Title: Confidence Bands for Coefficients in High Dimensional Linear Models with Error-in-variables,
Abstract: We study high-dimensional linear models with error-in-variables. Such models
are motivated by various applications in econometrics, finance and genetics.
These models are challenging because of the need to account for measurement
errors to avoid non-vanishing biases in addition to handle the high
dimensionality of the parameters. A recent growing literature has proposed
various estimators that achieve good rates of convergence. Our main
contribution complements this literature with the construction of simultaneous
confidence regions for the parameters of interest in such high-dimensional
linear models with error-in-variables.
These confidence regions are based on the construction of moment conditions
that have an additional orthogonal property with respect to nuisance
parameters. We provide a construction that requires us to estimate an
additional high-dimensional linear model with error-in-variables for each
component of interest. We use a multiplier bootstrap to compute critical values
for simultaneous confidence intervals for a subset $S$ of the components. We
show its validity despite of possible model selection mistakes, and allowing
for the cardinality of $S$ to be larger than the sample size.
We apply and discuss the implications of our results to two examples and
conduct Monte Carlo simulations to illustrate the performance of the proposed
procedure.