  Variable selection plays a fundamental role in high-dimensional data
analysis. Various methods have been developed for variable selection in recent
years. Well-known examples are forward stepwise regression (FSR) and least
angle regression (LARS), among others. These methods typically add variables
into the model one by one. For such selection procedures, it is crucial to find
a stopping criterion that controls model complexity. One of the most commonly
used techniques to this end is cross-validation (CV) which, in spite of its
popularity, has two major drawbacks: expensive computational cost and lack of
statistical interpretation. To overcome these drawbacks, we introduce a
flexible and efficient test-based variable selection approach that can be
incorporated into any sequential selection procedure. The test, which is on the
overall signal in the remaining inactive variables, is based on the maximal
absolute partial correlation between the inactive variables and the response
given active variables. We develop the asymptotic null distribution of the
proposed test statistic as the dimension tends to infinity uniformly in the
sample size. We also show that the test is consistent. With this test, at each
step of the selection, a new variable is included if and only if the $p$-value
is below some pre-defined level. Numerical studies show that the proposed
method delivers very competitive performance in terms of variable selection
accuracy and computational complexity compared to CV.
