Identification of Threshold Autoregressive Moving Average Models

Abstract

Due to the lack of a suitable modeling procedure and the difficulty to identify the threshold variable and estimate the threshold values, the threshold autoregressive moving average (TARMA) model with multi-regime has not attracted much attention in application. Therefore, the chief goal of our paper is to propose a simple and yet widely applicable modeling procedure for multi-regime TARMA models. Under no threshold case, we utilize extended least squares estimate (ELSE) and linear arranged regression to obtain a test statistic $\hat{F}$, which is proved to follow an approximate F distribution. And then, based on the statistic $\hat{F}$, we employ some scatter plots to identify the number and locations of the potential thresholds. Finally, the procedures are considered to build a TARMA model by these statistics and the Akaike information criterion (AIC). Simulation experiments and the application to a real data example demonstrate that both the power of the test statistic and the model-building can work very well in the case of TARMA models.

Appendix: Proofs of Theorems

Proof

(Theorem 1) For each regime of model (1.1), under the Assumption 4.1, we substitute the fitted residuals $\{\hat{\varepsilon }^{(j)}_{t-i}, i=1,\ldots ,q_j\}$ for $\{\varepsilon ^{(j)}_{t-i}, i=1,\ldots ,q_j\}$ using ELSE. Then, in every regime, model (1.1) is the linear regression model, we can obtain least squares estimate $\hat{\varPhi }^{(j)}$ of jth regime. Under the condition of Assumptions 1–4, they are fulfilled to the condition of Theorem 1 of Lai and Wei [17] and Theorem 2 of Liang et al. [21]. Therefore, for given k, d, and the threshold values $r_j$, the least squares estimates $\{\hat{\varPhi }^{(j)}, j=1,2,\ldots ,l\}$ converge to $\{\varPhi ^{(j)}, j=1,2,\ldots ,l\}$ almost surely. $\square$

Proof

(Theorem 2) Consider the observation $\{y_t, t=1, 2, \ldots , n\}$ and define

$$X_t = (1,y_{t-1},\ldots ,y_{t-p},\hat{\varepsilon }_{t-1},\ldots ,\hat{\varepsilon }_{t-q}),$$

with $\hat{\varepsilon }_{t-i}$’s being the residuals for model (1.1) fitted by the Hannan–Rissanen algorithm or ELSE.

Also define $\varPhi , A_n, V_n$ by

$$\varPhi ' = (\phi _0, \phi _1, \ldots , \phi _p, \theta _1, \ldots , \theta _q),$$

$$A_n = (n-p-q)^{-1}\sum X_t'X_t, \quad V_n = (n-p-q)^{-1}\sum X_t'y_t,$$

Therefore, without loss of generality, the least squares estimate of $\varPhi$ and the residuals are

$$\hat{\varPhi } = {A^{-1}_n}V_n,$$

$$\hat{e}_t = y_t - \hat{y}_t = X_t\varPhi + e_t - X_t\hat{\varPhi } = X_t(\varPhi - \hat{\varPhi }) + e_t.$$

Also define $\varPsi _n$ and $\hat{a}_t$ by

$$\varPsi _n = (n-p-q)^{-1}\sum X_t'\hat{e}_t$$

$$= (n-p-q)^{-1}\sum X_t'X_t(\varPhi - \hat{\varPhi })+ (n-p-q)^{-1}\sum X_t'e_t$$

$$= {A_n}(\varPhi - \hat{\varPhi })+ (n-p-q)^{-1}\sum X_t'e_t,$$

$$ \hat{a}_t = \hat{e}_t - X_t{A^{-1}_n}\varPsi _n.\quad \quad \quad \quad \quad \quad $$

Hence,

$$\begin{aligned} \begin{array}{llllllll} \big (\sum \hat{e}^2_t-\sum \hat{a}^2_t\big )/(n-p-q) \\ = \big [\sum \hat{e}^2_t - \sum (\hat{e}_t - X_t{A^{-1}_n}\varPsi _n)^2 \big ]/(n-p-q)\\ = \big [\sum \hat{e}_t'\hat{e}_t - \sum (\hat{e}_t - X_t{A^{-1}_n}\varPsi _n)'(\hat{e}_t - X_t{A^{-1}_n}\varPsi _n)\big ]/(n-p-q)\\ = \big [2\varPsi _n'{A^{-1}_n}\sum X_t'\hat{e}_t - (n-p-q) \varPsi _n'{A^{-1}_n}\varPsi _n\big ]/(n-p-q)\\ =\varPsi _n'{A^{-1}_n}\varPsi _n\\ = \big [{A_n}(\varPhi - \hat{\varPhi }) + (n-p-q)^{-1}\sum X_t'e_t\big ]'{A^{-1}_n}\big [{A_n}(\varPhi - \hat{\varPhi }) + (n-p-q)^{-1}\sum X_t'e_t\big ] \end{array} \end{aligned}$$

(3.6)

Because $X_t$ depends on $\{y_{t-k}; \hat{\varepsilon }_{t-l}, k = 1,\ldots ,p, l = 1,\ldots ,q\}$, which is independent of $e_t$. $(n-p-q)^{-\frac{1}{2}}\sum X_t'e_t$ forms a stationary and ergodic martingale difference process. Then $(n-p-q)^{-\frac{1}{2}}\sum X_t'e_t$ follows asymptotic normality according to a multivariate version of a martingale central limit theorem [2]. Theorem 2.1 shows $\varPhi - \hat{\varPhi } \rightarrow ^{a.s.} o(1)$. $X_t$ is $p+q+1$ dimensional, therefore, (4) follows approximately an F random variable with degrees of freedom $p + q + 1$ and $n-d-b-p-q-h$. As another point of view, it is obvious that $\dfrac{\sum \hat{a}^2_t}{(n-p-q)\sigma ^2}$ is a chi-square random variable with degrees of freedom $n-d-b-h-p-q$. Also, the numerator and denominator of (4) have the same asymptotic variance $\sigma ^2$. Then $(p + q + 1)\hat{F}(p, q, d)$ is asymptotically a chi-square random variable with degrees of freedom $p + q + 1$, which is a straightforward generalization of Corollary 3.1 of Keenan [16] or Tsay [32]. Theorem 2.2 is proved. $\square$