Kernel Regression Estimation with Time Series Errors

SUMMARY Suppose that data y 1, ..., yn are observed according to the model yi=f((i−12)/n)+ϵi,i=1,…,n, where f is a smooth function and {ϵ1,…,ϵn} is a sample from a zero-mean, covariance stationary time series. The problem of objectively choosing the bandwidth of a kernel estimate for f is addressed. It is shown both theoretically and by simulation that cross-validation produces extremely rough kernel estimates when the data are sufficiently positively correlated. This makes it inadvisable to use residuals from a cross-validated kernel estimate as a means of estimating the covariance function of the errors. Alternative methods of estimating the covariance function are proposed. In a simulation study, incorporating these estimated covariances into a risk estimation procedure leads to more efficient smoothing of positively correlated data.