Upcoming third-quarter earnings reports of North American public companies will provide a critical test of whether equity markets on both sides of the Canada-U.S. border will continue rising, according to CIBC World Markets.Analysts expect TSX earnings to fall by 31% on average, net of one-off items, the report said. While the average estimate is more than the 25% year-over-year decline in the second quarter, it is less than the 34% decline anticipated three months ago.The report suggested companies in the health, gold and IT sectors will lead in year-on-year growth in the third quarter. The financial sector should also benefit from firmer earnings growth by both banks and insurers. Rail and aerospace, on the other hand, are expected to drag down the Canadian industrial sector.Domestically-focused companies should fair better than U.S. counterparts due to Canada's more vigorous economic recovery.Canadian consumers and businesses still look healthier, based on a range of indicators, including wealth, income growth and credit availability.Retail sales, which have dropped this year, are still holding up slightly better than in the U.S.