Simply said, in finance for a freely convertible currency, you can either fix the interest rate and let exchange rate float, or you can fix exchange rate and let interest rate float.
HK's case, due to the currency regime, chooses to set exchange rate and leave interest rate floating.
So daily the interest rate between banks are market driven.
HK government has one interest rate set that's applicable, its the Base Rate, where if a bank really have no place to borrow money, they can put up asset to borrow from HKMA at US Fed fund rate + 0.50% (IIRC, on the spread). But this rate generally do not affect money market rates in hk (more like just a cap keeping overnight rates below the Base Rate)
Over the past 2 years, demand for HKD was high (don't ask me why, but just factually). In total more than HK$ 300 or 400b were converted from USD into HKD, when USD-HKD rate was at 7.75, and that grew something called the 'Aggregate Balance' from a low double digit to more than 400b (HKMA were selling HKD to the USD holders at 7.75.)
'Aggregate Balance' is the floating liquidity in the HK banking system (I will leave out the part where HKMA sometimes uses exchange bill to suck liquidity from the aggregate balance when its really overflowing) and currently still stays at HK$ 338b minus what was used early this week (when the world screamed about HKMA intervening in the market)
During normal business, the 338b is being lent by banks to banks, to corporates, to etc.. these cause the HKD interest rate to stay low. Overnight rates this week still stay at less than 0.1%, even though US has hiked their target rate range to 0.75~1.00%.
What would cause the rates to go up --> mainly when the Aggregate balance falls to close to 0.. that signals no excess liquidity in the HKD market and banks will have to borrow/lend to each other at rate closer to USD rates. Aggregate balance can fall in two ways, 1. HKMA can suck more liquidity from the Aggregate balance by issuing more bonds, or 2. selling of HKD continues and drains the liquidity in the aggregate balance.
Bank researchers are saying rates will start to follow US rates when Aggregate balance falls till low double digit.. well, nobody knows so we can only watch and see.