  This paper considers the problem of variable selection allowing for parameter
instability. It distinguishes between signal and pseudo-signal variables that
are correlated with the target variable, and noise variables that are not, and
investigate the asymptotic properties of the One Covariate at a Time Multiple
Testing (OCMT) method proposed by Chudik et al. (2018) under parameter
insatiability. It is established that OCMT continues to asymptotically select
an approximating model that includes all the signals and none of the noise
variables. Properties of post selection regressions are also investigated, and
in-sample fit of the selected regression is shown to have the oracle property.
The theoretical results support the use of unweighted observations at the
selection stage of OCMT, whilst applying down-weighting of observations only at
the forecasting stage. Monte Carlo and empirical applications show that OCMT
without down-weighting at the selection stage yields smaller mean squared
forecast errors compared to Lasso, Adaptive Lasso, and boosting.
