Hi,
If William was born in Canada, and stayed in the Prairies (Alberta, Saskatchewan, Manitoba) he
should appear on the 1926 Prairie Census. There is one in Saskatchewan that is a possible.
1926 McKenzie, Quill Lake, Saskatchewan
Purse, George, age 22, b. Quebec
Purse, Gladys, age 23, b. England
Purse, William, age 1, b. Sask.
Purse, Margaret, age 5/12 Sask.
https://www.familysearch.org/ark:/61903/3:1:3Q9M-C395-GBPW?i=5&cc=3005862&personaUrl=%2Fark%3A%2F61903%2F1%3A1%3AQPFX-57LRSaskatchewan birth records are privacy protected for 100 years.

There are border crossing records in 1938 crossing from Canada to the USA at Blaine, Washington,
they were together.
Purse, William Joseph, age 14, b. Sept. 13, 1924 Alsask, Sask. student
Purse, Marguerite Isabelle, age 12, b. Dec. 29, 1925, Quill Lake, Sask. student
Purse, George John, age 34, b. Dec. 10, 1903, Huntington, Quebec, bookkeeper
They each give an address of 2418 Brunswick St. Vancouver, BC. They each give John W. Purse
as father/grandfather 465 Cumberland Ave. Winnipeg, Man. They were all debarred from
crossing as they were deemed LPC (likely public charge).
Marguerite Isobelle Purse, age 19, of 465 Cumberland Ave. Winnipeg, father George John Purse,
mother Gladys Gray marries Malcolm Ernest Hall, age 19, pilot engineer RCAF in 1945
http://search-collections.royalbcmuseum.bc.ca/Image/Genealogy/803192e7-8261-47c4-94d8-850e8b321085Marguerite Isobelle (Purse) Hall dies in Nanaimo BC in 1994, father George John Purse, mother
Gladys Gray
http://search-collections.royalbcmuseum.bc.ca/Image/Genealogy/44a798dd-4a86-40f8-8843-0fe7429449d3DB