Does immigration help alleviate economy-wide labour shortages?

I study the impact of Canada's expansive immigration policy launched in 2016 on labour shortages in six regions of the country, particularly in Quebec, which enjoys some autonomy of management in this area. I look at movements of the Beveridge curve, which draws the classical inverse relation between the job vacancy rate and the unemployment rate, before, during, and after the 2020-2021 pandemic. Since immigration not only expands the supply of labour, but also adds to the demand for labour in the overall economy, its net effect on job vacancies in the aggregate is a priori uncertain. To clarify matters, I present a statistical analysis of pre- and post-pandemic data in the six Canadian regions. Results suggest that the common-sense belief that more immigration contributes to reducing economy-wide labour scarcity is wrong and constitutes a dangerous fallacy of composition.