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_kx and  . where ρ_k = corr (X_tX_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.