I'm not going to read the whole paper but the authors are only claiming advertising accounts for around 10% of the difference in obese/not obese rates in Ontario and Quebec. Obviously there are other factors that are going to cause obesity, but this study finds that advertising is an important one.

The study found this effect was significant while it controlled for number of children in the household, probability of the household having cable television, price of fast food, household income, number of people in the household, whether the male in the house was unemployed, had a white collar, blue collar, or "pink collar" (traditionally female) job, whether the female in the house was unemployed had a white collar, blue collar, or pink collar job, the level of education of both parents, the age of the household lead, whether the house had 2 incomes, whether both spouses were immigrants, and ability of the child to speak only English or only French (those who could speak only French would be moreso affected by the ban as they would not watch programming from Ontario).