Apartment rents are continuing their steady climb, as the growing demand for residential rental space keeps pushing up prices.

The average monthly apartment rent in the U.S. was $1,048 in the last quarter of 2012. That’s an increase of 3.8% over a year earlier, according to research firm Reis, Inc.,

The year-over-year increase is the largest since 2007.

A number of large communities saw year-over-year increases of more than 5% last year, including Seattle, San Francisco, Houston, San Jose and Baltimore.

Meanwhile, the national vacancy rate fell to 4.5%, from 4.7% in the third quarter. That’s the lowest national vacancy rate since 2001.

New York City had the country’s lowest vacancy rate, at 2.1%. The average monthly rent in New York last quarter was $2,985.

For most of the country, it’s now cheaper to buy a home than to rent one, according to an analysis by Deutsche Bank.

The company compares the average rental rate for an area to the average after-tax cost of a mortgage. In the third quarter of 2012, it found, renters paid almost 8% more than homeowners with a mortgage.

By comparison, since 1991 renters have on average paid about 15% less than homeowners.