Differencing as an approximate de-trending device

Consider the model yj = [latin small letter f with hook](j/n) + [var epsilon]j, J = 1,..., n, where the yj's are observed, [latin small letter f with hook] is a smooth but unknown function, and the [var epsilon]j's are unobserved errors from a zero mean, strictly stationary process. The problem addressed is that of estimating the covariance function c(k) = E([var epsilon]0[var epsilon]k) from the observations y1,..., yn without benefit of an initial estimate of [latin small letter f with hook]. It is shown that under appropriate conditions on [latin small letter f with hook] and the error process, consistent estimators of c(k) can be constructed from second differences of the observed data. The estimators of c(k) utilize only periodogram ordinates at frequencies greater than some small positive number [delta] that tends to 0 as n --> [infinity]. Tapering the differenced data plays a crucial role in constructing an efficient estimator of c(k)