1. Cornell Law School, Legal Information Institute (LII). Tenancy at Will. "A tenancy at will is a property tenure... that can be terminated at any time by either the tenant or the owner (landlord). It exists without a contract or lease and usually does not specify the length of a tenant's duration or the exchange of payment... A tenancy at will may also be created by implication, such as when a tenant under a lease holds over with the landlord's consent after the lease has expired." Retrieved from https://www.law.cornell.edu/wex/tenancyatwill.
2. Massachusetts General Laws, Chapter 186, Section 12. Termination of lease at will. This statute outlines the notice requirements for terminating an estate at will, thereby legally defining and recognizing its existence in the Commonwealth. The law states, "Estates at will may be determined by either party by three months' notice in writing for that purpose given to the other party..." (or a shorter period equal to the rent payment interval). This confirms the legal status of the tenancy described in the question.
3. Dukeminier, J., Krier, J. E., Alexander, G. S., Schill, M. H., & Strahilevitz, L. J. (2017). Property (9th ed.). Aspen Casebook Series, Wolters Kluwer. In the chapter on "The Leasehold Estates," the text distinguishes the holdover tenant scenarios. When a landlord consents to the holdover, a tenancy at will or a periodic tenancy is created. A tenancy at sufferance arises only when the landlord does not consent. (Specific discussion typically found in Section 6.2, "The Leasehold Estates").