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**Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model.**
*(English)*
Zbl 0786.62089

The estimation of the parameters of the self exciting threshold autoregressive model (SETAR) is studied. The author established the strong consistency and the limiting distribution of the conditional least squares estimator (CLSE). In this paper, we can find some important results in the study of nonlinear time series.

The main result is into two theorems. Theorem 1 assumes that the model \((x_ n)\) is stationary ergodic, having finite second moments, and that the stationary distribution of \((x_ 1,\dots,x_ p)'\) admits a density positive everywhere. Then the CLSE of the parameter vector \(\theta\) is strongly consistent, and so are the CLSE’s of the variances of the noises. Theorem 2, under some regularity assumptions, shows that \(N(r_ N-r)\) converges in distribution to \(M_ -\), where \(r_ N\) is the CLSE of the threshold \(r\), and \([M_ -,M_ +)\) is the unique random interval over which a compound Poisson process attains its global minimum. Furthermore, the CLSE of the parameter vector \(\theta\) is asymptotically normal.

This is a very good paper, and I think that in theorem 1 one can establish the speed of convergence of the CLSE to the true parameter, and the speed will depend on the stability of the autoregressive function.

The main result is into two theorems. Theorem 1 assumes that the model \((x_ n)\) is stationary ergodic, having finite second moments, and that the stationary distribution of \((x_ 1,\dots,x_ p)'\) admits a density positive everywhere. Then the CLSE of the parameter vector \(\theta\) is strongly consistent, and so are the CLSE’s of the variances of the noises. Theorem 2, under some regularity assumptions, shows that \(N(r_ N-r)\) converges in distribution to \(M_ -\), where \(r_ N\) is the CLSE of the threshold \(r\), and \([M_ -,M_ +)\) is the unique random interval over which a compound Poisson process attains its global minimum. Furthermore, the CLSE of the parameter vector \(\theta\) is asymptotically normal.

This is a very good paper, and I think that in theorem 1 one can establish the speed of convergence of the CLSE to the true parameter, and the speed will depend on the stability of the autoregressive function.

Reviewer: M.Boutahar (Marseille)

### MSC:

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

62J05 | Linear regression; mixed models |