Generally, the more savings and investments you have, the more stable your household finances are. The issue is, when do you say, "That's enough in cash, now I'm going to start investing for a higher return."

If you're debt-free, then I think six months' minimum monthly expenses is about the tipping point where the higher return of investing in stocks outweighs the advantages of yet more cash in your emergency fund. Any reasonably safe, reasonably liquid investments (such as an S&P 500 index fund) could be quickly turned into cash, if it looked like you were going to exhaust your initial 6 months' of emergency cash.

Your numbers would imply an extra $6000 to $12,000 of emergency cash (for a 9 to 12 month emergency fund). If equities return an average of 5% more per year than cash, that would mean that investing everything beyond a 6-month emergency fund in an S&P 500 index fund would bring an extra $300 to $600 a year in return. That's an extra month's worth of expenses every 4 years, just from the higher return on equities!

Of course the calculation changes completely if you have some reason to suspect that you'll be relying on the emergency fund for more than 6 months. But, unless you have some specific danger in mind, I think the extra return of equities outweighs the safety of cash, once you've got six months expenses stashed away.