If you're in a rent controlled apartment and you're paying about $1/sf, then you should probably stay put. You might still want to buy an investment, but you really should keep your apartment because $1/sf is too good to throw away.
However, if you're renting somewhere around $3/sf, which is about $3000 for a 1000sf pad, then it might be a better bet buy while interest rates are still at historic lows. Reasons why:
1. Interest rates are expected to hike up at least twice in 2017, then again in 2018. I don't think we'll be going into the 5% arena but it could go up to about 4.65% for a SFR.
2. If you're in a rent-controlled building, then the Landlord can hike up rents 2-3% per year. If you're in a non-rent controlled building, then the Landlord can raise the rents to whatever the market rate is after your lease is up.
3. If you get a 30-year fixed mortgage, then your "rent" is fixed for 30 years. Duh.
4. Rents are so high in LA right now, that if you ever decide to keep the home and rent it out, it's likely that someone else's rent money will pay for your mortgage. No brainer.