When we talk about interest rates, theoretically we say that lower interest rates lead to higher investment, thus higher GDP. My question is, using data from World Bank or other institution, what sort of interest rate should we consider (there are many of them, deposit interest rate, reference interest rate) and theoretically seems not to be well specified. For instance, in regression analysis, if we want to compute the impact of interest rate to GDP, what kind of rate are we paying most attention to while using data?
As is usually the case in economics and economic data analysis, the answer is "it depends."
From a theoretical standpoint "The" interest rate is not really something that exists in the real world, it is more of a conceptual tool that economists use to create a simple enough model to think about the basic economic relationships. Is this idea how much would it cost to anticipate the resources from the future so that you could consume or invest more, or on the flip side, how much a representative person would charge to give up consuming today, delaying it to the future. Even if you move to more practical analysis you will note that we usually talk about interest rates as if there were just a couple of them, and about capital as if it was this fungible thing (as if you transform a piece of machinery at a GM factory in Detroit on the tarmac of an airport in Minneapolis easily).
In reality there are many, many interest rates which vary depending on who is lending, who is borrowing, for what, for how long, if they have collateral, a history of previous loans, etc., etc., etc.
One way we get around that complexity without losing all the richness of reality is by recognizing that a lot of these differences are due to standard causes that are common to many loans, say length of the loan, the "risk rating" of the borrower and the type of loan. That is usually what we hear about looking at interest rates, that they are determined by the "risk free rate" (the most essential price of transferring resources over time as I mentioned above) plus a "risk premium," or due to the length as reflected in the "yield curve." It is common to refer to short-term US government bonds (say a 3-month treasury bill) as the risk-free asset, and, hence, use their implicit interest rate as the basic interest rate faced by the economy and that all the other interest rates would be impacted by changes in this rate, with the premiums being added on top of it. Usually, conventional monetary policy targets the rates for these short-term bonds, being the main lever for a Central Bank to affect the economy (at least in theory--as we have seen in the last 16 years the link is not as tight as some would believe).
All that to tell you that if you are interested in evaluating the impact of monetary policy on GDP growth it would be traditional to use the Fed Funds Rate or a Short Term US Treasury bill, like the 3-month or 1-year. If your analysis is about investment (the economic type, not financial) it could be argued that a longer term interest rate is more relevant, 10- or 30-year yields, say. If you are interested in investment by companies, it might be wiser to use the yield of some medium- or long-term maturity corporate debt composite, for example.
There is no simple answer, it really depends on your goals in doing the regression and what your question is. Naturally, if you go to the FED website (https://www.federalreserve.gov/econres.htm ) there are plenty of reports and papers doing this type of analysis with many different types of rates. You can learn a lot just by looking at that.