θ^{∗})|^{γ}

If \overline{δ}(x, ν) = (\sqrt{2}∥B∥I(|x_{c} − s| ≤ ν) + ν) \sqrt{1 + ∥x∥^{2}}, then by (3.9)

Δρ(θ, θ^{∗}) ≤ K_{1}|\overline{δ}(X_{0}, ν)|^{γ} + K_{2}|ε_{1}|^{β}|\overline{δ}(X_{0}, ν)|^{γ} + K_{3}(1 + ∥X_{0}∥^{2})^{β/2}|\overline{δ}(X_{0}, ν)|^{γ}. By the inequalities

E |ε_{1}|^{β}|\overline{δ}|^{γ} ≤ (E ε_{1}^{2β})^{1/2} · (E \overline{δ}^{2γ})^{1/2}
E (1 + ∥X_{0}∥^{2})^{β/2} |\overline{δ}|^{γ} ≤ (E (1 + ∥X_{0}∥^{2})^{β})^{1/2} · (E \overline{δ}^{2γ})^{1/2}

and the fact that E\overline{δ}^{2γ}(X_{0}, ν) → 0, (ν → 0) we conclude the statement for s ∈ R.

In the case s = ∞, for x ∈ R^{p},
|δ(x, θ, θ^{∗})| ≤ (\sqrt{2}∥B∥I(x_{c} > s^{∗}) + ν) \sqrt{1 + ∥x∥^{2}}
≤ (\sqrt{2}∥B∥I(d(x_{c}, ∞) < ν) + ν) \sqrt{1 + ∥x∥^{2}} = \overline{δ}_{∞}(x, ν)

and again, E\overline{δ}_{∞}^{2γ}(X_{0}, ν) → 0, (ν → 0).

The proof is similar in the case s = −∞. Thus, the lemma is proved.

Now, we are ready to state the main result.

Theorem 1. Under the assumptions of Lemma 1 and Condition 2

θ̂_{n} ⟶^{a.s.} θ_{0}, n → ∞.

Proof. Let the function f: Θ → R be defined by f(θ) = E ρ (e_{1}(θ)). Condition 2. implies

f(θ) = ∫_{R^{p}} E (ε_{1} + h(y, θ_{0}) − h(y, θ)) dF_{X_{0}}(y) > E ρ(ε_{1}) = f(θ_{0}),  (3.10)

where F_{X_{0}} is distribution function for X_{0}.

Define, then, the sequence of functions f_{n}: Θ → R by

f_{n}(θ) = \frac{1}{n − p + 1} ∑_{k=p}^{n} ρ(X_{k+1} − h(X_{0}, θ)).

Liebscher (2000) proved that, under the stationarity and ergodicity of the process {X_{t}},

sup_{θ∈Θ} |f_{n}(θ) − f(θ)| ⟶^{a.s.} 0, n → ∞

As, by definition of the functions f_{n} and the estimates θ̂_{n}, θ̂_{n} ∈ argmin_{θ∈Θ} f_{n}(θ) and, by (3.10), θ_{0} = argmin_{θ∈Θ} f(θ), then the result of Lemma 1 gives sufficient conditions for the convergence of θ̂_{n} to θ_{0} as shown by Vogel (1994). This completes the proof.

REFERENCES

[1] K. S. Chan, Consistency and limiting distributions of the least squares estimator of a threshold autoregressive model, Annals of Statistics, 21 (1993), 520–533.

[2] L. A. Klimko and P. L. Nelson, On conditional least-squares estimation for stochastic processes, Annals of Statistics, 6 (1978), 629–642.

[3] E. Liebscher, Strong convergence of estimators in nonlinear autoregressive models, Working paper, (2000), Technical University Ilmenau.

[4] D. J. Petruccelli, On the approximation of the time series by threshold autoregressive models, Sankhya, Series B, 54 (1992), 106–113.

[5] L. Qian, On maximum likelihood estimators for a threshold autoregression, J. Statistical Planning and Inference, 75 (1998), 21–46.

[6] D. Tjostheim, Estimation in nonlinear time series models, Stochastic Processes Appl., 21 (1986), 251–273.

[7] H. Tong, Nonlinear Tome Series: A Dynamical Approach, Oxford University Press.

[8] S. Vogel, A stochastic approach to stability in stochastic programming, J. Comp. Appl. Math., 56 (1994), 65–96.

(received 06.03.2002)

University of Belgrade, Faculty of Organizational Sciences, Jove Ilića 154, Belgrade, Yugoslavia E-mail: djoricd@fon.bg.ac.yu