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Quantile self-exciting threshold autoregressive time series models. (English) Zbl 1165.62062
Let \(-\infty< r_{0}<r_1<\dots<r_{m+1}<\infty\) be threshold values, and let \(\Omega_{i}=(r_{i-1},r_{i}],\;i=1,\dots,m\), and \(\Omega_{m+1}=(r_{m},r_{m+1})\). The quantile self-exciting threshold autoregressive time series (QSETAR) model is defined by the relationship \[ q^{\theta}_{x_{t}| {\mathbf x}_{t-1}}=\sum_{i=1}^{m+1}(\beta_{i0}^{\theta}+\beta_{i1}^{\theta}x_{t-1}+\dots+ \beta_{ip}^{\theta}x_{t-p})I_{[x_{t-d^{\theta}}\in\Omega_{i}]}, \] where \(q^{\theta}_{x_{t}| {\mathbf x}_{t-1}}\) is the \(\theta\) th \((0<\theta<1)\) conditional quantile of \(x_{t}\) given the previous values \({\mathbf x}_{t-1}=(x_{t-1},x_{t-2},\dots,,x_0)\). The authors present a Bayesian approach to quantile self-exciting threshold autoregressive time series models. The simulations show that the method can deal very well with non-stationary time series with very large, but not necessarily symmetric, variations. An interesting application of the proposed methodology to the growth rate of the US real GNP from the first quarter of 1947 to the first quarter of 1991 is discussed.

MSC:
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F15 Bayesian inference
62P20 Applications of statistics to economics
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