LONDON (Reuters) - Britain's government must share more responsibility with the Bank of England for overseeing the economy in order to reduce the BoE's vulnerability to political attack, a former politician who gave the BoE its independence nearly 20 years ago said.

Ed Balls, who served as chief adviser to former finance minister and later prime monister Gordon Brown, told BBC radio on Thursday that the job of ensuring the operational independence of the BoE and other central banks was still "unfinished business."

Prime Minister Theresa May took the rare step of criticizing BoE policy last month when she talked about the "bad side effects" of low interest rates. That prompted BoE Governor Mark Carney to say he would not take instruction from politicians on how the Bank should meet its 2 percent inflation target.

In the United States, President-elect Donald Trump criticized the low interest rates of the Federal Reserve during campaigning for the Nov. 8 election. And the European Central Bank has faced pressure from politicians in Germany, the euro zone's biggest economy, over its rock-bottom borrowing costs.

Balls was one of the main figures behind the decision of British government in 1997 to give the BoE operational independence about how to meet its inflation target, taking decisions over interest rates away from Westminster.

The government sets the Bank's mandate, however.

Balls told the BBC that the case for operational independence for central banks remained strong. But their growing powers called for a new relationship with governments.

He said he would like Britain's finance minister to chair a systemic risk body and set the priorities for ensuring the stability of the banking system in order to reduce the risk of the BoE "taking the rap" when its policies anger voters.

"If in the end, when things start to go wrong, the Bank is solely responsible for spotting risks, that is politically very dangerous for the Bank. So, in order to protect its independence the bank needs more critical support and accountability."

As well as the criticism from May, the Bank has come under attack from other British lawmakers, many of them supporters of Britain's decision to leave the European Union who were angered by the BoE's warnings of a Brexit hit to the economy.

Finance minister Philip Hammond said last month the government did not plan to change the BoE's independence to set monetary policy. Hammond will set out the new government's fiscal plans on Nov. 23 in a half-yearly budget statement which typically provides an update on the BoE's mandate.

Balls was speaking to the BBC after the publication of a paper on central bank independence which he co-wrote for the Harvard Kennedy School with James Howat, a BoE economist, and Anna Stansbury, an economics PhD candidate at Harvard.

The paper said central banks in advanced economies should "sacrifice some political independence without undermining the operational independence that is important in both their monetary policy and financial stability function."

(Reporting by Elisabeth O'Leary; Writing by William Schomberg Editing by Jeremy Gaunt)