Time series data are often well modeled by using the device of an autoregressive root that is local to unity. Unfortunately, the localizing parameter (c) is not consistently estimable using existing time series econometric techniques and the lack of a consistent estimator complicates inference. This paper develops procedures for the estimation of a common localizing parameter using panel data. Pooling information across individuals in a panel aids the identification and estimation of the localizing parameter and leads to consistent estimation in simple panel models. However, in the important case of models with concomitant deterministic trends, it is shown that pooled panel estimators of the localizing parameter are asymptotically biased. Some techniques are developed to overcome this difficulty, and consistent estimators of c in the region c < 0 are developed for panel models with deterministic and stochastic trends. A limit distribution theory is also established, and test statistics are constructed for exploring interesting hypotheses, such as the equivalence of local to unity parameters across subgroups of the population. The methods are applied to the empirically important problem of the efficient extraction of deterministic trends. They are also shown to deliver consistent estimates of distancing parameters in nonstationary panel models where the initial conditions are in the distant past. In the development of the asymptotic theory this paper makes use of both sequential and joint limit approaches. An important limitation in the operation of the joint asymptotics that is sometimes needed in our development is the rate condition n/T → 0. So the results in the paper are likely to be most relevant in panels where T is large and n is moderately large.