The partially linear additive model arises in many scientific endeavors. In this paper, we look at inference given panel data and a serially correlated error component structure. By combining polynomial spline series approximation with least squares and the estimation of correlation, we propose a weighted semiparametric least squares estimator (WSLSE) for the parametric components, and a weighted polynomial spline series estimator (WPSSE) for the nonparametric components. The WSLSE is shown to be asymptotically normal and more efficient than the unweighted one. In addition, based on the WSLSE and WPSSE, a two-stage local polynomial estimator (TSLLE) of the nonparametric components is proposed that takes both contemporaneous correlation and additive structure into account. The TSLLE has several advantages, including higher asymptotic efficiency and an oracle property that achieves the same asymptotic distribution of each additive component as if the parametric and other nonparametric components were known with certainty. Some simulation studies were conducted to illustrate the finite sample performance of the proposed procedure. An example of application to a set of panel data from a wage study is illustrated.