The purpose of this paper is to propose methodologies for statistical inference of low
dimensional parameters with high dimensional data.We focus on constructing confidence intervals
for individual coefficients and linear combinations of several of them in a linear regression
model, although our ideas are applicable in a much broader context.The theoretical results that
are presented provide sufficient conditions for the asymptotic normality of the proposed estimators
along with a consistent estimator for their finite dimensional covariance matrices. These
sufficient conditions allow the number of variables to exceed the sample size and the presence
of many small non-zero coefficients. Our methods and theory apply to interval estimation of a
preconceived regression coefficient or contrast as well as simultaneous interval estimation of
many regression coefficients. Moreover, the method proposed turns the regression data into
an approximate Gaussian sequence of point estimators of individual regression coefficients,
which can be used to select variables after proper thresholding. The simulation results that are
presented demonstrate the accuracy of the coverage probability of the confidence intervals
proposed as well as other desirable properties, strongly supporting the theoretical results.