Least-squares estimates of time series models from NxT panel data with fixed effects (i.e. N different constant terms) are severely biased when T is small, even as N tends to infinity, and are vulnerable to outliers. Efficient GMM estimates are asymptotically unbiased, but remain fragile
to outliers. In this paper, we propose an estimator that is unbiased and robust at once, in the simple setting of estimating the AR(1) coefficient from stationary Gaussian data. In its simplest form, the estimator is (a linear transformation of) the median of the ratios (y(i,t)-y(i,t-1))/(y(i,t-1)-y(i,t-2)). The estimator is exactly unbiased (for any N and T) if there is no contamination, and is
asymptotically sign-robust to independent additive outlier (AO) contamination at any rate and regardless of the AO distribution. (Hence under such contamination the breakdown point is 1). The asymptotic bias is always towards zero. We also derive the influence functional, the asymptotic bias, and maxbias under independent AO contamination with point-mass distribution.