Diagnostic Tests for Non-causal Time Series with Infinite Variance
  We study goodness-of-fit testing for non-causal autoregressive time series with non-Gaussian stable noise. To model time series exhibiting sharp spikes or occasional bursts of outlying observations, the exponent of the non-Gaussian stable variables is assumed to be less than two. Under such conditions, the innovation variables have no finite second moment. We proved that the sample autocorrelation functions of the trimmed residuals are asymptotically normal. Nonparametric tests are also investigated. The rank correlations of the residuals or the squared residuals are shown to be asymptotically normal. Thus, an assortment of portmanteau statistics are available for model assessment. 
Yunwei Cui, Rongning Wu, Thomas J. Fisher
math.ST stat.TH