  This paper is concerned with high-dimensional panel data models where the
number of regressors can be much larger than the sample size. Under the
assumption that the true parameter vector is sparse we propose a panel-Lasso
estimator and establish finite sample upper bounds on its estimation error
under two different sets of conditions on the covariates as well as the error
terms. In particular, we allow for heteroscedastic and non-gaussian error terms
which are weakly dependent over time. Upper bounds on the estimation error of
the unobserved heterogeneity are also provided under the assumption of
sparsity. Next, we show that our upper bounds are essentially optimal in the
sense that they can only be improved by multiplicative constants. These results
are then used to show that the Lasso can be consistent in even very large
models where the number of regressors increases at an exponential rate in the
sample size. Conditions under which the Lasso does not discard any relevant
variables asymptotically are also provided.
  In the second part of the paper we give lower bounds on the probability with
which the adaptive Lasso selects the correct sparsity pattern in finite
samples. These results are then used to give conditions under which the
adaptive Lasso can detect the correct sparsity pattern asymptotically. We
illustrate our finite sample results by simulations and apply the methods to
search for covariates explaining growth in the G8 countries.
