  The Whittle likelihood is a widely used and computationally efficient
pseudo-likelihood. However, it is known to produce biased parameter estimates
for large classes of models. We propose a method for de-biasing Whittle
estimates for second-order stationary stochastic processes. The de-biased
Whittle likelihood can be computed in the same $\mathcal{O}(n\log n)$
operations as the standard approach. We demonstrate the superior performance of
the method in simulation studies and in application to a large-scale
oceanographic dataset, where in both cases the de-biased approach reduces bias
by up to two orders of magnitude, achieving estimates that are close to exact
maximum likelihood, at a fraction of the computational cost. We prove that the
method yields estimates that are consistent at an optimal convergence rate of
$n^{-1/2}$, under weaker assumptions than standard theory, where we do not
require that the power spectral density is continuous in frequency. We describe
how the method can be easily combined with standard methods of bias reduction,
such as tapering and differencing, to further reduce bias in parameter
estimates.
