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 establish

finite sample upper bounds on the estimation error of the Lasso under two

different sets of conditions on the covariates as well as the error terms. 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 selectis 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.