This paper concerns statistical inference for longitudinal data with ultrahigh dimensional covariates. We first study the problem of constructing confidence intervals and hypothesis tests for a low dimensional parameter of interest. The major challenge is how to construct a powerful test statistic in the presence of high-dimensional nuisance parameters and sophisticated within-subject correlation of longitudinal data. To deal with the challenge, we propose a new quadratic decorrelated inference function approach, which simultaneously removes the impact of nuisance parameters and incorporates the correlation to enhance the efficiency of the estimation procedure. When the parameter of interest is of fixed dimension, we prove that the proposed estimator is asymptotically normal and attains the semiparametric information bound, based on which we can construct an optimal Wald test statistic. We further extend this result and establish the limiting distribution of the estimator under the setting with the dimension of the parameter of interest growing with the sample size at a polynomial rate. Finally, we study how to control the false discovery rate (FDR) when a vector of high-dimensional regression parameters is of interest. We prove that applying the Storey (2002)'s procedure to the proposed test statistics for each regression parameter controls FDR asymptotically in longitudinal data. We conduct simulation studies to assess the finite sample performance of the proposed procedures. Our simulation results imply that the newly proposed procedure can control both Type I error for testing a low dimensional parameter of interest and the FDR in the multiple testing problem. We also apply the proposed procedure to a real data example.
Keywords: False discovery rate; generalized estimating equation; quadratic inference function.