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A test for nonlinearity of time series with infinite variance. (English) Zbl 0971.62049
For a time series $$X_1$$, $$X_2$$, $$\dots$$ a linearity hypothesis $$H_0:$$ $$X_t=\sum_{j=0}^\infty c_j Z_{t-j}$$ is considered, where $$c_j$$ are real-valued coefficients, and $$Z_j$$ are i.i.d. heavy-tailed with infinite variance. Even in this case, the sample autocovariance function (ACF) of $$X_i$$, namely, $\hat\rho_n(h)=\{\sum_{t=1}^{n-h}X_t X_{t+h}\} \{\sum_{t=1}^n X_t^2\}^{-1}$ converges (as $$n\to\infty$$) to an analogue of ACF for $$X_j$$: $\rho(h)=\{\sum_{j=0}^\infty c_j c_{j+h}\} \{\sum_{j=0}^\infty c_j^2\}^{-1}.$ For heavy tailed nonlinear series (i.e. if $$H_0$$ doesn’t hold) the empirical ACF does not converge to any fixed number but in many cases it converges to some nondegenerate random variable. The authors propose a test in which estimates of the empirical ACF stability are used as test statistics for $$H_0$$ and investigate it’s properties.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62G10 Nonparametric hypothesis testing
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