GIVE statistic for goodness of fit in instrumental variables models with application to COVID data

Since COVID-19 outbreak, scientists have been interested to know whether there is any impact of the Bacillus Calmette–Guerin (BCG) vaccine against COVID-19 mortality or not. It becomes more relevant as a large population in the world may have latent tuberculosis infection (LTBI), for which a person may not have active tuberculosis but persistent immune responses stimulated by Mycobacterium tuberculosis antigens, and that means, both LTBI and BCG generate immunity against COVID-19. In order to understand the relationship between LTBI and COVID-19 mortality, this article proposes a measure of goodness of fit, viz., Goodness of Instrumental Variable Estimates (GIVE) statistic, of a model obtained by Instrumental Variables estimation. The GIVE statistic helps in finding the appropriate choice of instruments, which provides a better fitted model. In the course of study, the large sample properties of the GIVE statistic are investigated. As indicated before, the COVID-19 data is analysed using the GIVE statistic, and moreover, simulation studies are also conducted to show the usefulness of the GIVE statistic along with analysis of well-known Card data.