The Swiss National Bank already does this. They have tens of billions of US equities in their portfolio. The SNB prints up money (francs) sells those francs on the open market for dollars, puts those dollars in some sort of brokerage account and buys US equities.
Everybody already knows of the BoJ, which owns majority positions in several key ETFs for Japanese shares.
Less well known are the indirect methods, which include the fact that central banks are such quantity purchasers of CME products that they get top tier volume discounts.
The CME, of course, is where options and futures on commodites, stocks and bonds trade. Not one single central banks admits publicly to owning any CME paper products, but one does not get top tier volume discounts for doing nothing.
This would qualify as “direct” participation in my book because the higher leveraged products offered on the CME move the prices of stocks and bonds and commodities with outsized effects.
But, again, nobody talks about this in the press and no central bank has any such products listed anywhere on their balance sheet or in the footnotes. At least that I could find.
The BIS has tons of working papers out on how central banks can best go about “influencing” currency markets to control exchange rates. The methods include directly intervening, jawboning, and the use of proxy agents (like key wall street firms, PE groups, etc) to hide their actions.
Not very much of a hop, skip and a jump to imagine that happening for equities too.