Nonparametric statistics for testing of linearity and serial independence

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Abstract
For a series of independent identically distributed random variables {X t } the conditional mean and the conditional variance are given by M t (x)=E(X t ) and V k (x) = var(X t ). respectively This is used to construct a test of serial independence for a time series via a functional involving nonparametric estimates of M k (x) and V k (x). The resulting test is compared to a number of existing tests of serial independence, including the so-called BDS test, A linearity test can similarly be obtained by comparing M k (x) and V k, e (x) to ρ k x and . where ρ k = corr (X t X t−k ), and where V k.e (x) and are the conditional variance and the variance for the residual process from a linear fit. Resampling is essential to obtain an approximately correct size of the test, as asymptotic theory performs poorly. The tests are illustrated in a number of simulation experiments and on two real data examples.