  We propose a novel framework of the model specification test in regression
using unlabeled test data. In many cases, we have conducted statistical
inferences based on the assumption that we can correctly specify a model.
However, it is difficult to confirm whether a model is correctly specified. To
overcome this problem, existing works have devised statistical tests for model
specification. Existing works have defined a correctly specified model in
regression as a model with zero conditional mean of the error term over train
data only. Extending the definition in conventional statistical tests, we
define a correctly specified model as a model with zero conditional mean of the
error term over any distribution of the explanatory variable. This definition
is a natural consequence of the orthogonality of the explanatory variable and
the error term. If a model does not satisfy this condition, the model might
lack robustness with regards to the distribution shift. The proposed method
would enable us to reject a misspecified model under our definition. By
applying the proposed method, we can obtain a model that predicts the label for
the unlabeled test data well without losing the interpretability of the model.
In experiments, we show how the proposed method works for synthetic and
real-world datasets.
